Example 89
For a random sample of ninety-six smokers, the mean amount of short-term absenteeism from work was 2.15 hours per month, and the sample standard deviation was 2.09 hours per month. For an independent random sample of 206 employees who had never smoked, the mean amount of absenteeism was 1.69 hours per month, and the sample standard deviation was 1.91 hours per month.21 Find a 99% confidence interval for the difference between the two population means.
For the smokers, we have x = 2.15 nx - 96 sx = 2.09 and for those who never smoked y = 1.69 ny = 206 = 1.91
Since the sample sizes are large, we can use the sample variances in place of the unknown population variances in the formula given in the box. Confidence intervals for the difference between the population means then take the form
— + — < fJLx - fly < (x - y) + zaf2 — + —
where, for a 99% interval
20 Thirty observations in each sample are generally adequate for this approximation.
21 Reported in M. R. Manning, J. S. Osland, and A. Osland, "Work-related consequences of smoking cessation," Academy of Management Journal, 32 (1989), 606-21.
The required interval is then r>is (2-09)2 (1.91)2
96 206
which is
Since 0 is inside the 99% confidence interval for the difference in population means, the evidence in the data against the conjecture that the mean absentee rates for the two groups is the same is not overwhelming.
EXAMPLE Independent random samples of professors and chief executive officers were g 2Q asked to evaluate the relevance to managerial practice of strategic management re search over the past decade on a scale from one (declined substantially) to five (improved substantially).22 The sample of 321 professors produced a mean rating of 3.01, and sample standard deviation 1.09. For the sample of 94 chief executive officers, the mean rating was 2.88 and the sample standard deviation was 1.01. Denoting by fix the population mean for professors and by fiY the population mean for chief executive officers, find a 95% confidence interval for (fix — fiY).
Again, since the sample sizes are large, we can use the sample variances in place of the population variances and obtain intervals from where
(x - y) - za/2 / — + — < ¡Xx - fly < (x - y) + zal2 / — + —
V Wv «v V nx nv nx = 321 x = 3.01 sx = 1.09 ny = 94 y = 2.88 s, = 1.01
and for a 95% confidence interval
The interval is then nm ^ oo\ ! o* /a-09)2 . (i-oo2 ^
/ (1.09)2 (1.01)2 < (3.01 - 2.88) + 1.96 /--- + ^ '
321 94
12 S. A. Zahra and J. A. Pearce, "Priorities of CEOs and strategic management professors for future academic research," Journal of Managerial Issues, 4 (1992), 171 -89.
This interval includes zero, indicating an absence of strong evidence that the population means are different. Figure 8.13 shows this confidence interval, together with 80%, 90%, and 99% confidence intervals for the difference in the population means.
We now have to consider the case where the sample sizes are not large, and a confidence interval is needed for the difference between the means of two normal populations based on independent random samples from the two populations. In fact, when the population variances are unknown, there is considerable difficulty in attacking this general problem. However, in one special case, where it can be assumed that the two population variances are equal,23 a fairly straightforward method is available.
Suppose again that we have independent random samples of n.v and nY observations from normal populations with means fxx and ¡iY and that the populations have a common (unknown) variance cr. Inference about the population means is, as before, based on the difference (X — Y) between the two sample means. This random variable has a normal distribution with mean (fxx — jjly) and variance
- nx ny
FIGURE 8.13 80%, 90%, 95%, and 99% confidence intervals for difference in population means based on the data of Example 8.10
I_80% Confidence interval_^
I_90% Confidence interval_
I_95% Confidence interval ^
I_99% Confidence interval_^
23 We will see in Chapter 9 how the data can be used to check this assumption.
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