{"id":258,"date":"2021-12-15T22:07:30","date_gmt":"2021-12-15T22:07:30","guid":{"rendered":"https:\/\/pressbooks.palomar.edu\/introtostats\/chapter\/chapter-8\/"},"modified":"2025-08-28T23:56:15","modified_gmt":"2025-08-28T23:56:15","slug":"chapter-8","status":"publish","type":"chapter","link":"https:\/\/pressbooks.palomar.edu\/introtostats\/chapter\/chapter-8\/","title":{"raw":"Chapter 8: Introduction to t Tests and Confidence Intervals","rendered":"Chapter 8: Introduction to t Tests and Confidence Intervals"},"content":{"raw":"
\r\n

Key Terms<\/h3>\r\n<\/header>\r\n
\r\n

\r\nconfidence interval<\/span><\/a><\/p>\r\n

margin of error<\/span><\/a><\/p>\r\n

point estimate<\/span><\/a><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

In Chapter 7<\/span><\/a>, we made a big leap from basic descriptive statistics into full hypothesis testing and inferential statistics. For the rest of the unit, we will be learning new tests, each of which is just a small adjustment on the test before it. In this chapter, we will learn about the first of three t<\/span>\u00a0tests, and we will learn a new method of testing the null hypothesis: confidence intervals.<\/p>\r\nSocial Justice and the t Test<\/strong>
In the real world, we rarely know the exact parameters of a population \u2014 we only have samples. That is especially true in social justice research, where communities are often underrepresented in data collection. The t test gives us a way to use limited information from a sample to make inferences about larger populations. Combined with confidence intervals, it allows us to estimate not just a single number, but a plausible range of values. These tools give us more reliable ways to test whether differences we observe \u2014 such as disparities in wages, housing access, or health outcomes \u2014 are random or evidence of systemic inequality.\r\n

The t<\/span> Statistic<\/h3>\r\n

In Chapter 7<\/span><\/a>, we were introduced to hypothesis testing using the z<\/span>\u00a0statistic for sample means that we learned in Unit 1<\/span><\/a>. This was a useful way to link the material and ease us into the new way to looking at data, but it isn\u2019t a very common test because it relies on knowing the population\u2019s standard deviation, s<\/span>, which is rarely going to be the case. Instead, we will estimate that parameter s<\/span> using the sample statistic s<\/span> in the same way that we estimate \"mu\" using \"Upper (\"mu\" will still appear in our formulas because we suspect something about its value and that is what we are testing). Our new statistic is called t<\/span>, and for testing one population mean using a single sample (called a one-sample t<\/span>\u00a0test) it takes the form:<\/p>\r\n

\"\"<\/p>\r\n

Notice that t <\/span>looks almost identical to z<\/span>; this is because they test the exact same thing: the value of a sample mean compared to what we expect of the population. The only difference is that the standard error is now denoted \"\" to indicate that we use the sample statistic for standard deviation, s<\/span>, instead of the population parameter s<\/span>. The process of using and interpreting the standard error and the full test statistic remain exactly the same.<\/p>\r\n

In Chapter 3<\/span><\/a> we learned that the formulas for sample standard deviation and population standard deviation differ by one key factor: the denominator for the parameter is \"Upper but the denominator for the statistic is N <\/span>\u2212 1, also known as degrees of freedom, d<\/span>f<\/span>. Because we are using a new measure of spread, we can no longer use the standard normal distribution and the z<\/span>\u00a0table to find our critical values. For t<\/span>\u00a0tests, we will use the t<\/span>\u00a0distribution and t<\/span>\u00a0table to find these values.<\/p>\r\n

The t<\/span>\u00a0distribution, like the standard normal distribution, is symmetric and normally distributed with a mean of 0 and standard error (as the measure of standard deviation for sampling distributions) of 1. However, because the calculation of standard error uses degrees of freedom, there will be a different t<\/span>\u00a0distribution for every degree of freedom. Luckily, they all work exactly the same, so in practice this difference is minor.<\/p>\r\n

Figure 8.1<\/span><\/a> shows four curves: a normal distribution curve labeled z<\/span>, and three t<\/span>\u00a0distribution curves for 2, 10, and 30 degrees of freedom. Two things should stand out: First, for lower degrees of freedom (e.g., 2), the tails of the distribution are much fatter, meaning the a larger proportion of the area under the curve falls in the tail. This means that we will have to go farther out into the tail to cut off the portion corresponding to 5% or a<\/span> = .05, which will in turn lead to higher critical values. Second, as the degrees of freedom increase, we get closer and closer to the z <\/span>curve. Even the distribution with d<\/span>f<\/span> = 30, corresponding to a sample size of just 31 people, is nearly indistinguishable from z<\/span>. In fact, a t<\/span>\u00a0distribution with infinite degrees of freedom (theoretically, of course) is exactly the standard normal distribution. Because of this, the bottom row of the t<\/span>\u00a0table also includes the critical values for z<\/span>\u00a0tests at the specific significance levels. Even though these curves are very close, it is still important to use the correct table and critical values, because small differences can add up quickly.<\/p>\r\n\r\n

\r\n
\r\n

<\/a>Figure 8.1.<\/span> Distributions comparing effects of degrees of freedom. (\u201cDistributions Comparing Effects of Degrees of Freedom<\/span><\/a>\u201d by Judy Schmitt is licensed under CC BY-NC-SA 4.0<\/span><\/a>.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\"\"<\/div>\r\n<\/div>\r\n

The t<\/span>\u00a0distribution table lists critical values for one- and two-tailed tests at several levels of significance, arranged into columns. The rows of the t<\/span>\u00a0table list degrees of freedom up to d<\/span>f <\/span>= 100 in order to use the appropriate distribution curve. It does not, however, list all possible degrees of freedom in this range, because that would take too many rows. Above d<\/span>f <\/span>=\u00a040, the rows jump in increments of 10. If a problem requires you to find critical values and the exact degrees of freedom is not listed, you always round down to the next smallest number. For example, if you have 48 people in your sample, the degrees of freedom are N <\/span>\u2212 1 = 48 \u2212 1 = 47; however, 47 doesn\u2019t appear on our table, so we round down and use the critical values for d<\/span>f <\/span>= 40, even though 50 is closer. We do this because it avoids inflating Type I error (false positives, see Chapter 7<\/span><\/a>) by using criteria that are too\u00a0lax.<\/p>\r\nExample:<\/strong>
Suppose we\u2019re studying high school graduation rates among Indigenous students in a small district. With only a few students in the sample, our degrees of freedom are low. The t distribution accounts for this extra uncertainty by using \u201cfatter tails.\u201d This means we need stronger evidence to reject the null when our sample is small. In practice, it keeps us cautious about making sweeping claims from limited data, while still giving us a method to test hypotheses.\r\n

Hypothesis Testing with t<\/span><\/h3>\r\n

Hypothesis testing with the t<\/span>\u00a0statistic works exactly the same way as z<\/span>\u00a0tests did, following the four-step process of (1) stating the hypotheses, (2) finding the critical values, (3) computing the test statistic and effect size, and (4) making the decision.<\/p>\r\n

Example <\/span> Oil Change Speed<\/p>\r\n

We will work though an example: Let\u2019s say that the welfare office changed application systems.\u00a0 The old system approved benefits in about 30 days and you suspect that the new system takes much longer. After four people applied for welfare benefits, you think you have enough evidence to demonstrate this.<\/p>\r\n\r\n

Step 1:<\/span> State the Hypotheses<\/h5>\r\n

Our hypotheses for one-sample t<\/span>\u00a0tests are identical to those we used for z<\/span>\u00a0tests. We still state the null and alternative hypotheses mathematically in terms of the population parameter and written out in readable English. For our example:<\/p>\r\nHo <\/sub>:There is no difference in the average process time for welfare application between the new and old system.<\/strong>\r\n\r\nHo\u00a0<\/sub>: \u03bc = 30 days<\/strong>\r\n\r\nHa<\/sub>\u00a0<\/sub>:The average process time for welfare application takes longer with the new system.\u00a0<\/strong>\r\n\r\nHa<\/sub>\u00a0<\/sub>: \u03bc > 30 days<\/strong>\r\n

Step 2:<\/span> Find the Critical Values<\/h5>\r\n

As noted above, our critical values still delineate the area in the tails under the curve corresponding to our chosen level of significance. Because we have no reason to change significance levels, we will use a<\/span> = .05, and because we suspect a direction of effect, we have a one-tailed test. To find our critical values for t<\/span>, we need to add one more piece of information: the degrees of freedom. For this example:<\/p>\r\n

\"\"<\/p>\r\n

Going to our t<\/span>\u00a0table, a portion of which is found in Table 8.1<\/span><\/a>, we locate the column corresponding to our one-tailed significance level of .05 and find where it intersects with the row for 3 degrees of freedom. As we can see in Table 8.1<\/span><\/a>, our critical value is t<\/span>* = 2.353. (The complete t<\/span> table can be found in Appendix B<\/span><\/a>.)<\/p>\r\n\r\n

\r\n
\r\n

<\/a>Table 8.1.<\/span> t<\/span> distribution table (t<\/span> table).<\/p>\r\n

Adapted from \u201cTabla t<\/span><\/a>\u201d by Jsmura\/Wikimedia Commons, CC BY-SA 4.0<\/span><\/a>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

We can then shade this region on our t<\/span>\u00a0distribution to visualize our rejection region (Figure 8.2<\/span><\/a>).<\/p>\r\n\r\n

\r\n
\r\n

<\/a>Figure 8.2.<\/span> Rejection region. (\u201cRejection Region t2.353<\/span><\/a>\u201d by Judy Schmitt is licensed under CC BY-NC-SA 4.0<\/span><\/a>.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\"\"<\/div>\r\n<\/div>\r\n
Step 3:<\/span> Calculate the Test Statistic and Effect Size<\/h5>\r\n

The four processing times recipients experienced were 46 days, 58 days, 40 days, and 71 days. We will use these to calculate \"Upper and s<\/span> by first filling in the sum of squares in Table 8.2<\/span><\/a>.<\/p>\r\n\r\n

\r\n
\r\n

<\/a>Table 8.2.<\/span> Sum of squares.<\/p>\r\n\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

X<\/span><\/p>\r\n<\/td>\r\n

\r\n

X<\/span> \u2212 M<\/span><\/p>\r\n<\/td>\r\n

\r\n

(X<\/span> \u2212 M<\/span>)2<\/span><\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/thead>\r\n

\r\n

46<\/p>\r\n<\/td>\r\n

\r\n

\u22127.75<\/p>\r\n<\/td>\r\n

\r\n

60.06<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

58<\/p>\r\n<\/td>\r\n

\r\n

4.25<\/p>\r\n<\/td>\r\n

\r\n

18.06<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

40<\/p>\r\n<\/td>\r\n

\r\n

\u221213.75<\/p>\r\n<\/td>\r\n

\r\n

189.06<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

71<\/p>\r\n<\/td>\r\n

\r\n

17.25<\/p>\r\n<\/td>\r\n

\r\n

297.56<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

\u03a3<\/span> = 215<\/p>\r\n<\/td>\r\n

\r\n

\u03a3<\/span> = 0<\/p>\r\n<\/td>\r\n

\r\n

\u03a3<\/span> = 564.74<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n

After filling in the first row to get \u03a3<\/span>X<\/span> = 215, we find that the mean is M<\/span> = 53.75 (215 divided by sample size 4), which allows us to fill in the rest of the table to get our sum of squares SS<\/span> = 564.74, which we then plug in to the formula for standard deviation from Chapter 3<\/span><\/a>:<\/p>\r\n

\"\"<\/p>\r\n

Next, we take this value and plug it in to the formula for standard error:<\/p>\r\n

\"\"<\/p>\r\n

And, finally, we put the standard error, sample mean, and null hypothesis value into the formula for our test statistic t<\/span>:<\/p>\r\n

\"\"<\/p>\r\n

This may seem like a lot of steps, but it is really just taking our raw data to calculate one value at a time and carrying that value forward into the next equation: data \u2192<\/span> sample size\/degrees of freedom \u2192<\/span> mean \u2192<\/span> sum of squares \u2192<\/span> standard deviation \u2192<\/span> standard error \u2192<\/span> test statistic. At each step, we simply match the symbols of what we just calculated to where they appear in the next formula to make sure we are plugging everything in correctly.<\/p>\r\n\r\n

Step 4:<\/span> Make the Decision<\/h5>\r\n

Now that we have our critical value and test statistic, we can make our decision using the same criteria we used for a z<\/span>\u00a0test. Our obtained t<\/span>\u00a0statistic was t <\/span>= 3.46 and our critical value was t<\/span>*\u00a0=\u00a02.353: t <\/span>> t<\/span>*, so we reject the null hypothesis and conclude:<\/p>\r\n

Based on our four applicants, the new processing time to receive welfare benefits takes longer on average (M<\/span> = 53.75, SD<\/span> = 13.72) than the old processing system, t<\/span>(3) = 3.46, p<\/span>\u00a0<\u00a0.05, d<\/span> = 1.74.<\/p>\r\n

Notice that we also include the degrees of freedom in parentheses next to t<\/span>. Figure 8.3<\/span><\/a> shows the output from JASP.<\/p>\r\n\r\n

\r\n
FIGURE 8.3<\/span><\/span><\/a>.<\/span>\u00a0Output from JASP for the one-sample\u00a0t<\/span>\u00a0test described in this example. The output provides the\u00a0t<\/span>\u00a0value (3.462), degrees of freedom (3), and the exact\u00a0p<\/span> value (.020, which is less than .05). Note that the mean (53.750) and standard deviation for the sample are also provided (13.720). Based on our four applicants processing time, the new system takes longer on average (M<\/span>\u00a0= 53.75,\u00a0SD<\/span> = 13.72) than the\u00a0 old system to receive benefits, t<\/span>(3)\u00a0=\u00a03.46,\u00a0p<\/span> =\u00a0.02. \u00a0(\u201cJASP 1-sample t test<\/span><\/a>\u201d by Rupa G. Gordon\/Judy Schmitt is licensed under\u00a0CC BY-NC-SA 4.0<\/span><\/a>.)<\/h5>\r\n<\/div>\r\n
\r\n
\"\"<\/div>\r\n<\/div>\r\n

<\/h3>\r\nExample:<\/strong>
Imagine a city rolls out a new public housing policy. The old average wait time for housing assistance was 12 months. A researcher collects a sample of 20 applicants under the new system and finds an average wait of 14 months. A one-sample t test allows us to ask: is this longer wait likely due to chance, or does it suggest the new policy increases delays? This mirrors the way we use welfare processing times in the current example \u2014 testing whether new systems help or harm marginalized communities.\r\n

Confidence Intervals<\/h3>\r\n

Up to this point, we have learned how to estimate the population parameter for the mean using sample data and a sample statistic. From one point of view, this makes sense: we have one value for our parameter so we use a single value (called a [pb_glossary id=\"678\"]<\/a>[\/pb_glossary]point estimate<\/span>) to estimate it. However, we have seen that all statistics have sampling error and that the value we find for the sample mean will bounce around based on the people in our sample, simply due to random chance. Thinking about estimation from this perspective, it would make more sense to take that error into account rather than relying just on our point estimate. To do this, we calculate what is known as a confidence interval.<\/p>\r\n

A [pb_glossary id=\"676\"]<\/a>[\/pb_glossary]confidence interval<\/span> starts with our point estimate and then creates a range of scores considered plausible based on our standard deviation, our sample size, and the level of confidence with which we would like to estimate the parameter. This range, which extends equally in both directions away from the point estimate, is called the [pb_glossary id=\"677\"]<\/a>[\/pb_glossary]margin of error<\/span>. We calculate the margin of error by multiplying our two-tailed critical value by our standard error:<\/p>\r\n

\"\"<\/p>\r\n

One important consideration when calculating the margin of error is that it can only be calculated using the critical value for a two-tailed test. This is because the margin of error moves away from the point estimate in both directions, so a one-tailed value does not make sense.<\/p>\r\n

The critical value we use will be based on a chosen level of confidence, which is equal to 1 \u2212 a<\/span>. Thus, a 95% level of confidence corresponds to a<\/span> = .05. Thus, at the .05 level of significance, we create a 95% confidence interval. How to interpret that is discussed further on.<\/p>\r\n

Once we have our margin of error calculated, we add it to our point estimate for the mean to get an upper bound to the confidence interval and subtract it from the point estimate for the mean to get a lower bound for the confidence interval:<\/p>\r\n

\"\"<\/p>\r\n

or simply:<\/p>\r\n

\"\"<\/p>\r\n

To write out a confidence interval, we always use round brackets (i.e., parentheses) and put the lower bound, a comma, and the upper bound:<\/p>\r\n

\"\"<\/p>\r\n

Let\u2019s see what this looks like with some actual numbers by taking our welfare application data and using it to create a 95% confidence interval estimating the average length of time it takes to receive benefits. We already found that our average was M<\/span> = 53.75 days and our standard error was \"\" = 6.86. We also found a critical value to test our hypothesis, but remember that we were testing a one-tailed hypothesis, so that critical value won\u2019t work. To see why that is, look at the column headers on the t<\/span>\u00a0table. The column for one-tailed a<\/span> = .05 is the same as a two-tailed a<\/span> = .10. If we used the old critical value, we\u2019d actually be creating a 90% confidence interval (1.00 \u2212 0.10 = 0.90, or 90%). To find the correct value, we use the column for two-tailed a<\/span> = .05 and, again, the row for 3 degrees of freedom, to find t<\/span>* = 3.182.<\/p>\r\n

Now we have all the pieces we need to construct our confidence interval:<\/p>\r\n\"\"\r\n

\"\"<\/p>\r\n

\"\"<\/p>\r\n

\"\"<\/p>\r\n

\"\"<\/p>\r\n

So we find that our 95% confidence interval runs from 31.92 days to 75.58 days, but what does that actually mean? The range (31.92, 75.58) represents values of the mean that we consider reasonable or plausible based on our observed data. It includes our point estimate of the mean, M<\/span>\u00a0= 53.75, in the center, but it also has a range of values that could also have been the case based on what we know about how much these scores vary (i.e., our standard error).<\/p>\r\n

It is very tempting to also interpret this interval by saying that we are 95% confident that the true population mean falls within the range (31.92, 75.58), but this is not true. The reason it is not true is that phrasing our interpretation this way suggests that we have firmly established an interval and the population mean does or does not fall into it, suggesting that our interval is firm and the population mean will move around. However, the population mean is an absolute that does not change; it is our interval that will vary from data collection to data collection, even taking into account our standard error. The correct interpretation, then, is that we are 95% confident that the range (31.92, 75.58) brackets the true population mean. This is a very subtle difference, but it is an important one.<\/p>\r\nConsider studying wage gaps for women of color compared to national averages. Instead of estimating the population wage with a single sample mean, we construct a confidence interval. If our 95% CI for the average wage is ($37,000, $42,000), we can say this range of values is plausible based on the data. If the national mean is $45,000, and that value falls outside our confidence interval, we reject the null hypothesis and conclude there is a real gap. Confidence intervals thus help quantify inequity while accounting for uncertainty in our sample.\r\n

Hypothesis Testing with Confidence Intervals<\/h4>\r\n

As a function of how they are constructed, we can also use confidence intervals to test hypotheses. However, we are limited to testing two-tailed hypotheses only, because of how the intervals work, as discussed above.<\/p>\r\n

Once a confidence interval has been constructed, using it to test a hypothesis is simple. If the range of the confidence interval brackets (or contains, or is around) the null hypothesis value, we fail to reject the null hypothesis. If it does not bracket the null hypothesis value (i.e., if the entire range is above the null hypothesis value or below it), we reject the null hypothesis. The reason for this is clear if we think about what a confidence interval represents. Remember: a confidence interval is a range of values that we consider reasonable or plausible based on our data. Thus, if the null hypothesis value is in that range, then it is a value that is plausible based on our observations. If the null hypothesis is plausible, then we have no reason to reject it. Thus, if our confidence interval brackets the null hypothesis value, thereby making it a reasonable or plausible value based on our observed data, then we have no evidence against the null hypothesis and fail to reject it. However, if we build a confidence interval of reasonable values based on our observations and it does not contain the null hypothesis value, then we have no empirical (observed) reason to believe the null hypothesis value and therefore reject the null hypothesis.<\/p>\r\n

Example <\/span> Friendliness<\/p>\r\n

You hear that the national average on a measure of friendliness is 38 points. You want to know if people in your community are more or less friendly than people nationwide, so you collect data from 30 random people in town to look for a difference. We\u2019ll follow the same four-step hypothesis-testing procedure as before.<\/p>\r\n\r\n

Step 1:<\/span> State the Hypotheses<\/h5>\r\n

We will start by laying out our null and alternative hypotheses:<\/p>\r\n

\"\"<\/p>\r\n

\"\"<\/p>\r\n\r\n

Step 2:<\/span> Find the Critical Values<\/h5>\r\n

We need our critical values in order to determine the width of our margin of error. We will assume a significance level of a<\/span> = .05 (which will give us a 95% CI). From the t<\/span>\u00a0table, a two-tailed critical value at a<\/span> = .05 with 29 degrees of freedom (N <\/span>\u2212 1 = 30 \u2212 1 = 29) is t<\/span>* = 2.045.<\/p>\r\n\r\n

Step 3:<\/span> Calculate the Confidence Interval<\/h5>\r\n

Now we can construct our confidence interval. After we collect our data, we find that the average person in our community scored 39.85, or M<\/span> = 39.85, and our standard deviation was s<\/span> = 5.61. First, we need to use this standard deviation, plus our sample size of N <\/span>= 30, to calculate our standard error:<\/p>\r\n

\"\"<\/p>\r\n

Now we can put that value, our point estimate for the sample mean, and our critical value from Step\u00a02 into the formula for a confidence interval:<\/p>\r\n\"\"\r\n

\"\"<\/p>\r\n

\"\"<\/p>\r\n

\"\"<\/p>\r\n

\"\"<\/p>\r\n\r\n

Step 4:<\/span> Make the Decision<\/h5>\r\n

Finally, we can compare our confidence interval to our null hypothesis value. The null value of 38 is higher than our lower bound of 37.76 and lower than our upper bound of 41.94. Thus, the confidence interval brackets our null hypothesis value, and we fail to reject the null hypothesis:<\/p>\r\n

Fail to reject H<\/span>0<\/span>. Based on our sample of 30 people, our community is not different in average friendliness (M<\/span> = 39.85, SD<\/span> = 5.61) than the nation as a whole, 95% CI = (37.76, 41.94).<\/p>\r\n

Note that we don\u2019t report a test statistic or p<\/span> value because that is not how we tested the hypothesis, but we do report the value we found for our confidence interval.<\/p>\r\n

An important characteristic of hypothesis testing is that both methods will always give you the same result. That is because both are based on the standard error and critical values in their calculations. To check this, we can calculate a t<\/span>\u00a0statistic for the example above and find it to be t <\/span>= 1.81, which is smaller than our critical value of 2.045 and fails to reject the null hypothesis.<\/p>\r\nExample:<\/strong>
Suppose researchers examine whether LGBTQ+ youth report different levels of school safety than the national average. They collect survey data from 40 youth and construct a 95% CI for the mean safety score: (2.8, 3.4). If the national mean is 3.7, it lies outside the interval, suggesting these youth experience significantly lower safety. This approach uses the same logic as a t test but communicates the result as a plausible range \u2014 often more intuitive for policymakers and advocates.\r\n\r\nT Tests, Confidence Intervals, and Justice<\/strong>
T tests and confidence intervals give us ways to work with incomplete information, which is nearly always the case in real-world research. They allow us to test whether observed disparities are likely due to chance and to estimate the plausible size of those disparities. For social justice work, these methods are invaluable: they give voice to underrepresented groups by showing that even small samples can provide meaningful evidence, and they help ensure that findings are presented not as absolutes but as ranges that reflect uncertainty and complexity.\r\n

Exercises<\/h3>\r\n
    \r\n \t
  1. What is the difference between a z<\/span>\u00a0test and a one-sample t<\/span>\u00a0test?<\/li>\r\n \t
  2. What does a confidence interval represent?<\/li>\r\n \t
  3. What is the relationship between a chosen level of confidence for a confidence interval and how wide that interval is? For instance, if you move from a 95% CI to a 90% CI, what happens? Hint: look at the t<\/span>\u00a0table to see how critical values change when you change levels of significance.<\/li>\r\n \t
  4. Construct a confidence interval around the sample mean M<\/span> = 25 for the following conditions:\r\n
      \r\n \t
    1. N <\/span>= 25, s<\/span> = 15, 95% confidence level<\/li>\r\n \t
    2. N <\/span>= 25, s<\/span> = 15, 90% confidence level<\/li>\r\n \t
    3. \"\" = 4.5, a<\/span> = .05, d<\/span>f<\/span> = 20<\/li>\r\n \t
    4. s<\/span> = 12, d<\/span>f<\/span> = 16 (yes, that is all the information you need)<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
    5. True or false: A confidence interval represents the most likely location of the true population mean.<\/li>\r\n \t
    6. You hear that college campuses may differ from the general population in terms of political affiliation, and you want to use hypothesis testing to see if this is true and, if so, how big the difference is. You know that the average political affiliation in the nation is \"mu\" = 4.00 on a scale of 1.00 to 7.00, so you gather data from 150 college students across the nation to see if there is a difference. You find that the average score is 3.76 with a standard deviation of 1.52. Use a one-sample t<\/span>\u00a0test to see if there is a difference at the a<\/span> = .05 level.<\/li>\r\n \t
    7. You hear a lot of talk about increasing global temperature, so you decide to see for yourself if there has been an actual change in recent years. You know that the average land temperature from 1951-1980 was 8.79 degrees Celsius. You find annual average temperature data from 1981\u20132017 and decide to construct a 99% confidence interval (because you want to be as sure as possible and look for differences in both directions, not just one) using this data to test for a difference from the previous average.\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      \r\n

      Year<\/p>\r\n<\/td>\r\n

      		\r\n

      Temp<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/thead>\r\n

      \r\n

      1981<\/p>\r\n<\/td>\r\n

      		\r\n

      9.301<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1982<\/p>\r\n<\/td>\r\n

      		\r\n

      8.788<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1983<\/p>\r\n<\/td>\r\n

      		\r\n

      9.173<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1984<\/p>\r\n<\/td>\r\n

      		\r\n

      8.824<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1985<\/p>\r\n<\/td>\r\n

      		\r\n

      8.799<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1986<\/p>\r\n<\/td>\r\n

      		\r\n

      8.985<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1987<\/p>\r\n<\/td>\r\n

      		\r\n

      9.141<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1988<\/p>\r\n<\/td>\r\n

      		\r\n

      9.345<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1989<\/p>\r\n<\/td>\r\n

      		\r\n

      9.076<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1990<\/p>\r\n<\/td>\r\n

      		\r\n

      9.378<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1991<\/p>\r\n<\/td>\r\n

      		\r\n

      9.336<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1992<\/p>\r\n<\/td>\r\n

      		\r\n

      8.974<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1993<\/p>\r\n<\/td>\r\n

      		\r\n

      9.008<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1994<\/p>\r\n<\/td>\r\n

      		\r\n

      9.175<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1995<\/p>\r\n<\/td>\r\n

      		\r\n

      9.484<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1996<\/p>\r\n<\/td>\r\n

      		\r\n

      9.168<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1997<\/p>\r\n<\/td>\r\n

      		\r\n

      9.326<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1998<\/p>\r\n<\/td>\r\n

      		\r\n

      9.660<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      1999<\/p>\r\n<\/td>\r\n

      		\r\n

      9.406<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2000<\/p>\r\n<\/td>\r\n

      		\r\n

      9.332<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2001<\/p>\r\n<\/td>\r\n

      		\r\n

      9.542<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2002<\/p>\r\n<\/td>\r\n

      		\r\n

      9.695<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2003<\/p>\r\n<\/td>\r\n

      		\r\n

      9.649<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2004<\/p>\r\n<\/td>\r\n

      		\r\n

      9.451<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2005<\/p>\r\n<\/td>\r\n

      		\r\n

      9.829<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2006<\/p>\r\n<\/td>\r\n

      		\r\n

      9.662<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2007<\/p>\r\n<\/td>\r\n

      		\r\n

      9.876<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2008<\/p>\r\n<\/td>\r\n

      		\r\n

      9.581<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2009<\/p>\r\n<\/td>\r\n

      		\r\n

      9.657<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2010<\/p>\r\n<\/td>\r\n

      		\r\n

      9.828<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2011<\/p>\r\n<\/td>\r\n

      		\r\n

      9.650<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2012<\/p>\r\n<\/td>\r\n

      		\r\n

      9.635<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2013<\/p>\r\n<\/td>\r\n

      		\r\n

      9.753<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2014<\/p>\r\n<\/td>\r\n

      		\r\n

      9.714<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2015<\/p>\r\n<\/td>\r\n

      		\r\n

      9.962<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2016<\/p>\r\n<\/td>\r\n

      		\r\n

      10.160<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      \r\n

      2017<\/p>\r\n<\/td>\r\n

      		\r\n

      10.049<\/p>\r\n<\/td>\r\n<\/tr>\r\n

      <\/td>\r\n<\/td>\r\n<\/tr>\r\n
      <\/td>\r\n<\/td>\r\n<\/tr>\r\n
      <\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t
    8. Determine whether you would reject or fail to reject the null hypothesis in the following situations:\r\n
        \r\n \t
      1. t<\/span> = 2.58, N <\/span>= 21, two-tailed test at a<\/span> = .05<\/li>\r\n \t
      2. t<\/span> = 1.99, N <\/span>= 49, one-tailed test at a<\/span> = .01<\/li>\r\n \t
      3. \"mu\" = 47.82, 99% CI = (48.71, 49.28)<\/li>\r\n \t
      4. \"mu\" = 0, 95% CI = (\u22120.15, 0.20)<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
      5. You are curious about how people feel about craft beer, so you gather data from 55 people in the city on whether or not they like it. You code your data so that 0 is neutral, positive scores indicate liking craft beer, and negative scores indicate disliking craft beer. You find that the average opinion was M<\/span> = 1.10 and the spread was s<\/span> = 0.40, and you test for a difference from 0 at the a<\/span> = .05 level.<\/li>\r\n \t
      6. You want to know if college students have more stress in their daily lives than the general population (\"mu\" = 12), so you gather data from 25 people to test your hypothesis. Your sample has an average stress score of M<\/span> = 13.11 and a standard deviation of s<\/span> = 3.89. Use a one-sample t<\/span>\u00a0test to see if there is a difference.<\/li>\r\n<\/ol>\r\n
        \r\n

        Answers to Odd-Numbered Exercises<\/h3>\r\n<\/header>\r\n
        \r\n\r\n1)\r\nA z<\/span>\u00a0test uses population standard deviation for calculating standard error and gets critical values based on the standard normal distribution. A t<\/span>\u00a0test uses sample standard deviation as an estimate when calculating standard error and gets critical values from the t<\/span>\u00a0distribution based on degrees of freedom.\r\n\r\n3)\r\nAs the level of confidence gets higher, the interval gets wider. In order to speak with more confidence about having found the population mean, you need to cast a wider net. This happens because critical values for higher confidence levels are larger, which creates a wider margin of error.<\/span>\r\n\r\n5)\r\nFalse. A confidence interval is a range of plausible scores that may or may not bracket the true population mean.<\/span>\r\n\r\n7)\r\nM<\/span> = 9.44, <\/span>s<\/span> = 0.35, <\/span>\"\" = 0.06, <\/span>d<\/span>f <\/span>= 36, <\/span>t<\/span>* = 2.719, 99% CI = (9.28, 9.60); CI does not bracket <\/span>\"mu\", reject null hypothesis; <\/span>d <\/span>= 1.83<\/span>\r\n\r\n \r\n\r\n9)\r\n\r\nStep 1:<\/span> H<\/span>0<\/span>: <\/span>\"mu\" = 0 \u201cThe average person has a neutral opinion toward craft beer,\u201d <\/span>H<\/span>A<\/span>: <\/span>\"mu\" \u2260 0 \u201cOverall, people will have an opinion about craft beer, either good or bad.\u201d<\/span>\r\n\r\nStep 2:<\/span> Two-tailed test, <\/span>d<\/span>f<\/span> = 54, <\/span>t<\/span>* = 2.009<\/span>\r\n\r\nStep 3:<\/span> M<\/span> = 1.10, <\/span>\"\" = 0.05, <\/span>t <\/span>= 22.00<\/span>\r\n\r\nSte<\/span>p 4:<\/span> t <\/span>> <\/span>t<\/span>*, reject <\/span>H<\/span>0<\/span>. Based on opinions from 55 people, we can conclude that the average opinion of craft beer (<\/span>M<\/span> = 1.10) is positive, <\/span>t<\/span>(54) = 22.00, <\/span>p<\/span> < .05, <\/span>d<\/span> = 2.75.<\/span>\r\n

        <\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n \r\n

        \"\"<\/h3>\r\n\"P-Values<\/a>\" by Randall Munroe\/xkcd.com is licensed under CC BY-NC 2.5<\/a>.)\r\n\r\n\"\"<\/a>","rendered":"
        \n
        \n

        Key Terms<\/h3>\n<\/header>\n
        \n


        \nconfidence interval<\/span><\/a><\/p>\n

        margin of error<\/span><\/a><\/p>\n

        point estimate<\/span><\/a><\/p>\n<\/div>\n<\/div>\n

        In Chapter 7<\/span><\/a>, we made a big leap from basic descriptive statistics into full hypothesis testing and inferential statistics. For the rest of the unit, we will be learning new tests, each of which is just a small adjustment on the test before it. In this chapter, we will learn about the first of three t<\/span>\u00a0tests, and we will learn a new method of testing the null hypothesis: confidence intervals.<\/p>\n

        Social Justice and the t Test<\/strong>
        In the real world, we rarely know the exact parameters of a population \u2014 we only have samples. That is especially true in social justice research, where communities are often underrepresented in data collection. The t test gives us a way to use limited information from a sample to make inferences about larger populations. Combined with confidence intervals, it allows us to estimate not just a single number, but a plausible range of values. These tools give us more reliable ways to test whether differences we observe \u2014 such as disparities in wages, housing access, or health outcomes \u2014 are random or evidence of systemic inequality.<\/p>\n

        The t<\/span> Statistic<\/h3>\n

        In Chapter 7<\/span><\/a>, we were introduced to hypothesis testing using the z<\/span>\u00a0statistic for sample means that we learned in Unit 1<\/span><\/a>. This was a useful way to link the material and ease us into the new way to looking at data, but it isn\u2019t a very common test because it relies on knowing the population\u2019s standard deviation, s<\/span>, which is rarely going to be the case. Instead, we will estimate that parameter s<\/span> using the sample statistic s<\/span> in the same way that we estimate \"mu\" using \"Upper (\"mu\" will still appear in our formulas because we suspect something about its value and that is what we are testing). Our new statistic is called t<\/span>, and for testing one population mean using a single sample (called a one-sample t<\/span>\u00a0test) it takes the form:<\/p>\n

        \"\"<\/p>\n

        Notice that t <\/span>looks almost identical to z<\/span>; this is because they test the exact same thing: the value of a sample mean compared to what we expect of the population. The only difference is that the standard error is now denoted \"\" to indicate that we use the sample statistic for standard deviation, s<\/span>, instead of the population parameter s<\/span>. The process of using and interpreting the standard error and the full test statistic remain exactly the same.<\/p>\n

        In Chapter 3<\/span><\/a> we learned that the formulas for sample standard deviation and population standard deviation differ by one key factor: the denominator for the parameter is \"Upper but the denominator for the statistic is N <\/span>\u2212 1, also known as degrees of freedom, d<\/span>f<\/span>. Because we are using a new measure of spread, we can no longer use the standard normal distribution and the z<\/span>\u00a0table to find our critical values. For t<\/span>\u00a0tests, we will use the t<\/span>\u00a0distribution and t<\/span>\u00a0table to find these values.<\/p>\n

        The t<\/span>\u00a0distribution, like the standard normal distribution, is symmetric and normally distributed with a mean of 0 and standard error (as the measure of standard deviation for sampling distributions) of 1. However, because the calculation of standard error uses degrees of freedom, there will be a different t<\/span>\u00a0distribution for every degree of freedom. Luckily, they all work exactly the same, so in practice this difference is minor.<\/p>\n

        Figure 8.1<\/span><\/a> shows four curves: a normal distribution curve labeled z<\/span>, and three t<\/span>\u00a0distribution curves for 2, 10, and 30 degrees of freedom. Two things should stand out: First, for lower degrees of freedom (e.g., 2), the tails of the distribution are much fatter, meaning the a larger proportion of the area under the curve falls in the tail. This means that we will have to go farther out into the tail to cut off the portion corresponding to 5% or a<\/span> = .05, which will in turn lead to higher critical values. Second, as the degrees of freedom increase, we get closer and closer to the z <\/span>curve. Even the distribution with d<\/span>f<\/span> = 30, corresponding to a sample size of just 31 people, is nearly indistinguishable from z<\/span>. In fact, a t<\/span>\u00a0distribution with infinite degrees of freedom (theoretically, of course) is exactly the standard normal distribution. Because of this, the bottom row of the t<\/span>\u00a0table also includes the critical values for z<\/span>\u00a0tests at the specific significance levels. Even though these curves are very close, it is still important to use the correct table and critical values, because small differences can add up quickly.<\/p>\n

        \n
        \n

        <\/a>Figure 8.1.<\/span> Distributions comparing effects of degrees of freedom. (\u201cDistributions Comparing Effects of Degrees of Freedom<\/span><\/a>\u201d by Judy Schmitt is licensed under CC BY-NC-SA 4.0<\/span><\/a>.)<\/p>\n<\/div>\n<\/div>\n

        \n
        \"\"<\/div>\n<\/div>\n

        The t<\/span>\u00a0distribution table lists critical values for one- and two-tailed tests at several levels of significance, arranged into columns. The rows of the t<\/span>\u00a0table list degrees of freedom up to d<\/span>f <\/span>= 100 in order to use the appropriate distribution curve. It does not, however, list all possible degrees of freedom in this range, because that would take too many rows. Above d<\/span>f <\/span>=\u00a040, the rows jump in increments of 10. If a problem requires you to find critical values and the exact degrees of freedom is not listed, you always round down to the next smallest number. For example, if you have 48 people in your sample, the degrees of freedom are N <\/span>\u2212 1 = 48 \u2212 1 = 47; however, 47 doesn\u2019t appear on our table, so we round down and use the critical values for d<\/span>f <\/span>= 40, even though 50 is closer. We do this because it avoids inflating Type I error (false positives, see Chapter 7<\/span><\/a>) by using criteria that are too\u00a0lax.<\/p>\n

        Example:<\/strong>
        Suppose we\u2019re studying high school graduation rates among Indigenous students in a small district. With only a few students in the sample, our degrees of freedom are low. The t distribution accounts for this extra uncertainty by using \u201cfatter tails.\u201d This means we need stronger evidence to reject the null when our sample is small. In practice, it keeps us cautious about making sweeping claims from limited data, while still giving us a method to test hypotheses.<\/p>\n

        Hypothesis Testing with t<\/span><\/h3>\n

        Hypothesis testing with the t<\/span>\u00a0statistic works exactly the same way as z<\/span>\u00a0tests did, following the four-step process of (1) stating the hypotheses, (2) finding the critical values, (3) computing the test statistic and effect size, and (4) making the decision.<\/p>\n

        Example <\/span> Oil Change Speed<\/p>\n

        We will work though an example: Let\u2019s say that the welfare office changed application systems.\u00a0 The old system approved benefits in about 30 days and you suspect that the new system takes much longer. After four people applied for welfare benefits, you think you have enough evidence to demonstrate this.<\/p>\n

        Step 1:<\/span> State the Hypotheses<\/h5>\n

        Our hypotheses for one-sample t<\/span>\u00a0tests are identical to those we used for z<\/span>\u00a0tests. We still state the null and alternative hypotheses mathematically in terms of the population parameter and written out in readable English. For our example:<\/p>\n

        Ho <\/sub>:There is no difference in the average process time for welfare application between the new and old system.<\/strong><\/p>\n

        Ho\u00a0<\/sub>: \u03bc = 30 days<\/strong><\/p>\n

        Ha<\/sub>\u00a0<\/sub>:The average process time for welfare application takes longer with the new system.\u00a0<\/strong><\/p>\n

        Ha<\/sub>\u00a0<\/sub>: \u03bc > 30 days<\/strong><\/p>\n

        Step 2:<\/span> Find the Critical Values<\/h5>\n

        As noted above, our critical values still delineate the area in the tails under the curve corresponding to our chosen level of significance. Because we have no reason to change significance levels, we will use a<\/span> = .05, and because we suspect a direction of effect, we have a one-tailed test. To find our critical values for t<\/span>, we need to add one more piece of information: the degrees of freedom. For this example:<\/p>\n

        \"\"<\/p>\n

        Going to our t<\/span>\u00a0table, a portion of which is found in Table 8.1<\/span><\/a>, we locate the column corresponding to our one-tailed significance level of .05 and find where it intersects with the row for 3 degrees of freedom. As we can see in Table 8.1<\/span><\/a>, our critical value is t<\/span>* = 2.353. (The complete t<\/span> table can be found in Appendix B<\/span><\/a>.)<\/p>\n

        \n
        \n

        <\/a>Table 8.1.<\/span> t<\/span> distribution table (t<\/span> table).<\/p>\n

        Adapted from \u201cTabla t<\/span><\/a>\u201d by Jsmura\/Wikimedia Commons, CC BY-SA 4.0<\/span><\/a>.<\/p>\n<\/div>\n<\/div>\n

        We can then shade this region on our t<\/span>\u00a0distribution to visualize our rejection region (Figure 8.2<\/span><\/a>).<\/p>\n

        \n
        \n

        <\/a>Figure 8.2.<\/span> Rejection region. (\u201cRejection Region t2.353<\/span><\/a>\u201d by Judy Schmitt is licensed under CC BY-NC-SA 4.0<\/span><\/a>.)<\/p>\n<\/div>\n<\/div>\n

        \n
        \"\"<\/div>\n<\/div>\n
        Step 3:<\/span> Calculate the Test Statistic and Effect Size<\/h5>\n

        The four processing times recipients experienced were 46 days, 58 days, 40 days, and 71 days. We will use these to calculate \"Upper and s<\/span> by first filling in the sum of squares in Table 8.2<\/span><\/a>.<\/p>\n

        \n
        \n

        <\/a>Table 8.2.<\/span> Sum of squares.<\/p>\n\n\n\n\n<\/colgroup>\n\n\n\n\n\n\n\n\n
        \n

        X<\/span><\/p>\n<\/td>\n

        		\n

        X<\/span> \u2212 M<\/span><\/p>\n<\/td>\n

        		\n

        (X<\/span> \u2212 M<\/span>)2<\/span><\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

        \n

        46<\/p>\n<\/td>\n

        		\n

        \u22127.75<\/p>\n<\/td>\n

        		\n

        60.06<\/p>\n<\/td>\n<\/tr>\n

        \n

        58<\/p>\n<\/td>\n

        		\n

        4.25<\/p>\n<\/td>\n

        		\n

        18.06<\/p>\n<\/td>\n<\/tr>\n

        \n

        40<\/p>\n<\/td>\n

        		\n

        \u221213.75<\/p>\n<\/td>\n

        		\n

        189.06<\/p>\n<\/td>\n<\/tr>\n

        \n

        71<\/p>\n<\/td>\n

        		\n

        17.25<\/p>\n<\/td>\n

        		\n

        297.56<\/p>\n<\/td>\n<\/tr>\n

        \n

        \u03a3<\/span> = 215<\/p>\n<\/td>\n

        		\n

        \u03a3<\/span> = 0<\/p>\n<\/td>\n

        		\n

        \u03a3<\/span> = 564.74<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

        After filling in the first row to get \u03a3<\/span>X<\/span> = 215, we find that the mean is M<\/span> = 53.75 (215 divided by sample size 4), which allows us to fill in the rest of the table to get our sum of squares SS<\/span> = 564.74, which we then plug in to the formula for standard deviation from Chapter 3<\/span><\/a>:<\/p>\n

        \"\"<\/p>\n

        Next, we take this value and plug it in to the formula for standard error:<\/p>\n

        \"\"<\/p>\n

        And, finally, we put the standard error, sample mean, and null hypothesis value into the formula for our test statistic t<\/span>:<\/p>\n

        \"\"<\/p>\n

        This may seem like a lot of steps, but it is really just taking our raw data to calculate one value at a time and carrying that value forward into the next equation: data \u2192<\/span> sample size\/degrees of freedom \u2192<\/span> mean \u2192<\/span> sum of squares \u2192<\/span> standard deviation \u2192<\/span> standard error \u2192<\/span> test statistic. At each step, we simply match the symbols of what we just calculated to where they appear in the next formula to make sure we are plugging everything in correctly.<\/p>\n

        Step 4:<\/span> Make the Decision<\/h5>\n

        Now that we have our critical value and test statistic, we can make our decision using the same criteria we used for a z<\/span>\u00a0test. Our obtained t<\/span>\u00a0statistic was t <\/span>= 3.46 and our critical value was t<\/span>*\u00a0=\u00a02.353: t <\/span>> t<\/span>*, so we reject the null hypothesis and conclude:<\/p>\n

        Based on our four applicants, the new processing time to receive welfare benefits takes longer on average (M<\/span> = 53.75, SD<\/span> = 13.72) than the old processing system, t<\/span>(3) = 3.46, p<\/span>\u00a0<\u00a0.05, d<\/span> = 1.74.<\/p>\n

        Notice that we also include the degrees of freedom in parentheses next to t<\/span>. Figure 8.3<\/span><\/a> shows the output from JASP.<\/p>\n

        \n
        FIGURE 8.3<\/span><\/span><\/a>.<\/span>\u00a0Output from JASP for the one-sample\u00a0t<\/span>\u00a0test described in this example. The output provides the\u00a0t<\/span>\u00a0value (3.462), degrees of freedom (3), and the exact\u00a0p<\/span> value (.020, which is less than .05). Note that the mean (53.750) and standard deviation for the sample are also provided (13.720). Based on our four applicants processing time, the new system takes longer on average (M<\/span>\u00a0= 53.75,\u00a0SD<\/span> = 13.72) than the\u00a0 old system to receive benefits, t<\/span>(3)\u00a0=\u00a03.46,\u00a0p<\/span> =\u00a0.02. \u00a0(\u201cJASP 1-sample t test<\/span><\/a>\u201d by Rupa G. Gordon\/Judy Schmitt is licensed under\u00a0CC BY-NC-SA 4.0<\/span><\/a>.)<\/h5>\n<\/div>\n
        \n
        \"\"<\/div>\n<\/div>\n

        <\/h3>\n

        Example:<\/strong>
        Imagine a city rolls out a new public housing policy. The old average wait time for housing assistance was 12 months. A researcher collects a sample of 20 applicants under the new system and finds an average wait of 14 months. A one-sample t test allows us to ask: is this longer wait likely due to chance, or does it suggest the new policy increases delays? This mirrors the way we use welfare processing times in the current example \u2014 testing whether new systems help or harm marginalized communities.<\/p>\n

        Confidence Intervals<\/h3>\n

        Up to this point, we have learned how to estimate the population parameter for the mean using sample data and a sample statistic. From one point of view, this makes sense: we have one value for our parameter so we use a single value (called a <\/a><\/a>point estimate<\/span>) to estimate it. However, we have seen that all statistics have sampling error and that the value we find for the sample mean will bounce around based on the people in our sample, simply due to random chance. Thinking about estimation from this perspective, it would make more sense to take that error into account rather than relying just on our point estimate. To do this, we calculate what is known as a confidence interval.<\/p>\n

        A <\/a><\/a>confidence interval<\/span> starts with our point estimate and then creates a range of scores considered plausible based on our standard deviation, our sample size, and the level of confidence with which we would like to estimate the parameter. This range, which extends equally in both directions away from the point estimate, is called the <\/a><\/a>margin of error<\/span>. We calculate the margin of error by multiplying our two-tailed critical value by our standard error:<\/p>\n

        \"\"<\/p>\n

        One important consideration when calculating the margin of error is that it can only be calculated using the critical value for a two-tailed test. This is because the margin of error moves away from the point estimate in both directions, so a one-tailed value does not make sense.<\/p>\n

        The critical value we use will be based on a chosen level of confidence, which is equal to 1 \u2212 a<\/span>. Thus, a 95% level of confidence corresponds to a<\/span> = .05. Thus, at the .05 level of significance, we create a 95% confidence interval. How to interpret that is discussed further on.<\/p>\n

        Once we have our margin of error calculated, we add it to our point estimate for the mean to get an upper bound to the confidence interval and subtract it from the point estimate for the mean to get a lower bound for the confidence interval:<\/p>\n

        \"\"<\/p>\n

        or simply:<\/p>\n

        \"\"<\/p>\n

        To write out a confidence interval, we always use round brackets (i.e., parentheses) and put the lower bound, a comma, and the upper bound:<\/p>\n

        \"\"<\/p>\n

        Let\u2019s see what this looks like with some actual numbers by taking our welfare application data and using it to create a 95% confidence interval estimating the average length of time it takes to receive benefits. We already found that our average was M<\/span> = 53.75 days and our standard error was \"\" = 6.86. We also found a critical value to test our hypothesis, but remember that we were testing a one-tailed hypothesis, so that critical value won\u2019t work. To see why that is, look at the column headers on the t<\/span>\u00a0table. The column for one-tailed a<\/span> = .05 is the same as a two-tailed a<\/span> = .10. If we used the old critical value, we\u2019d actually be creating a 90% confidence interval (1.00 \u2212 0.10 = 0.90, or 90%). To find the correct value, we use the column for two-tailed a<\/span> = .05 and, again, the row for 3 degrees of freedom, to find t<\/span>* = 3.182.<\/p>\n

        Now we have all the pieces we need to construct our confidence interval:<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        So we find that our 95% confidence interval runs from 31.92 days to 75.58 days, but what does that actually mean? The range (31.92, 75.58) represents values of the mean that we consider reasonable or plausible based on our observed data. It includes our point estimate of the mean, M<\/span>\u00a0= 53.75, in the center, but it also has a range of values that could also have been the case based on what we know about how much these scores vary (i.e., our standard error).<\/p>\n

        It is very tempting to also interpret this interval by saying that we are 95% confident that the true population mean falls within the range (31.92, 75.58), but this is not true. The reason it is not true is that phrasing our interpretation this way suggests that we have firmly established an interval and the population mean does or does not fall into it, suggesting that our interval is firm and the population mean will move around. However, the population mean is an absolute that does not change; it is our interval that will vary from data collection to data collection, even taking into account our standard error. The correct interpretation, then, is that we are 95% confident that the range (31.92, 75.58) brackets the true population mean. This is a very subtle difference, but it is an important one.<\/p>\n

        Consider studying wage gaps for women of color compared to national averages. Instead of estimating the population wage with a single sample mean, we construct a confidence interval. If our 95% CI for the average wage is ($37,000, $42,000), we can say this range of values is plausible based on the data. If the national mean is $45,000, and that value falls outside our confidence interval, we reject the null hypothesis and conclude there is a real gap. Confidence intervals thus help quantify inequity while accounting for uncertainty in our sample.<\/p>\n

        Hypothesis Testing with Confidence Intervals<\/h4>\n

        As a function of how they are constructed, we can also use confidence intervals to test hypotheses. However, we are limited to testing two-tailed hypotheses only, because of how the intervals work, as discussed above.<\/p>\n

        Once a confidence interval has been constructed, using it to test a hypothesis is simple. If the range of the confidence interval brackets (or contains, or is around) the null hypothesis value, we fail to reject the null hypothesis. If it does not bracket the null hypothesis value (i.e., if the entire range is above the null hypothesis value or below it), we reject the null hypothesis. The reason for this is clear if we think about what a confidence interval represents. Remember: a confidence interval is a range of values that we consider reasonable or plausible based on our data. Thus, if the null hypothesis value is in that range, then it is a value that is plausible based on our observations. If the null hypothesis is plausible, then we have no reason to reject it. Thus, if our confidence interval brackets the null hypothesis value, thereby making it a reasonable or plausible value based on our observed data, then we have no evidence against the null hypothesis and fail to reject it. However, if we build a confidence interval of reasonable values based on our observations and it does not contain the null hypothesis value, then we have no empirical (observed) reason to believe the null hypothesis value and therefore reject the null hypothesis.<\/p>\n

        Example <\/span> Friendliness<\/p>\n

        You hear that the national average on a measure of friendliness is 38 points. You want to know if people in your community are more or less friendly than people nationwide, so you collect data from 30 random people in town to look for a difference. We\u2019ll follow the same four-step hypothesis-testing procedure as before.<\/p>\n

        Step 1:<\/span> State the Hypotheses<\/h5>\n

        We will start by laying out our null and alternative hypotheses:<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        Step 2:<\/span> Find the Critical Values<\/h5>\n

        We need our critical values in order to determine the width of our margin of error. We will assume a significance level of a<\/span> = .05 (which will give us a 95% CI). From the t<\/span>\u00a0table, a two-tailed critical value at a<\/span> = .05 with 29 degrees of freedom (N <\/span>\u2212 1 = 30 \u2212 1 = 29) is t<\/span>* = 2.045.<\/p>\n

        Step 3:<\/span> Calculate the Confidence Interval<\/h5>\n

        Now we can construct our confidence interval. After we collect our data, we find that the average person in our community scored 39.85, or M<\/span> = 39.85, and our standard deviation was s<\/span> = 5.61. First, we need to use this standard deviation, plus our sample size of N <\/span>= 30, to calculate our standard error:<\/p>\n

        \"\"<\/p>\n

        Now we can put that value, our point estimate for the sample mean, and our critical value from Step\u00a02 into the formula for a confidence interval:<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        Step 4:<\/span> Make the Decision<\/h5>\n

        Finally, we can compare our confidence interval to our null hypothesis value. The null value of 38 is higher than our lower bound of 37.76 and lower than our upper bound of 41.94. Thus, the confidence interval brackets our null hypothesis value, and we fail to reject the null hypothesis:<\/p>\n

        Fail to reject H<\/span>0<\/span>. Based on our sample of 30 people, our community is not different in average friendliness (M<\/span> = 39.85, SD<\/span> = 5.61) than the nation as a whole, 95% CI = (37.76, 41.94).<\/p>\n

        Note that we don\u2019t report a test statistic or p<\/span> value because that is not how we tested the hypothesis, but we do report the value we found for our confidence interval.<\/p>\n

        An important characteristic of hypothesis testing is that both methods will always give you the same result. That is because both are based on the standard error and critical values in their calculations. To check this, we can calculate a t<\/span>\u00a0statistic for the example above and find it to be t <\/span>= 1.81, which is smaller than our critical value of 2.045 and fails to reject the null hypothesis.<\/p>\n

        Example:<\/strong>
        Suppose researchers examine whether LGBTQ+ youth report different levels of school safety than the national average. They collect survey data from 40 youth and construct a 95% CI for the mean safety score: (2.8, 3.4). If the national mean is 3.7, it lies outside the interval, suggesting these youth experience significantly lower safety. This approach uses the same logic as a t test but communicates the result as a plausible range \u2014 often more intuitive for policymakers and advocates.<\/p>\n

        T Tests, Confidence Intervals, and Justice<\/strong>
        T tests and confidence intervals give us ways to work with incomplete information, which is nearly always the case in real-world research. They allow us to test whether observed disparities are likely due to chance and to estimate the plausible size of those disparities. For social justice work, these methods are invaluable: they give voice to underrepresented groups by showing that even small samples can provide meaningful evidence, and they help ensure that findings are presented not as absolutes but as ranges that reflect uncertainty and complexity.<\/p>\n

        Exercises<\/h3>\n
          \n
        1. What is the difference between a z<\/span>\u00a0test and a one-sample t<\/span>\u00a0test?<\/li>\n
        2. What does a confidence interval represent?<\/li>\n
        3. What is the relationship between a chosen level of confidence for a confidence interval and how wide that interval is? For instance, if you move from a 95% CI to a 90% CI, what happens? Hint: look at the t<\/span>\u00a0table to see how critical values change when you change levels of significance.<\/li>\n
        4. Construct a confidence interval around the sample mean M<\/span> = 25 for the following conditions:\n
            \n
          1. N <\/span>= 25, s<\/span> = 15, 95% confidence level<\/li>\n
          2. N <\/span>= 25, s<\/span> = 15, 90% confidence level<\/li>\n
          3. \"\" = 4.5, a<\/span> = .05, d<\/span>f<\/span> = 20<\/li>\n
          4. s<\/span> = 12, d<\/span>f<\/span> = 16 (yes, that is all the information you need)<\/li>\n<\/ol>\n<\/li>\n
          5. True or false: A confidence interval represents the most likely location of the true population mean.<\/li>\n
          6. You hear that college campuses may differ from the general population in terms of political affiliation, and you want to use hypothesis testing to see if this is true and, if so, how big the difference is. You know that the average political affiliation in the nation is \"mu\" = 4.00 on a scale of 1.00 to 7.00, so you gather data from 150 college students across the nation to see if there is a difference. You find that the average score is 3.76 with a standard deviation of 1.52. Use a one-sample t<\/span>\u00a0test to see if there is a difference at the a<\/span> = .05 level.<\/li>\n
          7. You hear a lot of talk about increasing global temperature, so you decide to see for yourself if there has been an actual change in recent years. You know that the average land temperature from 1951-1980 was 8.79 degrees Celsius. You find annual average temperature data from 1981\u20132017 and decide to construct a 99% confidence interval (because you want to be as sure as possible and look for differences in both directions, not just one) using this data to test for a difference from the previous average.
            \n\n\n\n<\/colgroup>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
            \n

            Year<\/p>\n<\/td>\n

            		\n

            Temp<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

            \n

            1981<\/p>\n<\/td>\n

            		\n

            9.301<\/p>\n<\/td>\n<\/tr>\n

            \n

            1982<\/p>\n<\/td>\n

            		\n

            8.788<\/p>\n<\/td>\n<\/tr>\n

            \n

            1983<\/p>\n<\/td>\n

            		\n

            9.173<\/p>\n<\/td>\n<\/tr>\n

            \n

            1984<\/p>\n<\/td>\n

            		\n

            8.824<\/p>\n<\/td>\n<\/tr>\n

            \n

            1985<\/p>\n<\/td>\n

            		\n

            8.799<\/p>\n<\/td>\n<\/tr>\n

            \n

            1986<\/p>\n<\/td>\n

            		\n

            8.985<\/p>\n<\/td>\n<\/tr>\n

            \n

            1987<\/p>\n<\/td>\n

            		\n

            9.141<\/p>\n<\/td>\n<\/tr>\n

            \n

            1988<\/p>\n<\/td>\n

            		\n

            9.345<\/p>\n<\/td>\n<\/tr>\n

            \n

            1989<\/p>\n<\/td>\n

            		\n

            9.076<\/p>\n<\/td>\n<\/tr>\n

            \n

            1990<\/p>\n<\/td>\n

            		\n

            9.378<\/p>\n<\/td>\n<\/tr>\n

            \n

            1991<\/p>\n<\/td>\n

            		\n

            9.336<\/p>\n<\/td>\n<\/tr>\n

            \n

            1992<\/p>\n<\/td>\n

            		\n

            8.974<\/p>\n<\/td>\n<\/tr>\n

            \n

            1993<\/p>\n<\/td>\n

            		\n

            9.008<\/p>\n<\/td>\n<\/tr>\n

            \n

            1994<\/p>\n<\/td>\n

            		\n

            9.175<\/p>\n<\/td>\n<\/tr>\n

            \n

            1995<\/p>\n<\/td>\n

            		\n

            9.484<\/p>\n<\/td>\n<\/tr>\n

            \n

            1996<\/p>\n<\/td>\n

            		\n

            9.168<\/p>\n<\/td>\n<\/tr>\n

            \n

            1997<\/p>\n<\/td>\n

            		\n

            9.326<\/p>\n<\/td>\n<\/tr>\n

            \n

            1998<\/p>\n<\/td>\n

            		\n

            9.660<\/p>\n<\/td>\n<\/tr>\n

            \n

            1999<\/p>\n<\/td>\n

            		\n

            9.406<\/p>\n<\/td>\n<\/tr>\n

            \n

            2000<\/p>\n<\/td>\n

            		\n

            9.332<\/p>\n<\/td>\n<\/tr>\n

            \n

            2001<\/p>\n<\/td>\n

            		\n

            9.542<\/p>\n<\/td>\n<\/tr>\n

            \n

            2002<\/p>\n<\/td>\n

            		\n

            9.695<\/p>\n<\/td>\n<\/tr>\n

            \n

            2003<\/p>\n<\/td>\n

            		\n

            9.649<\/p>\n<\/td>\n<\/tr>\n

            \n

            2004<\/p>\n<\/td>\n

            		\n

            9.451<\/p>\n<\/td>\n<\/tr>\n

            \n

            2005<\/p>\n<\/td>\n

            		\n

            9.829<\/p>\n<\/td>\n<\/tr>\n

            \n

            2006<\/p>\n<\/td>\n

            		\n

            9.662<\/p>\n<\/td>\n<\/tr>\n

            \n

            2007<\/p>\n<\/td>\n

            		\n

            9.876<\/p>\n<\/td>\n<\/tr>\n

            \n

            2008<\/p>\n<\/td>\n

            		\n

            9.581<\/p>\n<\/td>\n<\/tr>\n

            \n

            2009<\/p>\n<\/td>\n

            		\n

            9.657<\/p>\n<\/td>\n<\/tr>\n

            \n

            2010<\/p>\n<\/td>\n

            		\n

            9.828<\/p>\n<\/td>\n<\/tr>\n

            \n

            2011<\/p>\n<\/td>\n

            		\n

            9.650<\/p>\n<\/td>\n<\/tr>\n

            \n

            2012<\/p>\n<\/td>\n

            		\n

            9.635<\/p>\n<\/td>\n<\/tr>\n

            \n

            2013<\/p>\n<\/td>\n

            		\n

            9.753<\/p>\n<\/td>\n<\/tr>\n

            \n

            2014<\/p>\n<\/td>\n

            		\n

            9.714<\/p>\n<\/td>\n<\/tr>\n

            \n

            2015<\/p>\n<\/td>\n

            		\n

            9.962<\/p>\n<\/td>\n<\/tr>\n

            \n

            2016<\/p>\n<\/td>\n

            		\n

            10.160<\/p>\n<\/td>\n<\/tr>\n

            \n

            2017<\/p>\n<\/td>\n

            		\n

            10.049<\/p>\n<\/td>\n<\/tr>\n

            <\/td>\n<\/td>\n<\/tr>\n
            <\/td>\n<\/td>\n<\/tr>\n
            <\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n
          8. Determine whether you would reject or fail to reject the null hypothesis in the following situations:\n
              \n
            1. t<\/span> = 2.58, N <\/span>= 21, two-tailed test at a<\/span> = .05<\/li>\n
            2. t<\/span> = 1.99, N <\/span>= 49, one-tailed test at a<\/span> = .01<\/li>\n
            3. \"mu\" = 47.82, 99% CI = (48.71, 49.28)<\/li>\n
            4. \"mu\" = 0, 95% CI = (\u22120.15, 0.20)<\/li>\n<\/ol>\n<\/li>\n
            5. You are curious about how people feel about craft beer, so you gather data from 55 people in the city on whether or not they like it. You code your data so that 0 is neutral, positive scores indicate liking craft beer, and negative scores indicate disliking craft beer. You find that the average opinion was M<\/span> = 1.10 and the spread was s<\/span> = 0.40, and you test for a difference from 0 at the a<\/span> = .05 level.<\/li>\n
            6. You want to know if college students have more stress in their daily lives than the general population (\"mu\" = 12), so you gather data from 25 people to test your hypothesis. Your sample has an average stress score of M<\/span> = 13.11 and a standard deviation of s<\/span> = 3.89. Use a one-sample t<\/span>\u00a0test to see if there is a difference.<\/li>\n<\/ol>\n
              \n
              \n

              Answers to Odd-Numbered Exercises<\/h3>\n<\/header>\n
              \n

              1)
              \nA z<\/span>\u00a0test uses population standard deviation for calculating standard error and gets critical values based on the standard normal distribution. A t<\/span>\u00a0test uses sample standard deviation as an estimate when calculating standard error and gets critical values from the t<\/span>\u00a0distribution based on degrees of freedom.<\/p>\n

              3)
              \nAs the level of confidence gets higher, the interval gets wider. In order to speak with more confidence about having found the population mean, you need to cast a wider net. This happens because critical values for higher confidence levels are larger, which creates a wider margin of error.<\/span><\/p>\n

              5)
              \nFalse. A confidence interval is a range of plausible scores that may or may not bracket the true population mean.<\/span><\/p>\n

              7)
              \nM<\/span> = 9.44, <\/span>s<\/span> = 0.35, <\/span>\"\" = 0.06, <\/span>d<\/span>f <\/span>= 36, <\/span>t<\/span>* = 2.719, 99% CI = (9.28, 9.60); CI does not bracket <\/span>\"mu\", reject null hypothesis; <\/span>d <\/span>= 1.83<\/span><\/p>\n

               <\/p>\n

              9)<\/p>\n

              Step 1:<\/span> H<\/span>0<\/span>: <\/span>\"mu\" = 0 \u201cThe average person has a neutral opinion toward craft beer,\u201d <\/span>H<\/span>A<\/span>: <\/span>\"mu\" \u2260 0 \u201cOverall, people will have an opinion about craft beer, either good or bad.\u201d<\/span><\/p>\n

              Step 2:<\/span> Two-tailed test, <\/span>d<\/span>f<\/span> = 54, <\/span>t<\/span>* = 2.009<\/span><\/p>\n

              Step 3:<\/span> M<\/span> = 1.10, <\/span>\"\" = 0.05, <\/span>t <\/span>= 22.00<\/span><\/p>\n

              Ste<\/span>p 4:<\/span> t <\/span>> <\/span>t<\/span>*, reject <\/span>H<\/span>0<\/span>. Based on opinions from 55 people, we can conclude that the average opinion of craft beer (<\/span>M<\/span> = 1.10) is positive, <\/span>t<\/span>(54) = 22.00, <\/span>p<\/span> < .05, <\/span>d<\/span> = 2.75.<\/span><\/p>\n

              \n<\/div>\n<\/div>\n

               <\/p>\n

              \"\"<\/h3>\n

              “P-Values<\/a>” by Randall Munroe\/xkcd.com is licensed under CC BY-NC 2.5<\/a>.)<\/p>\n

              \"\"<\/a><\/p>\n

              definition<\/span>