Probability and Statistics (Answer up to 7th decimal point**)

And please give the answer step-by-step as I want to know how to calculate the answer, thanks.

Q1. (16%) A researcher conducts a study to investigate the number of hours a university student prepares for the "Statistics and Probability" final exam. A random sample of 20 students was selected. Below are their preparation times in hours.

9 & 12 & 12 & 12 & 15 & 17 & 18 & 20 & 21 & 22 & 23 & 24 & 24 & 25 & 25 & 25 & 26 & 26 & 27 & 30

(a) Determine the five-point-summary for the above sample and draw the box-and-whisker plot. Find also the interquartile range.

(b) Find the mean of the above sample by any valid means. Assume that preparation times in hours of all students are normally distributed and the population standard deviation is 6.00. Use the sample mean to construct a 90% confidence interval for the average score. Give an interpretation for the confidence interval.

Q2. (14%) A study was conducted to find out how certain disease may affect the body temperature of a patient with the disease. Eight patients with the disease were randomly selected. Their body temperatures (in degrees Celsius) and the degrees of seriousness of the disease (measured with a scale from 1 to 10; 1 is the least serious and 10 is the most serious) were recorded as shown below.

temperature (seriousness): 37.5 (3) & 37.7 (5) & 37.2 (5) & 38.6 (6) & 38.5 (8) & 40.1 (8) & 39.7 (9) & 40.2 (10)

Regression analysis was used to determine how the degree of seriousness may affect body temperature of a patient.

(a) Let =₀+₁ be the regression line. What are the values of b₀ and b₁?

₀=___________ , ₁= ___________

(b) Use the regression line to estimate the body temperature of a patient with degree of seriousness about 7.

(c) The Health Department claims that the patient's body temperature (Y) and his/her degree of seriousness of the disease (X) are positively linearly related. Test this claim at 5% significance level.

Q3. (8%) In a factory, one worker produces an average of 175 units per day with a standard deviation of 20. Another worker produces at an average rate of 165 per day with a standard deviation of 21. What is the probability that the production of worker 1 will be larger than that of worker 2 by more than 400 units during 50 working days?

Q4. (14%) Suppose that the amount of time teenagers spend weekly working at part-time jobs is normally distributed. A random sample of 15 teenagers was drawn and each reported the amount of time spent at part-time jobs (in minutes). These are listed here.

180 & 130 & 150 & 165 & 90 & 130 & 120 & 60 & 200 & 180 & 80 & 240 & 210 & 150 & 125

(a) Given that the margin of error for the population mean is 27.7064, determine the corresponding critical value. What is the associated significance level?

(b) Based on (a), determine the associated confidence interval estimate of the population mean. State clearly the level of confidence in this problem.

(c) Determine the 90% confidence interval estimate of the population variance.

(d) Give an interpretation for the confidence intervals in (b) and (c).

Q5. (12%) The editor of a book publishing company decides whether to publish a textbook. Before reaching a publishing decision, the book will be reviewed for a positive or negative feedback. In the past, 99% of the huge successes received positive feedbacks, 70% of the moderate successes received positive feedbacks, 40% of the breakeven received positive feedbacks, and 20% of the losers received positive feedbacks. Based on the books published by the company, x% of them are huge successes, 20% of them are moderate successes, y% are breakeven, and 25% are losers.

(a) Given that the probability of a book that receives a positive feedback is 0.469, find x% and y%.

(b) Based on (a), if the book receives a positive feedback, what is the probability that it will be a huge success or a moderate success?

(c) Learning from the past experience, the editor sets up a new rule. For each book that receives negative feedback, there is no chance that the editor will publish it. On the other hand, for each book that receives a positive feedback, there is 80% chance that the editor will publish it. The editor receives 12 reviews and he makes the decision based on each feedback separately. Given that no more than 6 books will be published, what is the probability that no less than 3 books will be published?

Q6. (20%)

(a) In property management of shopping malls, a shopping mall is said to have a satisfactory customer service if the mean score of its customer service is greater than 85. The property management manager of a shopping mall claims that his mall is about to obtain a satisfactory customer service. Fifty customers are randomly selected to evaluate the customer service of the mall, their scores have a mean of 86 and a standard deviation of 14. Assume that the score of the customer service is normally distributed. Is there enough evidence to support the claim of the property management manager at 1% level of significance?

(b) The Human Resources Manager of an international company would like to compare the weekly working hours per person between employees in City A and City B. Random samples are drawn and the results are summarized below:

City A - Sample size: 15, Mean (hours): 42, Standard deviation (hours): 2.8

City B - Sample size: 16, Mean (hours): 44, Standard deviation (hours): 3.0

Can we conclude that the true mean weekly working hours per person between employees in City A and City B are different at 10% level of significance? Justify your assumptions.

Q7. (16%) A study was conducted to find out how certain disease may affect the blood pressures of patients in various age groups. Randomly selected samples of 80 patients with the disease were divided into three age groups, and their systolic blood pressures (measured in mmHg) were recorded. Below are the statistics.

Sample 1 (less than 25 years old) - n: a, : 3300, s: 10

Sample 2 (between 25 and 55) - n: b, : 3976, s: 8.5

Sample 3 (over 55) - n: 22, : 3256, s: 9

Assume the systolic pressures of patients in each of the three age groups were normally distributed and that the three groups had roughly the same variance in systolic blood pressures. Given that the sum of squares for errors is 6551.75. Perform ANOVA to determine if there are significant differences among the average systolic blood pressures of the individuals in these three groups at =0.05.