The z Distribution
The z statistic is normally distributed with a mean of zero and a standard deviation of one.
dard deviations of the average for the overall population. There can be 99 percent confidence that the sample is typical of the overall population if the sample average falls within roughly three sample standard deviations of the population average.
To illustrate, consider the case of a retailer that receives a large shipment of lightbulbs from a new supplier and wishes to learn if these new bulbs are of standard quality. Lightbulbs received from the retailer's current supplier have an average life of 2,000 hours, with a standard deviation of 200 hours. The retailer's null hypothesis is that the new bulbs are of equal quality, or H0: ^ = 2,000 hours. The alternate hypothesis is that the new bulbs are not of equal quality, or Ha: ^ ± 2,000. Obviously, all new bulbs cannot be tested. To test the null hypothesis, the retailer might decide to test the life of a random sample of 100 bulbs. The retailer would be inclined to reject the new bulbs if this sample had a dramatically shorter mean life than bulbs from its current supplier. To minimize the Type I error of incorrectly deciding to reject new bulbs of equal quality, the significance level of the hypothesis test might be set at a = 0.05 or a = 0.01. The retailer will purchase the new bulbs provided the chance of incorrectly rejecting equal quality bulbs is only 5 percent or 1 percent, respectively.
In the lightbulb example, the relevant test statistic z = (X - 2,000) 20; because ^ = 2,000 hours, ct = 200 hours, and n = 100 sample observations. So long as the computed value for this test statistic is within roughly ± 2, the retailer could safely infer with 95 percent confidence that the new bulbs are of the same quality as those obtained from current suppliers. The chance of incorrectly rejecting equal quality bulbs is 5 percent when the test statistic falls in the range between ± 2. Such a value for the test statistic requires a sample average bulb life within the range from 1,960 hours to 2,040. The 99 percent confidence interval requires the test statistic to fall within the range ± 3, and a sample average bulb life of 1,940 hours to 2,060 hours. By accepting bulbs with a sample average life that falls within this broader range, the chance of wrongly rejecting equal quality bulbs (Type I error) can be cut to 1 percent.
If the population standard deviation ct is unknown and the sample size is large, n > 30, the sample standard deviation s can be substituted for ct in the test statistic calculation:
s/wn where X is the sample mean, ^ is the known mean of the population, s is the sample standard deviation, and n is sample size. Again, a confidence interval for the true mean ^ is from X - z(s/Wn) to X + z(s/Wn), where z is from the normal table in Appendix C for the relevant confidence level. This test statistic formula, like that given in Equation 3.9, is based on the assumption that the sample is "small" relative to the size of the overall population. If sample size exceeds 5 percent of the overall population, then the denominator of each equation must be multiplied by what is known as the finite population correction factor, or MN - n)/(N - 1) where N is the size of the overall population and n is sample size.
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