Waiting Times, Sample Size and Statistical Significance
A public sector bank in Dispur advertises that the average waiting time at its main branch is 8 minutes. A consumer group records waiting times for 36 randomly selected customers and obtains a mean of 9.2 minutes with a standard deviation of 3.0 minutes. The group wants to test at the 5% level whether the true mean differs from 8 minutes, and to report a 95% confidence interval. A bank official says that a difference of 1.2 minutes is too small to matter even if it is statistically significant. A second group uses only 9 customers, gets the same mean and standard deviation, and finds that it cannot reject the claim. The consumer group must explain why the sample size changes the conclusion and what a Type I error means in this context.
1. Carry out the test at the 5% level for n = 36 and compute the 95% confidence interval. State the conclusion.
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Use a one-sample z-test because n = 36 is reasonably large and the sample SD estimates the population SD:
[ H_0:\mu=8,\qquad H_1:\mu\ne8 ] [ z=\frac{\bar{x}-\mu_0}{s/\sqrt{n}} =\frac{9.2-8}{3/\sqrt{36}} =\frac{1.2}{0.5}=2.4 ]
At the 5% level the critical values are ±1.96. Since 2.4 > 1.96, reject H0. The 95% confidence interval is
[ 9.2\pm1.96(0.5)=(8.22,,10.18) ]
Because 8 is not in the interval, the test and interval agree: there is statistically significant evidence that the mean waiting time differs from 8 minutes. The sample suggests the true mean is plausibly between about 8.2 and 10.2 minutes.
2. Explain why the second group with n = 9 cannot reject the claim even though the sample mean and SD are the same.
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With n = 9 the standard error becomes (3/\sqrt{9}=1) minute instead of 0.5 minute, and the appropriate test uses t with 8 degrees of freedom:
[ t=\frac{9.2-8}{3/\sqrt{9}}=\frac{1.2}{1}=1.2 ]
The 5% two-tailed critical value for t with 8 df is about 2.306. Since 1.2 < 2.306, the group fails to reject H0. The smaller sample gives less information, so the same 1.2-minute difference is only 1.2 standard errors rather than 2.4. The conclusion is not that the true mean is 8; it is that the sample is too small to rule out 8 at the 5% level. A confidence interval would be much wider and would show the uncertainty.
3. Distinguish statistical significance from practical importance here, and explain a Type I error in this setting.
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Statistical significance means the observed difference is unlikely under H0; practical importance means the difference is large enough to affect customers or the bank. Here the point estimate is 1.2 minutes, but the 95% interval runs from 8.22 to 10.18 minutes. The lower end is only 0.22 minutes above 8, so the true average could be barely different from the advertised figure. The bank official's claim that 1.2 minutes is “too small to matter” is a judgement about costs and customer tolerance, not a statistical conclusion.
A Type I error would be rejecting the claim that the mean is 8 minutes when it actually is 8. The test is set at 5%, so if H0 is true, there is about a 5% chance of such a false alarm. The consumer group should report the estimate, the interval, and the assumptions - random sampling and approximate normality or a large enough n.