8-5. Exercises

1. BASIC

On all exercises for this section you may assume that the sample is sufficiently large for the relevant test to be validly performed.

Ex 1. Compute the value of the test statistic for each test using the information given.

  1. Testing H0:p=0.50 vs. Ha:p>0.50,n=360,p^=0.56.H_0:p=0.50 \space vs. \space H_a:p>0.50, n = 360, \hat{p}=0.56.

  2. Testing H0:p=0.50 vs. Ha:p≠0.50,n=360,p^=0.56.H_0:p=0.50 \space vs. \space H_a:p\ne0.50, n = 360, \hat{p}=0.56.

  3. Testing H0:p=0.37 vs. Ha:p<0.37,n=1200,p^=0.35.H_0:p=0.37 \space vs. \space H_a:p<0.37, n = 1200, \hat{p}=0.35.

Ex 2. Compute the value of the test statistic for each test using the information given.

  1. Testing H0:p=0.72 vs. Ha:p<0.72,n=2100,p^=0.71.H_0:p=0.72 \space vs. \space H_a:p<0.72, n = 2100, \hat{p}=0.71.

  2. Testing H0:p=0.83 vs. Ha:p≠0.83,n=500,p^=0.86.H_0:p=0.83 \space vs. \space H_a:p\ne0.83, n = 500, \hat{p}=0.86.

  3. Testing H0:p=0.22 vs. Ha:p<0.22,n=750,p^=0.18.H_0:p=0.22 \space vs. \space H_a:p<0.22, n = 750, \hat{p}=0.18.

Ex 3. For each part of Exercise 1 construct the rejection region for the test for α=0.05α=0.05 and make the decision based on your answer to that part of the exercise.

Ex 4. For each part of Exercise 2 construct the rejection region for the test for α=0.05α=0.05 and make the decision based on your answer to that part of the exercise.

Ex 5. For each part of Exercise 1 compute the observed significance (p-value) of the test and compare it to α=0.05α=0.05 in order to make the decision by the p-value approach to hypothesis testing.

Ex 6. For each part of Exercise 2 compute the observed significance (p-value) of the test and compare it to α=0.05α=0.05 in order to make the decision by the p-value approach to hypothesis testing.

Ex 7. Perform the indicated test of hypotheses using the critical value approach.

  1. Testing H0:p=0.55 vs. Ha:p>0.55,α=0.05,n=300,p^=0.60.H_0:p=0.55 \space vs. \space H_a:p>0.55, α=0.05, n = 300, \hat{p}=0.60.

  2. Testing H0:p=0.47 vs. Ha:p≠0.47,α=0.01,n=9750,p^=0.46.H_0:p=0.47 \space vs. \space H_a:p\ne 0.47, α=0.01, n = 9750, \hat{p}=0.46.

Ex 8. Perform the indicated test of hypotheses using the critical value approach.

  1. Testing H0:p=0.15 vs. Ha:p≠0.15,α=0.001,n=1600,p^=0.18.H_0:p=0.15 \space vs. \space H_a:p\ne 0.15, α=0.001, n =1600, \hat{p}=0.18.

  2. Testing H0:p=0.90 vs. Ha:p>0.90,α=0.01,n=1100,p^=0.91.H_0:p=0.90 \space vs. \space H_a:p> 0.90, α=0.01, n = 1100, \hat{p}=0.91.

Ex 9. Perform the indicated test of hypotheses using the p-value approach.

  1. Testing H0:p=0.37 vs. Ha:p≠0.37,α=0.005,n=1300,p^=0.40.H_0:p=0.37 \space vs. \space H_a:p\ne 0.37, α=0.005, n = 1300, \hat{p}=0.40.

  2. Testing H0:p=0.94 vs. Ha:p>0.94,α=0.05,n=1200,p^=0.96.H_0:p=0.94 \space vs. \space H_a:p>0.94, α=0.05, n = 1200, \hat{p}=0.96.

Ex 10. Perform the indicated test of hypotheses using the p-value approach.

  1. Testing H0:p=0.25 vs. Ha:p<0.25,α=0.10,n=850,p^=0.23.H_0:p=0.25 \space vs. \space H_a:p < 0.25, α=0.10, n = 850, \hat{p}=0.23.

  2. Testing H0:p=0.33 vs. Ha:p≠0.33,α=0.05,n=1100,p^=0.30.H_0:p=0.33 \space vs. \space H_a:p\ne 0.33, α=0.05, n = 1100, \hat{p}=0.30.

2. APPLICATIONS

Ex 11. Five years ago 3.9% of children in a certain region lived with someone other than a parent. A sociologist wishes to test whether the current proportion is different. Perform the relevant test at the 5% level of significance using the following data: in a random sample of 2,759 children, 119 lived with someone other than a parent.

Ex 12. The government of a particular country reports its literacy rate as 52%. A nongovernmental organization believes it to be less. The organization takes a random sample of 600 inhabitants and obtains a literacy rate of 42%. Perform the relevant test at the 0.5% (one-half of 1%) level of significance.

Ex 13. Two years ago 72% of household in a certain county regularly participated in recycling household waste. The county government wishes to investigate whether that proportion has increased after an intensive campaign promoting recycling. In a survey of 900 households, 674 regularly participate in recycling. Perform the relevant test at the 10% level of significance.

Ex 14. Prior to a special advertising campaign, 23% of all adults recognized a particular company’s logo. At the close of the campaign the marketing department commissioned a survey in which 311 of 1,200 randomly selected adults recognized the logo. Determine, at the 1% level of significance, whether the data provide sufficient evidence to conclude that more than 23% of all adults now recognize the company’s logo.

Ex 15. A report five years ago stated that 35.5% of all state-owned bridges in a particular state were “deficient.” An advocacy group took a random sample of 100 state-owned bridges in the state and found 33 to be currently rated as being “deficient.” Test whether the current proportion of bridges in such condition is 35.5% versus the alternative that it is different from 35.5%, at the 10% level of significance.

Ex 16. In the previous year the proportion of deposits in checking accounts at a certain bank that were made electronically was 45%. The bank wishes to determine if the proportion is higher this year. It examined 20,000 deposit records and found that 9,217 were electronic. Determine, at the 1% level of significance, whether the data provide sufficient evidence to conclude that more than 45% of all deposits to checking accounts are now being made electronically.

Ex 17. According to the Federal Poverty Measure 12% of the U.S. population lives in poverty. The governor of a certain state believes that the proportion there is lower. In a sample of size 1,550, 163 were impoverished according to the federal measure.

  1. Test whether the true proportion of the state’s population that is impoverished is less than 12%, at the 5% level of significance.

  2. Compute the observed significance of the test.

Ex 18. An insurance company states that it settles 85% of all life insurance claims within 30 days. A consumer group asks the state insurance commission to investigate. In a sample of 250 life insurance claims, 203 were settled within 30 days.

  1. Test whether the true proportion of all life insurance claims made to this company that are settled within 30 days is less than 85%, at the 5% level of significance.

  2. Compute the observed significance of the test.

Ex 19. A special interest group asserts that 90% of all smokers began smoking before age 18. In a sample of 850 smokers, 687 began smoking before age 18.

  1. Test whether the true proportion of all smokers who began smoking before age 18 is less than 90%, at the 1% level of significance.

  2. Compute the observed significance of the test.

Ex 20. In the past, 68% of a garage’s business was with former patrons. The owner of the garage samples 200 repair invoices and finds that for only 114 of them the patron was a repeat customer.

  1. Test whether the true proportion of all current business that is with repeat customers is less than 68%, at the 1% level of significance.

  2. Compute the observed significance of the test.

3. ADDITIONAL EXERCISES

Ex 21. A rule of thumb is that for working individuals one-quarter of household income should be spent on housing. A financial advisor believes that the average proportion of income spent on housing is more than 0.25. In a sample of 30 households, the mean proportion of household income spent on housing was 0.285 with a standard deviation of 0.063. Perform the relevant test of hypotheses at the 1% level of significance. Hint: This exercise could have been presented in an earlier section.

Ex 22. Ice cream is legally required to contain at least 10% milk fat by weight. The manufacturer of an economy ice cream wishes to be close to the legal limit, hence produces its ice cream with a target proportion of 0.106 milk fat. A sample of five containers yielded a mean proportion of 0.094 milk fat with standard deviation 0.002. Test the null hypothesis that the mean proportion of milk fat in all containers is 0.106 against the alternative that it is less than 0.106, at the 10% level of significance. Assume that the proportion of milk fat in containers is normally distributed. Hint: This exercise could have been presented in an earlier section.

4. LARGE DATA SET EXERCISES

Ex 23. Large Data Sets 4 and 4A list the results of 500 tosses of a die. Let p denote the proportion of all tosses of this die that would result in a five. Use the sample data to test the hypothesis that p is different from 1/6, at the 20% level of significance.

Ex 24. Large Data Set 6 records results of a random survey of 200 voters in each of two regions, in which they were asked to express whether they prefer Candidate A for a U.S. Senate seat or prefer some other candidate. Use the full data set (400 observations) to test the hypothesis that the proportion p of all voters who prefer Candidate A exceeds 0.35. Test at the 10% level of significance.

Ex 25. Lines 2 through 536 in Large Data Set 11 is a sample of 535 real estate sales in a certain region in 2008. Those that were foreclosure sales are identified with a 1 in the second column. Use these data to test, at the 10% level of significance, the hypothesis that the proportion p of all real estate sales in this region in 2008 that were foreclosure sales was less than 25%. (The null hypothesis is H0:p=0.25.H_0:p=0.25. )

Ex 26. Lines 537 through 1106 in Large Data Set 11 is a sample of 570 real estate sales in a certain region in 2010. Those that were foreclosure sales are identified with a 1 in the second column. Use these data to test, at the 5% level of significance, the hypothesis that the proportion p of all real estate sales in this region in 2010 that were foreclosure sales was greater than 23%. (The null hypothesis is H0:p=0.23.H_0:p=0.23. )

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