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CJ: A random sample of 100 inmates at a maximum security prison shows that exactly 10 of the respondents had been the victims of violent crime during their incarceration. Estimate the proportion of victims for the population as a whole, using the 90% confidence level. (Hint Calculate the sample proportion Ps before using Formula 6.3. Remember that proportions are equal to frequency divided by N.)SOC: The survey mentioned in problem 8.5 found that 25 of the 178 households consisted of unmarried couples who were living together. What is your estimate of the population proportion? Use the 95%% level. Problem 6.5: SOC A researcher has gathered information from a random sample of 178 households. For each variable below, construct confidence intervals to estimate the population mean. Use the 90%% level. 3. An average of 2.3 people resides in each household. Standard deviation is .35. b. There was an average of 2.1 television sets (s = .10) and .78 telephones (s = .55) per household. c. The households averaged 8.0 hours of television viewing per day (s = 3.0).For each of the following three sample sizes, construct the 95% confidence interval for the population proportion. Use a sample proportion of 0.40 throughout. What happens to interval width as sample size increases? Why? P . = 0.40 Sample A: N = 100 Sample B: N = 1000 Sample C: N = 10.000PS: Two individuals are running for mayor of Shinbone. Kansas. You conduct an election survey a week before the election and find that 51% of the respondents prefer candidate A. Can you predict a winner? Use the 09% level. Hint: In a two-candidate race, what percentage of the vote would the winner need? Does the confidence interval indicate that candidate A has a sure margin of victory? Remember that while the population parameter is probably (a 5 .01) in the confidence interval, it may be anywhere in the interval.) P - 0.51 N - 5THSOC: The fraternities and sororities at St. Algebra College have been plagued by declining membership over the past several years and want to know if the incoming freshman class will be a fertile recruiting ground. Not having enough money to survey all 1,800 freshmen, they commission you to survey the interests of a random sample. You find that 35 of your 150 respondents are "extremely" interested in social clubs. At the 95%% level, what is your estimate of the number of freshmen who would be extremely interested? Hint: The high and low values of your final confidence interval are proportions. How can proportions also be expressed as numbers?)You are the automotive specialist for the monthly magazine Consumer Beware. Part of your job is to investigate the claims made by automakers in their TV commercials. You are particularly suspicious of a new economy car that the automaker claims will get 78 miles per gallon. After checking the mileage figures for a random sample of 125 owners of this car, you find an average miles per gallon of 75.5, with a standard deviation of 3.7. At the 99%% level, do your results tend to confirm or refute the manufacturer's claims?SOC: A school system has assigned several hundred "chronic and severe" underachievers to an alternative educational experience. To assess the program, a random sample of 35 students has been selected for comparison with all students in the system. a. In terms of GPA, did the program work? Systemwide GPA Program GRA A - 2.47 X - 255 5 - 0.70 b. In terms of absenteeism (number of days missed per year). what can be said about the success of the program? Systemwide Program A G.137 X = 4.76 8 - 1.11 N - 35 c. In terms of standardized test scores in math and reading, was the program a success? Math Test- Math Test- Systemwide Program P - 103 X = 106 9 - 20 N - 35 Reading Test- Reading Test- Systemwide Program A = 110 X = 113 5 - 20 N - 35 (Hint: Note the wording of the research questions. Is a one-failed fest justified? Is the program a success if the students in the program are no different from students systemwide? What if the program students were performing at lower levels? If a one-failed fest is used, what form should the research hypothesis take? Where will the critical region begin?A random sample of 113 felons convicted of nonviolent crimes in a state prison system completed a program designed to improve their employment possibilities before being released on parole. Fifty-eight eventually became repeat offenders. Is this recidivism rate significantly different from the rate for all offenders in that state (57%)? Summarize your conclusions in a sentence or two. (HINT: You must use the information given in the problem to compute a sample proportion. Remember to convert the population percentage to a proportion.)CJ: Statewide, the police clear by arrest 35% of the robberies and 42%% of the aggravated assaults reported to them. A researcher takes a random sample of all the robberies (N = 207) and aggravated assaults (N = 178) reported to a metropolitan police department in one year and finds that 83 of the robberies and 80 of the assaults were cleared by arrest. Are the local arrest rates significantly different from the statewide rate? Write a sentence or two interpreting your decision.SW: You are the head of an agency seeking funding for a program to reduce unemployment among teenage males. Nationally, the unemployment rate for this group is 18%. A random sample of 323 teenage males in your area reveals an unemployment rate of 21.7%. Is the difference significant? Can you demonstrate a need for the program? Should you use a one- tailed test in this situation? Why? Explain the result of your test of significance as you would to a funding agency.PA: A number of years ago, the road and highway maintenance department of Sin City, Nevada, began recruiting minority group members through an affirmative action program. In terms of efficiency ratings as compiled by their superiors, how do the affirmative action employees rate? The ratings of random samples of both groups were collected, and the results are reported here (higher ratings indicate greater efficiency)- Sample 1 Sample 2 ( Affirmative Action) (Regular) XA = 152 5 - 39 N 87 N- 100 Write a sentence or two of interpretation.Are college students who live in dormitories significantly more involved in campus life than students who commute to campus? The following data report the average number of hours per week students devote to extracurricular activities. Is the difference between these randomly selected samples of commuter and residential students significant? Sample 1 Sample 2 (Residential) (Commuter) X, - 12.4 X, - 102 8 - 20 N, - 172As the director of the local Boys Club. you have claimed for years that membership in your club reduces juvenile delinquency. Now a cynical member of your funding agency has demanded proof of your claim. Random samples of members and nonmembers are gathered and interviewed with respect to their involvement in delinquent activities. Each respondent is asked to enumerate the number of delinquent acts he has engaged in over the past year. The average numbers of admitted acts of delinquency are reported below. What can you tell the funding agency? Sample I Sample 2 (Members) (Nonmembers) X - 103 X - 123 S-42 N, - 40 N - 55For each problem, test the sample proportions for the significance of the difference. Sample 1 Sample 2 Pal = 0.20 N - 114 N - 101 Sample 1 Sample 2 Pa = 0.02 N- 478 N - 537CJ: About half of the police officers in Shinbone, Kansas, have completed a special course in investigative procedures. Has the course increased their efficiency in clearing crimes by arrest? The proportions of cases cleared by arrest for samples of trained and untrained officers are reported here. Sample 1 Sample 2 [Trained) (Untrained) H. - 0.47 H. - 0.43 N 157 M - 113The mayor of a large city started a "crimeline" program some years ago and wonders whether it is working. The program publicizes unsolved violent crimes in the local media and offers cash rewards for information leading to arrests. Are "featured" crimes more likely to be cleared by arrest than other violent crimes? Results from random samples of both types of crimes are reported as follows: Sample 1 Sample 2 (Crimeline Crimes (Non-Crimeline Crimes Cleared by Arest) Cleared by Arrest) Pal = 0.35 Px - 0.25 N - 178 N - 212