8-4. Exercises

1. BASIC

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

1. $H_0:μ=27 \space vs. H_a:μ<27, @ α=0.05, n = 12, σ = 2.2.$

2. $H_0:μ=52 \space vs. H_a:μ\ne 52, @ α=0.05, n = 6, σ \space unknown.$

3. $H_0:μ=-105 \space vs. H_a:μ>-105, @ α=0.10, n = 24, σ \space unknown.$

4. $H_0:μ=78 \space vs. H_a:μ \ne 78, @ α=0.10, n = 8, σ = 1.7.$

Ex 2. Find the rejection region (for the standardized test statistic) for each hypothesis test based on the information given. The population is normally distributed.

1. $H_0:μ=17 \space vs. H_a:μ<17, @ α=0.01, n = 26, σ = 0.94.$

2. $H_0:μ=880 \space vs. H_a:μ\ne 880, @ α=0.01, n = 4, σ \space unknown.$

3. $H_0:μ=-12 \space vs. H_a:μ>-12, @ α=0.05, n = 18, σ = 1.1.$

4. $H_0:μ=21 \space vs. H_a:μ\ne 21, @ α=0.05, n = 23, σ \space unknown.$

Ex 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.

1. $H_0:μ=141 \space vs. H_a:μ<141, @ α=0.20, n = 29, σ \space unknown.$

2. $H_0:μ=-54 \space vs. H_a:μ<-54, @ α=0.05, n = 15, σ = 1.9.$

3. $H_0:μ=98.6 \space vs. H_a:μ\ne 98.6, @ α=0.05, n = 12, σ \space unknown.$

4. $H_0:μ=3.8 \space vs. H_a:μ > 3.8, @ α=0.001, n = 27, σ \space unknown.$

Ex 4. 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.

1. $H_0:μ=-62 \space vs. H_a:μ\ne -62, @ α=0.005, n = 8, σ \space unknown.$

2. $H_0:μ=73 \space vs. H_a:μ >73, @ α=0.001, n = 22, σ \space unknown.$

3. $H_0:μ=1124 \space vs. H_a:μ<1124, @ α=0.001, n = 21, σ \space unknown.$

4. $H_0:μ=0.12 \space vs. H_a:μ\ne 0.12, @ α=0.001, n = 14, σ = 0.026$

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

1. Test $H_0:μ=50 \space vs. H_a:μ\ne 50, @ α=0.01.$

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

Ex 6. A random sample of size 16 drawn from a normal population yielded the following results: $\bar{x}=-0.96, s = 1.07.$

1. Test $H_0:μ=0 \space vs. H_a:μ< 5, @ α=0.001.$

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

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

1. Test $H_0:μ=250 \space vs. H_a:μ> 250, @ α=0.05.$

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

Ex 8. A random sample of size 12 drawn from a normal population yielded the following results: $\bar{x}=86.2, s = 0.63.$

1. Test $H_0:μ=85.5 \space vs. H_a:μ\ne 85.5, @ α=0.01.$

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

2. APPLICATIONS

Ex 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.

Ex 10. A dairy farm uses the somatic cell count (SCC) report on the milk it provides to a processor as one way to monitor the health of its herd. The mean SCC from five samples of raw milk was 250,000 cells per milliliter with standard deviation 37,500 cell/ml. Test whether these data provide sufficient evidence, at the 10% level of significance, to conclude that the mean SCC of all milk produced at the dairy exceeds that in the previous report, 210,250 cell/ml. Assume a normal distribution of SCC.

Ex 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.

Ex 12. An economist wishes to determine whether people are driving less than in the past. In one region of the country the number of miles driven per household per year in the past was 18.59 thousand miles. A sample of 15 households produced a sample mean of 16.23 thousand miles for the last year, with sample standard deviation 4.06 thousand miles. Assuming a normal distribution of household driving distances per year, perform the relevant test at the 5% level of significance.

Ex 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.)

Ex 14. The target temperature for a hot beverage the moment it is dispensed from a vending machine is 170°F. A sample of ten randomly selected servings from a new machine undergoing a pre-shipment inspection gave mean temperature 173°F with sample standard deviation 6.3°F.

1. Assuming that temperature is normally distributed, perform the test that the mean temperature of dispensed beverages is different from 170°F, at the 10% level of significance.

2. The sample mean is greater than 170, suggesting that the actual population mean is greater than 170°F. Perform this test, also at the 10% level of significance. (The computation of the test statistic done in part (a) still applies here.)

Ex 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: 128 135 121 142 126 151 123

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 $α.$

Ex 16. A 24-hour advance prediction of a day’s high temperature is “unbiased” if the long-term average of the error in prediction (true high temperature minus predicted high temperature) is zero. The errors in predictions made by one meteorological station for 20 randomly selected days were: 2 0 −3 1 −2 1 0 −1 1 −1 −4 1 1 −4 0 −4 −3 −4 2 2

Ex 17. Assuming a normal distribution of errors, test the null hypothesis that the predictions are unbiased (the mean of the population of all errors is 0) versus the alternative that it is biased (the population mean is not 0), at the 1% level of significance.

Ex 18. Would the decision be the same at the 5% level of significance? The 10% level of significance? Answer either by constructing new rejection regions (critical value approach) or by estimating the p-value of the test in part (a) and comparing it to α.α.

Ex 19. 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.

Ex 20. One water quality standard for water that is discharged into a particular type of stream or pond is that the average daily water temperature be at most 18°C. Six samples taken throughout the day gave the data: 16.8 21.5 19.1 12.8 18.0 20.7 The sample mean $\bar{x} = 18.15$ exceeds 18, but perhaps this is only sampling error. Determine whether the data provide sufficient evidence, at the 10% level of significance, to conclude that the mean temperature for the entire day exceeds 18°C.

3. ADDITIONAL EXERCISES

Ex 21. 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.

Ex 22. At every setting a high-speed packing machine delivers a product in amounts that vary from container to container with a normal distribution of standard deviation 0.12 ounce. To compare the amount delivered at the current setting to the desired amount 64.1 ounce, a quality inspector randomly selects five containers and measures the contents of each, obtaining sample mean 63.9 ounces and sample standard deviation 0.10 ounce. Test whether the data provide sufficient evidence, at the 5% level of significance, to conclude that the mean of all containers at the current setting is less than 64.1 ounces.

Ex 23. 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: 4,320 4,290 4,360 4,350 4,320

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 $α=0.10$ and make a decision based on the p-value approach. Explain fully.

Ex 24. A literary historian examines a newly discovered document possibly written by Oberon Theseus. The mean average sentence length of the surviving undisputed works of Oberon Theseus is 48.72 words. The historian counts words in sentences between five successive 101 periods in the document in question to obtain a mean average sentence length of 39.46 words with standard deviation 7.45 words. (Thus the sample size is five.)

1. Determine if these data provide sufficient evidence, at the 1% level of significance, to conclude that the mean average sentence length in the document is less than 48.72.

2. Estimate the p-value of the test.

3. Based on the answers to parts (a) and (b), state whether or not it is likely that the document was written by Oberon Theseus.