## 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.

### Small Sample Tests for a Population Mean

#### EXERCISES

BASIC

1. Find the rejection region (for the standardized test statistic) for each hypothesis test based on the information given. The population is normally distributed.

a. $H_{0}: \mu=27$ vs. $H_{a}: \mu < 27 @ \alpha=0.05, n=12, \sigma=2.2$.

b. $H_{0}: \mu=52$ vs. $H_{a}: \mu \neq 52 @ \alpha=0.05, n=6, \sigma$ unknown.

c. $H_{0}: \mu=-105$ vs. $H_{a}: \mu > -105 @ \alpha=0.10, n=24, \sigma$ unknown.

d. $H_{0}: \mu=78.8$ vs. $H_{a}: \mu \neq 78.8 @ \alpha=0.10, n=8, \sigma=1.7$

3. Find the rejection region (for the standardized test statistic) for each hypothesis test based on the information given. The population is normally distributed. Identify the test as left-tailed, right-tailed, or two-tailed.

a. $H_{0}: \mu=141$ vs. $H_{a}: \mu < 141 @ \alpha=0.20, n=29, \sigma$ unknown.

b. $H_{0}: \mu=-54$ vs. $H_{a}: \mu < -54 @ \alpha=0.05, n=15, \sigma=1.9$

c. $H_{0}: \mu=98.6$ vs. $H_{a}: \mu \neq 98.6 @ \alpha=0.05, n=12, \sigma$ unknown.

d. $H_{0}: \mu=3.8$ vs. $H_{a}: \mu > 3.8 @ \alpha=0.001, n=27, \sigma$ unknown.

5. 5. A random sample of size 20 drawn from a normal population yielded the following results: $\bar{x}=49.2, s=$ 1.33.

a. Test $H_{0}: \mu=50$ vs. $H_{a}: \mu \neq 50 @ \alpha=0.01$.

b. Estimate the observed significance of the test in part (a) and state a decision based on the $p$-value approach to hypothesis testing.

7. A random sample of size 8 drawn from a normal population yielded the following results: $\bar{x}=289, s=$ $46.$

a. Test $H_{0}: \mu=250$ vs. $H_{a}: \mu > 250 @ \alpha=0.05$.

b. Estimate the observed significance of the test in part (a) and state a decision based on the $p$-value approach to hypothesis testing.

#### APPLICATIONS

9. Researchers wish to test the efficacy of a program intended to reduce the length of labor in childbirth. The accepted mean labor time in the birth of a first child is 15.3 hours. The mean length of the labors of 13 first-time mothers in a pilot program was 8.8 hours with standard deviation 3.1 hours. Assuming a normal distribution of times of labor, test at the 10% level of significance test whether the mean labor time for all women following this program is less than 15.3 hours.

11. Six coins of the same type are discovered at an archaeological site. If their weights on average are significantly different from 5.25 grams then it can be assumed that their provenance is not the site itself. The coins are weighed and have mean 4.73 g with sample standard deviation 0.18 g. Perform the relevant test at the 0.1% (1/10th of 1%) level of significance, assuming a normal distribution of weights of all such coins.

13. The recommended daily allowance of iron for females aged 19–50 is 18 mg/day. A careful measurement of the daily iron intake of 15 women yielded a mean daily intake of 16.2 mg with sample standard deviation 4.7 mg.

1. Assuming that daily iron intake in women is normally distributed, perform the test that the actual mean daily intake for all women is different from 18 mg/day, at the 10% level of significance.
2. The sample mean is less than 18, suggesting that the actual population mean is less than 18 mg/day. Perform this test, also at the 10% level of significance. (The computation of the test statistic done in part (a) still applies here).

15. The average number of days to complete recovery from a particular type of knee operation is 123.7 days. From his experience a physician suspects that use of a topical pain medication might be lengthening the recovery time. He randomly selects the records of seven knee surgery patients who used the topical medication. The times to total recovery were:

$\begin{array}{lllllll}128 & 135 & 121 & 142 & 126 & 151 & 123\end{array}$

1. Assuming a normal distribution of recovery times, perform the relevant test of hypotheses at the 10% level of significance.
2. Would the decision be the same at the 5% level of significance? Answer either by constructing a new rejection region (critical value approach) or by estimating the p-value of the test in part (a) and comparing it to $\alpha$.

17. Pasteurized milk may not have a standardized plate count (SPC) above 20,000 colony-forming bacteria per milliliter (cfu/ml). The mean SPC for five samples was 21,500 cfu/ml with sample standard deviation 750 cfu/ml. Test the null hypothesis that the mean SPC for this milk is 20,000 versus the alternative that it is greater than 20,000, at the 10% level of significance. Assume that the SPC follows a normal distribution.

#### ADDITIONAL EXERCISES

19. A calculator has a built-in algorithm for generating a random number according to the standard normal distribution. Twenty-five numbers thus generated have mean 0.15 and sample standard deviation 0.94. Test the null hypothesis that the mean of all numbers so generated is 0 versus the alternative that it is different from 0, at the 20% level of significance. Assume that the numbers do follow a normal distribution.

21. A manufacturing company receives a shipment of 1,000 bolts of nominal shear strength 4,350 lb. A quality control inspector selects five bolts at random and measures the shear strength of each. The data are:

$\begin{array}{lllll}4,320 & 4,290 & 4,360 & 4,350 & 4,320\end{array}$

1. Assuming a normal distribution of shear strengths, test the null hypothesis that the mean shear strength of all bolts in the shipment is 4,350 lb versus the alternative that it is less than 4,350 lb, at the 10% level of significance.
2. Estimate the $p$-value (observed significance) of the test of part (a).
3. Compare the $p$-value found in part (b) to $\alpha=0.10$ and make a decision based on the $p$-value approach. Explain fully.