# What Assumption is Necessary to Ensure the Validity of This Confidence Interval?

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What conditions are required for the test results to be valid?

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1.1. A random sample of n measurements was selected from a population with unknown mean µ, and known standard deviation σ. Calculate a 95% confidence interval for µ. for each of the following situations:

a)n = 75,× = 28, σ2 = 12

b)n = 200,× = 102, σ2 = 22

c)n = 100,× = 15, σ= .3

d)n = 100,× = 4.05, o = .83

e)Is the assumption that the underlying population of measurements is normally distributed necessary to ensure the validity of the confidence intervals in parts a–d? Explain.

1.2 Aluminum cans contaminated by fire. A gigantic warren house located in Tampa, Florida. stores approximately 60 million empty aluminium beer and soda cans. Recent a fire occurred at the warehouse. The smoke from the fire contaminated many of the cans with blackspot rendering them unusable. A University of South Florida statistician was hired by the insurance company to estimate p, the true proportion of cans in the warehouse that were con­taminated by the fire. How many aluminium cans should be randomly sampled to estimate p to within .02 with 90% confidence?

1.3.Is caffeine addictive? Does the caffeine in coffee, tea, and cola induce an addiction similar to that induced by alcohol, tobacco, heroin, and cocaine? In an attempt to answer this question, researchers at Johns Hopkins University examined 27 caffeine drinkers and found 25 who displayed some type of withdrawal symptoms when abstaining from caffeine. [Note: The 27 caffeine drinkers volunteered for the study.] Furthermore, of 11 caffeine drinkers who were diagnosed as caffeine dependent, 8 displayed dramatic withdrawal symptoms (including impairment in normal functioning) when they consumed a caffeine-free diet in a controlled setting. The National Coffee Association claimed, however, that the study group was too small to draw conclusions. Is the sample large enough to estimate the true proportion of caffeine drinkers who are caffeine dependent to within .05 of the true value with 99% confi­dence? Explain.

1.4 Petroleum waste contamination. Accidental spillage and misguided disposal of petroleum wastes have resulted in extensive contamination of soils across the country. A common hazardous compound found in the contaminated soil is benzo(a)pyrene [B(a)p). An experiment was conducted to determine the effectiveness of a method designed to remove B(a)p from soil (Journal of Hazardous Materials, June 1995). Three soil specimens contaminated with a known amount of B(a)p were treated with a toxin that inhibits microbial growth. After 95 days of incubation, the percentage of B(a)p removed from each soil specimen was measured. The experiment produced the following summary statistics: X = 49.3 and s = 1.5.

a. Use a 99% confidence interval to estimate the mean percentage of B(a)p removed from a soil specimen in which the toxin was used.

b. Interpret the interval in terms of this application.

c. What assumption is necessary to ensure the validity of this confidence interval?

d. Comment on whether the true mean percent removed could be as high as 50%.

e. Find and interpret a 90% confidence interval for the true variance in the percentages of B(a)p removed.

1.5.IQ comparison of older vs. younger workers. The Age Discrimination in Employment Act (ADEA) made it illegal to discriminate against workers 40 years of age and older. Opponents of the law argue that there are sound economic reasons why employers would not want to hire and train workers who are very close to retirement. They also argue that people’s abilities tend to deteriorate with age. Do 25-year-olds score significantly higher than 60-year-olds on the Wechsler Adult Intelligence Scale, the most popular IQ test? The data in the next table are raw test scores (i.e., not the familiar normalised IQ scores) for a sample of thirty-six 25-year-olds and thirty-six 60-year-olds.

a)Estimate the mean raw test score for all 25-year-olds using a 99% confidence interval. Give a practical interpretation of the confidence interval.

b)What assumption(s) must hold for the method of estimation used in part a to be appropriate?

c)Find a 95% confidence interval for the mean raw Score of all 60-year-olds and interpret your result.

2.1 Student loan default rate. The national student loan default 1E1 rate has fluctuated over the last several years. A few years ago, the Department of Education reported the default rate (i.e., the proportion of college students who default on their loans) at .07 Set up the null and alternative hypotheses if you want to determine if the student loan default rate this year is less than .07.

2.2 Calories in school lunches. A University of Florida economist conducted a study of Virginia elementary school lunch menus. During the state-mandated testing period. school lunches averaged 863 calories (National Bureau of Economic Research, Nov. 2002). The economist claims that after the testing period ends, the average caloric content of Virginia school lunches drops significantly. Set up the null and alternative hypotheses to test the economist’s claim.

2.3 Revenue for a full-service funeral. According to the National Funeral Directors Association (NFDA), the nation’s 22,000 funeral homes collected an average of \$6,500 per full-service funeral in 2009 (www.nfda.org). A random sample of 36 funeral homes reported revenue data for the current year. Among other measures, each reported its average fee for a full-service funeral. These data (in thousands of dol­lars) are shown in the following table.

a)What are the appropriate null and alternative hypoth­eses to test whether the average full-service fee of U. S. funeral homes this year exceeds \$6,500?

b)Conduct the test at a = .05. Do the sample data pro­vide sufficient evidence to conclude that the average fee this year is higher than in 2009?

c)In conducting the test, was it necessary to assume that the population of average full-service fees was normally distributed? Justify your answer.

2.4 A new dental bonding agent. When bonding teeth, or­thodontists must maintain a dry field. A new bond­ing adhesive (called Star bond) has been developed to eliminate the necessity of a dry field. However, there is concern that the new bonding adhesive is not as strong as the current standard, a composite adhesive (Trends in Biomaterials & Artificial Organs, Jan. 2003). Tests on a sample of 10 extracted teeth bonded with the new adhesive resulted in a mean breaking strength (after 24 hours) of x = 5.07 Mpa and a standard deviation of s = .46 Mpa. Orthodontists want to know if the true mean breaking strength of the new bonding adhesive is less than 5.70 Mpa, the mean breaking strength of the composite adhesive.

a)Set up the null and alternative hypotheses for the test.

b)Find the rejection region for the test using α = .01.

c)Compute the test statistic.

d)Give the appropriate conclusion for the test.

e)What conditions are required for the test results to be valid?

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