4. a fitness magazine advertises that the mean monthly cost of joining a health club is less than $50. you work for a consumer advocacy group and find that a random sample of 30 clubs has a mean monthly cost of 48.25$ and a standard deviation of $5.23

1. construct a 99% confidence interval for the mean monthly cost of joining a health club

2. at a 0.1 level fo significance, do you have enough evidence to reject the advertisement's claim? your answer should include both hypotheses, the rejection region, the test statistic and a conclusive statement in non-technical terms.

3. based on your conclusion, what type of error are you possibly making (type 1 or 2)? can the probability of this error be easily measured?

4. if the number of fitness clubs you surveyed had been 10 instead of 30. with the same sample mean and standard deviation, would the conclusion of your test in (2) have been different? if so, is it surprising? explain. (assume the distribution of the population to be close to normal

A magazine uses a survey of readers to obtain customer satisfaction ratings for the nation's largest retailers. Each survey respondent is asked to rate a specified retailer in terms of six factors: quality of products, selection, value, checkout efficiency, service, and store layout. An overall satisfaction score summarizes the rating for each respondent with 100 meaning the respondent is completely satisfied in terms of all six factors. Sample data representative of independent samples of Retailer A and Retailer B customers are shown below.

a)

Formulate the null and alternative hypotheses to test whether there is a difference between the population mean customer satisfaction scores for the two retailers. (Let μ₁ = population mean satisfaction score for Retailer A customers and μ₂ = population mean satisfaction score for Retailer B customers.)

a)H₀: μ₁ − μ₂ ≠ 0

H_a: μ₁ − μ₂ = 0

b) H₀: μ₁ − μ₂ < 0

H_a: μ₁ − μ₂ = 0

c) H₀: μ₁ − μ₂ ≤ 0

H_a: μ₁ − μ₂ > 0

d) H₀: μ₁ − μ₂ = 0

H_a: μ₁ − μ₂ ≠ 0

e) H₀: μ₁ − μ₂ ≥ 0

H_a: μ₁ − μ₂ < 0

B) Assume that experience with the satisfaction rating scale of the magazine indicates that a population standard deviation of 11 is a reasonable assumption for both retailers. Conduct the hypothesis test.

Find the value of the test statistic. (Round your answer to two decimal places.) _________

Find the p-value. (Round your answer to four decimal places.)

p-value = __________

At a 0.05 level of significance what is your conclusion?

a) Reject H₀. There is insufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.

b) Do not Reject H₀. There is sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.      

c) Do not reject H₀. There is insufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.

d) Reject H₀. There is sufficient evidence to conclude that the population mean satisfaction scores differ for the two retailers.

C)

Provide a 95% confidence interval for the difference between the population mean customer satisfaction scores for the two retailers. (Round your answers to two decimal places.) _____to ______

Which retailer, if either, appears to have the greater customer satisfaction?

The 95% confidence interval  ---Select--- is completely below contains is completely above zero. This suggests that the Retailer A has a  ---Select--- higher lower population mean customer satisfaction score than Retailer B.

A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 25 weeks. Assume that the length of unemployment is normally distributed with population mean of 25 weeks and the population standard deviation of 9 weeks. Suppose you would like to select a random sample of 35 unemployed individuals for a follow-up study. Round the answers of following questions to 4 decimal places.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. What is the probability that one randomly selected individual found a job less than 27 weeks?
4. For 35 unemployed individuals, find the probability that the average time that they found the next job is less than 27 weeks.
5. For part d), is the assumption of normal necessary? NoYes

A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 33 weeks. Assume that for the population of all unemployed individuals the population mean length of unemployment is 33 weeks and that the population standard deviation is 3 weeks. Suppose you would like to select a random sample of 35 unemployed individuals for a follow-up study. Find the probability that a single randomly selected value is less than 34. P(X < 34) = Find the probability that a sample of size n = 35 is randomly selected with a mean less than 34. P( ¯ x < 34) =
Enter your answers as numbers accurate to 4 decimal places.

A new magazine, Cycling n’ Running NZ is about to be published. The magazines’ publishers are unsure whether readers of this magazine would be more interested in articles on cycling or articles on running. Accordingly, a study was conducted to find out how interested readers of this magazine would be in articles on either of these topics. The variables to use in answering this question are Cycling and Running. Potential interest in both topics was measured on a five-point semantic differential scale that was anchored 1=Very Uninterested to 5=Very Interested. Is there a difference in the extent of preference for articles about cycling compared to articles about running?

row 1(going down): cycling

Row 2 (going down): running

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A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 25 weeks. Assume that the length of unemployment is normally distributed with population mean of 25 weeks and the population standard deviation of 2 weeks. Suppose you would like to select a random sample of 39 unemployed individuals for a follow-up study. Round the answers of following questions to 4 decimal places.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. What is the probability that one randomly selected individual found a job more than 25 weeks?
4. For 39 unemployed individuals, find the probability that the average time that they found the next job is more than 25 weeks.
5. For part d), is the assumption of normal necessary? No Yes

A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 28 weeks. Assume that for the population of all unemployed individuals the population mean length of unemployment is 28 weeks and that the population standard deviation is 2.5 weeks. Suppose you would like to select a random sample of 72 unemployed individuals for a follow-up study.

Find the probability that a single randomly selected value is greater than 28.5.
P(X > 28.5) =  (Enter your answers as numbers accurate to 4 decimal places.)

Find the probability that a sample of size n=72n=72 is randomly selected with a mean greater than 28.5.
P(M > 28.5) =  (Enter your answers as numbers accurate to 4 decimal places.)

Suppose you wish to test the hypothesis that the number of print media (newspaper, magazine, and so forth) that people subscribe to is related to subscribers’ education levels. The following hypothetical data was gathered form five people.

                                Number of subscriptions, x :4, 5, 1, 2, 3

Education level, y (years): 16, 20, 8, 9, 12

H0: The number of print media that people subscribe to is not related to subscribers’ education levels.

H1: The number of print media that people subscribe to is related to subscribers’ education levels.

Question: If the computed t value is in the rejection region (r=.97, p<.01), can you reject the null hypothesis at the p<.01 level of significance? Why? Show ALL work for the t-test calculation

The consumer magazine also claims that the cinnamon rolls at STARBUCKS do not weigh at least 8 ounces. A random sample of 25 customers purchasing cinnamon rolls yields the following results: - the sample mean equals 7.87 ounces - it is known from previous studies that the population standard deviation equals 0.25 ounces.

a. Set up a 95% confidence interval for the true mean?

b. What sample size is required if you want to be 99% sure that the sample mean will be within 0.2 ounces of the true mean?

c. Test the hypothesis that the true population mean is less than 8 ounces. Set the type one error equal to 1%.

A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 40 weeks. Assume that for the population of all unemployed individuals the population mean length of unemployment is 40 weeks and that the population standard deviation is 2.8 weeks. Suppose you would like to select a random sample of 92 unemployed individuals for a follow-up study.

Find the probability that a single randomly selected value is greater than 40.6. P(X > 40.6) = (Enter your answers as numbers accurate to 4 decimal places.)

Find the probability that a sample of size n = 92 is randomly selected with a mean greater than 40.6. P(M > 40.6) = (Enter your answers as numbers accurate to 4 decimal places.)