After working at Guinness, Stella Cunliffe was hired by the British government’s criminology department. She noticed that adult male prisoners who had short prison sentences returned to prison at a very high rate—
11.2: As we can with the z test, the single-
Two ways that researchers can evaluate the findings of a hypothesis test are by calculating a confidence interval and an effect size.
Confidence intervals for the independent-
We use the formula for the independent-
We replace the population mean difference, (μX − μY), with the sample mean difference, (MX − MY)sample, because this is what the confidence interval is centered around. We also indicate that the first mean difference in the numerator refers to the bounds of the confidence intervals, the upper bound in this case:
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With algebra, we isolate the upper bound of the confidence interval to create the following formula:
(MX − MY)upper = t(sdifference) + (MX − MY)sample
We create the formula for the lower bound of the confidence interval in exactly the same way, using the negative version of the t statistic:
(MX − MY)lower = −t(sdifference) + (MX − MY)sample
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, sdifference 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:
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STEP 1: Draw a normal curve with the sample difference between means in the center (as shown in Figure 11-4).
Figure 11-
STEP 2: Indicate the bounds of the confidence interval on either end, and write the percentages under each segment of the curve.
(See Figure 11-4.)
STEP 3: Look up the t statistics for the lower and upper ends of the confidence interval in the t table.
Use a two-
Figure 11-
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:
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For the upper end, the formula is:
The confidence interval is [−2.96, −0.04], as shown in Figure 11-6.
Figure 11-
STEP 5: Check your answer.
Each end of the confidence interval should be exactly the same distance from the sample mean.
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-
As with all hypothesis tests, it is recommended that the results be supplemented with an effect size. For an independent-
Stage (a) (variance for each sample):
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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.
The test statistic that we calculated for this study was:
For Cohen’s d, we simply replace the denominator with standard deviation, spooled, instead of standard error, sdifference.
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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 shown again in Table 11-2, we learned in Chapter 8 (0.2 is a small effect; 0.5 is a medium effect; 0.8 is a large effect), this is a large effect.
Effect Size | Convention | Overlap |
---|---|---|
Small | 0.2 | 85% |
Medium | 0.5 | 67% |
Large | 0.8 | 53% |
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Data Transformations
When we conduct hypothesis tests, such as the independent-
Figure 11-
In a situation such as the childhood mortality data, we can transform skewed data into a more normal distribution. In fact, even when the sample is small and the population distribution is skewed, there are still a variety of ways for us to transform skewed data so that they are no longer skewed. One way is to transform scale data to ordinal data. For example, consider a sample of incomes of $24,000, $27,000, $35,000, $46,000, and $550,000. Here, the income of $550,000 is far higher than the next-
Scale: | $24,000 $27,000 $35,000 $46,000 $550,000 |
Ordinal: | 5 4 3 2 1 |
Now $550,000 is ranked first and $46,000 is ranked second. However, the large difference between the two scores disappeared when we transformed the data from a scale measure to an ordinal measure. The problem with transforming scale data to ordinal data is that a z test or a t test only works with scale data. (We will learn tests we can use with ordinal data in Chapters 17 and 18.) Fortunately, other transformations also can diminish skew while still allowing us to use the more powerful parametric tests we’ve learned so far.
A square root transformation reduces skew by compressing both the negative and positive sides of a skewed distribution.
A square root transformation reduces skew by compressing both the negative and positive sides of a skewed distribution. Let’s take the same five incomes on a scale measure, but instead of converting them to ranks, we’ll take the square root of each of them. As we will see, the effect is more dramatic on the higher values.
Scale: | $24,000 $27,000 $35,000 $46,000 $550,000 |
Square root: | $154.92 $164.32 $187.08 $214.48 $741.62 |
Now the severe outlier of $550,000, much higher than the next-
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Here we have discussed two ways to deal with skewed data:
Remember that we need to apply any kind of data transformation to every observation in the data set. Furthermore, data transformation should only be used if it isn’t possible to operationalize the variable of interest in a better way. You want to keep your statistical analyses as simple as possible and no simpler.
Reviewing the Concepts
Clarifying the Concepts
Calculating the Statistics
Applying the Concepts
Solutions to these Check Your Learning questions can be found in Appendix D.