Statistics 200 Assignment #2 Complete Solution

Question: #1538

Statistics 200 Assignment #2

Instructions:
Typewritten answers are preferred. If your handwriting is illegible or if the answers are not presented in a neat and organized manner, with answers clearly numbered and blank lines separating answers, you may lose 2 marks.
Please leave one-inch margins for marking.
If you use R Commander to compute probabilities for any of the questions, please state clearly you have done so and provide the parameter values that you input in R Commander.

1. A realtor knows that 15% of listed houses are sold while the rest are delisted. He does some research and finds the selling price of a house has a mean of $600(k) and standard deviation of $150(k). Prices of individual houses are independent of each other, and whether a house is sold or not is independent for different houses. Suppose the realtor has 80 houses to sell, and he makes 5% commission on each house sold.

(a) What is the probability that the realtor sells more than 10 houses? (3 marks)
(b) Suppose that the realtor has sold 36 houses. What is the probability that he makes more than $1400(k)? State any assumption you need to solve this question. Explain briefly why this assumption is needed. If no assumption is needed, please do state so. (4 marks)
(c) Suppose that the realtor has sold 12 houses. What is the probability that he makes more than $400(k)? State any assumption you need to solve this question. Explain briefly why this assumption is needed. If no assumption is needed, please do state so. (4 marks)

2. The editor of a Vancouver magazine wants to determine the proportion of magazines that have some sort of nonconforming attribute such as improper page setup, missing pages, duplicated pages, typographical errors, etc. The editor commissions a study of 200 magazines and finds that 35 contain some type of nonconformance.
(a) Construct a 99% confidence interval for the true proportion of magazines that contain some type of nonconformance. State all your assumptions. (3 marks)
(b) The editor is not satisfied with the length of the confidence interval as he considers it too wide. How many magazines would he need to sample in order to have a 95% confidence interval with a margin of error of 0.02? (3 marks)

3. A researcher wishes to conduct a hypothesis test to investigate whether the mean exercise time of office workers in Vancouver is less than 4 hours per week. A random sample of 36 office workers in Vancouver is taken and the exercise time per week of each of the sampled worked is recorded. Their average weekly exercise time is 3.5 hours, and the standard deviation is 1 hour. Twenty of the sampled office workers exercise for less than 4 hours per week.
(a) Construct a 95% confidence interval for the mean weekly exercise time among the all office workers in Vancouver. Also provide an interpretation to the interval in the context of this question. (3 marks)
(b) Write down the null and alternative hypotheses that the researcher should use in his hypothesis test. Define any notation used. (3 marks)
(c) Compute the test statistic based on the data collected from the sampled workers. (1 mark)
(d) Find the exact value or provide a range of values for the P-value. Clearly explain your steps.
(2 marks)
(e) Clearly state your conclusion in the context of this question in one or two sentences. Suppose that a significance level of 0.05 is used. (1 mark)
(f) Describe the Type I error in the context of this question. (1 mark)
(g) Using the data collected from the sampled workers, conduct another hypothesis test (at the 5% significance level) to investigate whether exactly half of the office worker population exercise less than 4 hours per week. Remember to include the hypotheses, obtain the test statistic and P-value, and draw a conclusion in the context of the question. Define any notation used. (7 marks)