10-2: As we can with the z test, the single-sample t test, and the paired-samples t test, we can determine a confidence interval and calculate a measure of effect size—Cohen’s d—when we conduct an independent-samples t test.

10-7: The formulas for the upper and lower bounds of the confidence interval are: (M_X − M_Y)_upper = t(s_difference) + (M_X − M_Y)_sample and (M_X − M_Y)_lower = −t(s_difference) + (M_X − M_Y)_sample, respectively. In each case, we multiply the t statistic that marks off the tail by the standard error for the differences between means and add it to the difference between means in the sample. Remember that one of the t statistics is negative.

Let’s calculate the confidence interval that parallels the hypothesis test we conducted earlier, comparing ratings of those who are told they are drinking wine from a $10 bottle and ratings of those told they are drinking wine from a $90 bottle (Plassmann et al., 2008). The difference between the means of these samples was calculated in the numerator of the t statistic. It is 2.5 − 4.0 = −1.5. (Note that the order of subtraction in calculating the difference between means is irrelevant; we could just as easily have subtracted 2.5 from 4.0 and gotten a positive result, 1.5.) The standard error for the differences between means, s_difference, was calculated to be 0.616. The degrees of freedom were determined to be 7. Here are the five steps for determining a confidence interval for a difference between means:

STEP 1: Draw a normal curve with the sample difference between means in the center (as shown in Figure 10-4).

STEP 2: Indicate the bounds of the confidence interval on either end, and write the percentages under each segment of the curve (see Figure 10-4).

STEP 3: Look up the t statistics for the lower and upper ends of the confidence interval in the t table.

Use a two-tailed test and a p level of 0.05 (which corresponds to a 95% confidence interval). Use the degrees of freedom—7—that we calculated earlier. The table indicates a t statistic of 2.365. Because the normal curve is symmetric, the bounds of the confidence interval fall at t statistics of −2.365 and 2.365. (Note that these cutoffs are identical to those used for the independent-samples t test because the p level of 0.05 corresponds to a confidence level of 95%.) We add those t statistics to the normal curve, as in Figure 10-5.

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STEP 4: Convert the t statistics to raw differences between means for the lower and upper ends of the confidence interval.

For the lower end, the formula is:

(M_X − M_Y)_lower = −t(s_difference) + (M_X − M_Y)_sample

= −2.365(0.616) + (−1.5) = −2.96

For the upper end, the formula is:

(M_X − M_Y)_upper = t(s_difference) + (M_X − M_Y)_sample

= 2.365(0.616) + (−1.5) = −0.04

confidence interval is [−2.96, −0.04], as shown in Figure 10-6.

STEP 5: Check your answer

Each end of the confidence interval should be exactly the same distance from the sample mean.

−2.96 − (−1.5) = −1.46

−0.04 − (−1.5) = 1.46

The interval checks out. The bounds of the confidence interval are calculated as the difference between sample means, plus or minus 1.46. Also, the confidence interval does not include 0; thus, it is not plausible that there is no difference between means. We can conclude that people told they are drinking wine from a $10 bottle give different ratings, on average, than those told they are drinking wine from a $90 bottle. When we conducted the independent-samples t test earlier, we rejected the null hypothesis and drew the same conclusion as we did with the confidence interval. But the confidence interval provides more information because it is an interval estimate rather than a point estimate.

As with all hypothesis tests, it is recommended that the results be supplemented with an effect size. For an independent-samples t test, as with other t tests, we can use Cohen’s d as the measure of effect size. The fictional data provided means of 2.5 for those told they were drinking wine from a $10 bottle and 4.0 for those told they were drinking wine from a $90 bottle (Plassmann et al., 2008). Previously, we calculated a standard error for the difference between means, s_difference, of 0.616. Here are the calculations we performed:

Stage (a) (variance for each sample):

Stage (b) (combining variances):

Stage (c) (variance form of standard error for each sample):

Stage (d) (combining variance forms of standard error):

Stage (e) (converting the variance form of standard error to the standard deviation form of standard error):

Because the goal is to disregard the influence of sample size in order to calculate Cohen’s d, we want to use the standard deviation, rather than the standard error, in the denominator. So we can ignore the last three stages, all of which contribute to the calculation of standard error. That leaves stages (a) and (b). It makes more sense to use the one that includes information from both samples, so we focus our attention on stage (b). Here is where many students make a mistake. What we have calculated in stage (b) is pooled variance, not pooled standard deviation. We must take the square root of the pooled variance to get the pooled standard deviation, the appropriate value for the denominator of Cohen’s d.

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The test statistic that we calculated for this study was:

For Cohen’s d, we simply replace the denominator with standard deviation, s_pooled, instead of standard error, s_difference.

For this study, the effect size is reported as: d = 1.63. The two sample means are 1.63 standard deviations apart. According to the conventions that we learned in Chapter 8 and are shown again in Table 10-2, this is a large effect.

Solutions to these Check Your Learning questions can be found in Appendix D.