# Statistics

Chapter 9, Section 2

Problem 12

Interpreting Displays - Conduct the hypothesis test by using the results from the given displays.

Bednets to Reduce Malaria - In a randomized controlled trial in Kenya, insecticide-treated bednets were tested as a way to reduce malaira. Among 343 infants using bednets, 15 developed malaria. Among 294 infants not using bednets, 27 developed malaria (based on data from Sustainability of Reductions in Malaria Transmission and Infant Mortality in Western Kenya with Use of Insecticide-Treated Bednets," by Lindblade, et al., Journal of the American Medical Association, Vol. 291, No. 21). Use a 0.01 significance level to test the claim that the incidence of malaria is lower or infants using bednets. Do the bednets appear to be effective?

MINITAB

Difference = p (1) - p (2)

Estimate for difference: -0.00125315

99% Upper bound for difference: -0.00125315

Test for difference = 0 (vs < 0): z = -2.44 P-value == 0.007

Problem 38

Equivalence of Hypothesis Test and Confidence Interval - Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 10 having a common attribute. The second sample consists of 2000 people with 1404 of them having the same common attribute. Compare the results from a hypothesis test of p1 = p2 (with a 0.05 significance level) and a 95% confidence interval estimate of p1 - p2.

Chapter 9, Section 3

Problem 18

Assume that the two samples are independent simple random samples selected from normally distrubted populations. Do not assume that th epopulation standard deviations are equal, unless your instructor stipulates otherwise.

Hypothesis Test for Braking Distances of Cars - Refer to the sample data given in Exercise 17 and use a 0.05 significance level to test the claim that the mean braking distance of four-cylinder cars is greater than the mean braking distance of six-cylinder cars.

Exercise 17 - Sample Data

A simple random sample of 13 four-cylinder cars is obtained, and the braking distances are measured. The mean braking distance is 137.5 ft and the standarad deviation is 5.8 ft. A simple random sample of 12 six-cylinder cars is obtained and the braking distances have a mean of 136.3 ft with a standard deviation of 9.7 ft (based on Data Set 16 in Appendix b ).

Problem 30

Radiation in Baby Teeth - Listed below are amounts of strontium-90 (in millibecquerels or mBq per gram of calcium) in a simple random sample of baby teeth obtained from Pennsylvantia residents and New York residents born after 1979 (based on data from "Au Unexpected Rise in Strontium-90 in U.S. Deciduous Teeth in the 1990s," by Mangano, et al., Science of the Total Environment).

a. Use a 0.05 significance level to test the claim that the mean amount of strontium-90 from Pennsylvania residents is greater than the mean amount from New York residents.

b. Construct a 90% confidence interval of the difference between the mean amount of strontium-90 from Pennsylvania residents and the mean amount from New York residents.

Pennsylvania: 155 142 149 130 151 163 151 142 156 133 138 161

New York: 133 140 142 131 134 129 128 140 140 140 137 143

Chapter 9, Section 4

Problem 12

Assume that the paired sample data are simple random samples and that the differences have a distribution that is approximately normal.

Are Flights Cheaper When Scheduled Earlier? - Listed below are the costs (in dollars) of flights from New York (JFK) to San Francisco for US Air, Continental, Delta, United, American, Alaska, and Northwest. Use a 0.01 significance level to test the claim that flights scheduled one day in advance cost more than flights scheduled 30 days in advance. What strategy appears to be effective in saving money when flying?

Flight scheduled one day in advance 456 614 628 1088 943 567 536

Flight scheduled 30 days in advance 244 260 264 264 278 318 280

Problem 20

Assume that the paired sample data are simple random samples and that the differences have a distribution that is approximately normal.

Heights of Winners and Runners-Up - Listed below are the heights (in inches) of candidates who won presidential elections and the heights of the candidates who were runners up. The data are in chronological order, so the corresponding heights from the two lists are matched. For candidates who won more than once, only the heights from the first election are included, and no elections before 1900 are included.

a. A well-known theory is that winning candidates tend to be taller than the corresponding losing candidates. Use a 0.05 significance level to test that theory. Does height appear to be an important factor in winning the presidency?

b. If you plan to test the claim in part (a) by using a confidence interval, what confidence level should be used? Construct a confidence interval using that confidence level, then interpret the result.

Won Presidency Runner-Up

71 74.5 74 73 69.5 71.5 75 72 73 74 68 69.5 72 71 72 71.5

70.5 69 74 70 71 72 70 67 70 68 71 72 70 72 72 72

Chapter 9, Section 5

Problem 10

Hypothesis Tests of Claims About Variation - Test the given claim. Assume that both samples are independent simple random samples from populations having normal distributions.

Braking Distances of Cars - A random sample of 13 four-cylinder cars is obtained, and the braking distances are measured and found to have a mean of 137.5 ft and a standard deviation of 5.8 ft. A random sample of 12 six-cylinder cars is obtained and the braking distances have a mean of 136.3 ft and a standard deviation of 9.7 ft (based on Data Set 16 in Appendix B). Use a 0.05 significance level to test the claim that braking distances of four-cylinder cars and breaking distances of six-cylinder cars have the same standard deviation.

Problem 16

Hypothesis Tests of Claims About Variation - Test the given claim. Assume that both samples are independent simple random samples from populations having normal distributions.

BMI for Miss America - Listed below are body mass indexes (BMI) for Miss America winners from two different time periods. Use a 0.05 significance level to test the calim that winners from both time periods have BMI values with the same amount of variation.

BMI (from recent winners): 19.5 20.3 19.6 20.2 17.8 17.9 19.1 18.8 17.6 16.8

BMI (from the 1920s and 1930s): 20.4 21.9 22.1 22.3 20.3 18.8 18.9 19.4 18.4 19.1

https://brainmass.com/statistics/hypothesis-testing/statistics-351217

#### Solution Summary

This solution is comprised of detailed step-by-step calculations and analysis of the given problems related to Statistics and provides students with a clear perspective of the underlying concepts.

Sufficiency and Order Statistics

Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the uniform distribution over the closed interval [-theta, theta ]

having pdf f(x; theta ) = (1/2(theta))I[-theta , theta ](x).

Argue that the mle of theta; equals theta;hat= max(-Y1, Yn).

Demonstrate that the mle theta;hat is a sufficient statistic for theta;.

Define at least two ancillary statistics for this distribution

See attachment for better symbol representation.

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