## Sample Tests for a Population Mean

This section talks about using the central limit theorem to test a population mean when the sample size is large. It also addresses how to interpret the test results in the application background. Then, it discusses testing a population mean when the sample size is small, outlines a five-step testing procedure, and illustrates the procedure with an example. Study the example carefully and complete the relevant exercises and applications. Finally, it talks about large sample tests for a population proportion. The critical value and p-value approach are introduced based on a standardized test statistic.

### Large Sample Tests for a Population Mean

#### BASIC

1. Find the rejection region (for the standardized test statistic) for each hypothesis test.

1. $H_{0}: \mu=27$ vs. $H_{a}: \mu$.
2. $H_{0}: \mu=52$ vs. $H_{a}: \mu \neq 52 @ \alpha=0.05$.
3. $H_{0}: \mu=-105$ vs. $H_{a}: \mu>-105 @ \alpha=0.10$.
4. $H_{0}: \mu=78.8$ vs. $H_{a}: \mu \neq 78.8 @ \alpha=0.10$.

3. Find the rejection region (for the standardized test statistic) for each hypothesis test. Identify the test as left-tailed, right-tailed, or two-tailed.

1. $H_{0}: \mu=141$ vs. $H_{a}: \mu < 141 @ \alpha=0.20$.
2. $H_{0}: \mu=-54$ vs. $H_{a}: \mu < -54 @ \alpha=0.05$.
3. $H_{0}: \mu=98.6$ vs. $H_{a}: \mu \neq 98.6 @ \alpha=0.05$.
4. $H_{0}: \mu=3.8$ vs. $H_{a}: \mu > 3.8 @ \alpha=0.001$

5. Compute the value of the test statistic for the indicated test, based on the information given.

1. Testing $H_{0}: \mu=72.2$ vs. $H_{a}: \mu > 72.2, \sigma$ unknown, $n=55, \bar{x}=75.1, s=9.25$
2. Testing $H_{0}: \mu=58$ vs. $H_{a}: \mu > 58, \sigma=1.22, n=40, \bar{x}=58.5, s=1.29$
3. Testing $H_{0}: \mu=-19.5$ vs. $H_{a}: \mu < -19.5, \sigma$ unknown, $n=30, \bar{x}=-23.2, s=9.55$
4. Testing $H_{0}: \mu=805$ vs. $H_{a}: \mu \neq 805, \sigma=37.5, n=75, \bar{x}=818, s=36.2$

7. Perform the indicated test of hypotheses, based on the information given.

1. Test $H_{0}: \mu=212$ vs. $H_{a}: \mu < 212 @ \alpha=0.10, \sigma$ unknown, $n=36, \bar{x}=211.2, s=2.2$
2. Test $H_{0}: \mu=-18$ vs. $H_{a}: \mu > -18 @ \alpha=0.05, \sigma=3.3, n=44, \bar{x}=-17.2, s=3.1$
3. Test $H_{0}: \mu=24$ vs. $H_{a}: \mu \neq 24 @ \alpha=0.02, \sigma$ unknown, $n=50, \bar{x}=22.8, s=1.9$

#### APPLICATIONS

9. In the past the average length of an outgoing telephone call from a business office has been 143 seconds. A manager wishes to check whether that average has decreased after the introduction of policy changes. A sample of 100 telephone calls produced a mean of 133 seconds, with a standard deviation of 35 seconds. Perform the relevant test at the 1% level of significance.

11. The average household size in a certain region several years ago was 3.14 persons. A sociologist wishes to test, at the 5% level of significance, whether it is different now. Perform the test using the information collected by the sociologist: in a random sample of 75 households, the average size was 2.98 persons, with sample standard deviation 0.82 person.

13. An automobile manufacturer recommends oil change intervals of 3,000 miles. To compare actual intervals to the recommendation, the company randomly samples records of 50 oil changes at service facilities and obtains sample mean 3,752 miles with sample standard deviation 638 miles. Determine whether the data provide sufficient evidence, at the 5% level of significance, that the population mean interval between oil changes exceeds 3,000 miles.

15. A grocery store chain has as one standard of service that the mean time customers wait in line to begin checking out not exceed 2 minutes. To verify the performance of a store the company measures the waiting time in 30 instances, obtaining mean time 2.17 minutes with standard deviation 0.46 minute. Use these data to test the null hypothesis that the mean waiting time is 2 minutes versus the alternative that it exceeds 2 minutes, at the 10% level of significance.

17. Authors of a computer algebra system wish to compare the speed of a new computational algorithm to the currently implemented algorithm. They apply the new algorithm to 50 standard problems; it averages 8.16 seconds with standard deviation 0.17 second. The current algorithm averages 8.21 seconds on such problems. Test, at the 1% level of significance, the alternative hypothesis that the new algorithm has a lower average time than the current algorithm.

19. The mean household income in a region served by a chain of clothing stores is $48,750. In a sample of 40 customers taken at various stores the mean income of the customers was$51,505 with standard deviation $6,852. Test at the 10% level of significance the null hypothesis that the mean household income of customers of the chain is$48,750 against that alternative that it is different from $48,750. The sample mean is greater than$48,750, suggesting that the actual mean of people who patronize this store is greater than \$48,750. Perform this test, also at the 10% level of significance. (The computation of the test statistic done in part (a) still applies here).

#### LARGE DATA SET EXERCISES

Note: All of the data sets associated with these questions are missing, but the questions themselves are included here for reference.

21. Large Data Set 1 records the SAT scores of 1,000 students. Regarding it as a random sample of all high school students, use it to test the hypothesis that the population mean exceeds 1,510, at the 1% level of significance. (The null hypothesis is that $\mu=1510$).

23. Large Data Set 1 lists the SAT scores of 1,000 students.

1. Regard the data as arising from a census of all students at a high school, in which the SAT score of every student was measured. Compute the population mean $\mu$.
2. Regard the first 50 students in the data set as a random sample drawn from the population of part (a) and use it to test the hypothesis that the population mean exceeds 1,510, at the 10% level of significance. (The null hypothesis is that $\mu=1510$).
3. Is your conclusion in part (b) in agreement with the true state of nature (which by part (a) you know), or is your decision in error? If your decision is in error, is it a Type I error or a Type II error?