Assignment 4A. Theory Component

Show your work for this component. Where relevant, keep all your middle work as well as your final answer to a minimum of four decimals, unless otherwise stated. For any test of hypothesis questions, show all steps in detail and provide a conclusion in terms of the context of the question.

Problem 1. Population data

A population consists of 5 units: X = 5, 7, 9, 11, and 13. All units are given equal probability of selection and a sample of size 2 is selected with replacement. Use the Central Limit Theorem to answer part (e) below.

a.Compute the population mean (), population variance (?2), and population standard deviation (?).

[3 marks]

b.Graph a probability histogram of the population values. Title this graph: Population Distribution. What is the shape of the histogram? [2 marks]

c.List all possible samples of size 2 and calculate the mean of each sample. [4 marks]

d.Graph a probability histogram of the sample means. Title this graph: Sampling Distribution of Sample Means. What is the shape of the histogram? [2 marks]

e.Find the mean, variance, and standard deviation of the sample means. [3 marks]

f.Compare your results from (a) and (e). What do you notice? [2 marks]

g.Compare the shapes of the probability histograms in (b) and (d). What do you notice? [2 marks]

Problem 2. Home cost

The average cost of a single detached home in Pricy Town for 2014 is $482,000 with a standard deviation of $36,000. Let be the mean price of 49 houses. Find the probability that :

a.is greater than $492,000. [3 marks]

b.will fall between $476,000 and $488,000. [4 marks]

Problem 3. Tuition fee

A random sample of 30 full-time students form a large post-secondary institution was asked about their average tuition fee for the first year. From the data it was noted that the mean is $7850.00. The population standard deviation of the tuition fee is $257.00. Assume that the tuition fee is approximately normally distributed.

a.Construct an 80% confidence interval for . [3 marks]

b.Construct a 97% confidence interval for . [3 marks]

c.Refer to (a) and (b). What happens to the width of the confidence interval as the confidence level is increased (and everything else is kept the same)? [1 mark]

d.Refer to (a). Suppose you were not happy with the width of the confidence interval and wanted to cut the size of the interval to half, what should the sample size be? (Note: No calculation necessary!)

[1 mark]

Problem 4. Insurance rates

In order to set rates, an insurance company is trying to estimate the number of sick days that full time welders in an oilfield welding company take per year. A previous study indicated that the standard deviation was 8.2 days. How large a sample must be selected if the company wants to be 90% confident that the true mean differs from the sample mean by no more than 2 days? [3 marks]

Problem 5. Systolic blood pressure

A random sample of 16 healthy adults over the age of 45 showed a mean systolic blood pressure of 129 mm Hg and a standard deviation of 11 mm Hg. Construct a 98% confidence interval for the population mean of systolic blood pressure. Assume that systolic blood pressure is approximately normally distributed. [4 marks]

Problem 6. Left-handedness

According to a survey with 900 males, 13% are left handed.

a.Construct a 97.5% confidence interval for population proportion of all males who are left handed.

[3 marks]

b.If both the sample size and the number of successes are doubled, what is the effect on the width of the confidence interval, if nothing else changed? [1 marks]

Problem 7. Sample size

In each case, find the appropriate sample size required to construct a 85% confidence interval for p (population proportion) that has a margin of error 0.05.

a.Assume that no preliminary estimate for p is available. [3 marks]

b.Assume that the preliminary estimate for p is 0.35. [3 marks]

Problem 8. Waiting time

A fast food outlet claims that the mean waiting time in line is less than 4 minutes. Alan is suspicious of this claim. He took a random sample of 25 customers and noted that the mean wait time is 4.2 minutes with a population standard deviation of 0.5 minute. If ? = 0.05, use the four-step P-value approach and test whether the mean wait time is less than 4 minutes. Assume that the mean waiting times follow normal distribution. [10 marks]

Problem 9. Generic medicine

A generic medicine that claims to reduce fever and pain comes in 5 oz. bottles. Ted was suspicious about the volume of contents. He measured the volume of 15 randomly selected bottles and noticed that the mean is only 4.6 oz. with a standard deviation of 0.7 oz. Is the volume of the content not 5 oz.? Use the critical value approach to arrive at a conclusion. Show all key steps of the test of hypothesis. Use 0.10 level of significance. Assume that the volume of the contents follows a normal distribution. [10 marks]

Problem 10. Telephone lines

A cell company claims that at least 15% of its customers have 2 or more lines. The company selects a random sample of 500 customers and finds that 65 have 2 or more telephone lines. If ? = 0.02, show all key steps and test whether the proportion is less than 15% using the critical value approach. [10 marks]

Assignment 4 Assignment 4 consists of a theory component (4A, worth 80%) and a computer component (4B, worth 20%). The grades allocated are summarized below. Assignment Grade weight Assignment 4A. Theory component 80 marks 80% Assignment 4B. Computer component 20 marks 20% 100 marks 100% Total marks You must submit Assignment 4A and Assignment 4B together as PDFs, through the appropriate drop box on the course home page. Submit: 1. One PDF solution file (file) entitled Assignment4A containing all your answers to Assignment 4A, presented in the proper order. Your name and student ID number must be at the top of the first page of your solution file. and 2. One PDF solution file (file) entitled Assignment4B containing all your answers to Assignment 4B, presented in the proper order. We suggest that you print the Assignment 4 questions, so that you can conveniently review the questions with their solutions when you prepare for your exams. 1 Assignment 4A. Theory Component Show your work for this component. Where relevant, keep all your middle work as well as your final answer to a minimum of four decimals, unless otherwise stated. For any test of hypothesis questions, show all steps in detail and provide a conclusion in terms of the context of the question. Problem 1. Population data A population consists of 5 units: X = 5, 7, 9, 11, and 13. All units are given equal probability of selection and a sample of size 2 is selected with replacement. Use the Central Limit Theorem to answer part (e) below. a. Compute the population mean (), population variance (2), and population standard deviation (). [3 marks] b. Graph a probability histogram of the population values. Title this graph: Population Distribution. What is the shape of the histogram? [2 marks] c. List all possible samples of size 2 and calculate the mean of each sample. [4 marks] d. Graph a probability histogram of the sample means. Title this graph: Sampling Distribution of Sample Means. What is the shape of the histogram? [2 marks] e. Find the mean, variance, and standard deviation of the sample means. [3 marks] f. Compare your results from (a) and (e). What do you notice? [2 marks] g. Compare the shapes of the probability histograms in (b) and (d). What do you notice? [2 marks] Problem 2. Home cost The average cost of a single detached home in Pricy Town for 2014 is $482,000 with a standard deviation of $36,000. Let be the mean price of 49 houses. Find the probability that : X a X is greater than $492,000. [3 marks] h. will fall between $476,000 and $488,000. [4 marks] Problem 3. Tuition fee A random sample of 30 full-time students form a large post-secondary institution was asked about their average tuition fee for the first year. From the data it was noted that the mean is $7850.00. The population standard deviation of the tuition fee is $257.00. Assume that the tuition fee is approximately normally distributed. a Construct an 80% confidence interval for . [3 marks] i. Construct a 97% confidence interval for . [3 marks] j. Refer to (a) and (b). What happens to the width of the confidence interval as the confidence level is increased (and everything else is kept the same)? [1 mark] k. Refer to (a). Suppose you were not happy with the width of the confidence interval and wanted to cut the size of the interval to half, what should the sample size be? (Note: No calculation necessary!) [1 mark] 2 Problem 4. Insurance rates In order to set rates, an insurance company is trying to estimate the number of sick days that full time welders in an oilfield welding company take per year. A previous study indicated that the standard deviation was 8.2 days. How large a sample must be selected if the company wants to be 90% confident that the true mean differs from the sample mean by no more than 2 days? [3 marks] Problem 5. Systolic blood pressure A random sample of 16 healthy adults over the age of 45 showed a mean systolic blood pressure of 129 mm Hg and a standard deviation of 11 mm Hg. Construct a 98% confidence interval for the population mean of systolic blood pressure. Assume that systolic blood pressure is approximately normally distributed. [4 marks] Problem 6. Left-handedness According to a survey with 900 males, 13% are left handed. a Construct a 97.5% confidence interval for population proportion of all males who are left handed. [3 marks] l. If both the sample size and the number of successes are doubled, what is the effect on the width of the confidence interval, if nothing else changed? [1 marks] Problem 7. Sample size In each case, find the appropriate sample size required to construct a 85% confidence interval for p (population proportion) that has a margin of error 0.05. a Assume that no preliminary estimate for p is available. [3 marks] m. Assume that the preliminary estimate for p is 0.35. [3 marks] Problem 8. Waiting time A fast food outlet claims that the mean waiting time in line is less than 4 minutes. Alan is suspicious of this claim. He took a random sample of 25 customers and noted that the mean wait time is 4.2 minutes with a population standard deviation of 0.5 minute. If = 0.05, use the four-step P-value approach and test whether the mean wait time is less than 4 minutes. Assume that the mean waiting times follow normal distribution. [10 marks] Problem 9. Generic medicine A generic medicine that claims to reduce fever and pain comes in 5 oz. bottles. Ted was suspicious about the volume of contents. He measured the volume of 15 randomly selected bottles and noticed that the mean is only 4.6 oz. with a standard deviation of 0.7 oz. Is the volume of the content not 5 oz.? Use the critical value approach to arrive at a conclusion. Show all key steps of the test of hypothesis. Use 0.10 level of significance. Assume that the volume of the contents follows a normal distribution. [10 marks] Problem 10. Telephone lines A cell company claims that at least 15% of its customers have 2 or more lines. The company selects a random sample of 500 customers and finds that 65 have 2 or more telephone lines. If = 0.02, show all key steps and test whether the proportion is less than 15% using the critical value approach. [10 marks] 3 Assignment 4B. Computer Component As you work through each computer problem, use StatCrunch to generate all computer-related solutions. Do not round off the results you get from StatCrunch. Make sure that, for each computer problem, you copy and paste the output generated by StatCrunch (as is) as requested, into a single word processing file called Assignment4B. Use a word processing program that allows you to convert to a PDF file after you have completed all your solutions. Be sure to type the appropriate problem subheading (e.g., Problem 1a) before you copy and paste the related StatCrunch output or type solutions to the related interpretation questions in the Assignment4B word processing file. Problem 1. Population experiment Consider the following population experiment: You write the population values 6, 8, 10, 12, 14 on separate slips of paper and place these in a hat. You then select one slip of paper in a random manner. Let the population random variable, X, be the number that you observe on the slip of paper selected. The probability distribution of X is described in Figure 1 below. X 6 8 10 12 14 P(X) .20 .20 .20 .20 .20 Figure 1. Population distribution. a Open a new data table in StatCrunch. Create the two variables X and P(X), and enter the values for these two variables, as shown in Figure 1. Create a new word document called Assignment4B. Copy and paste the probability distribution table that you created in StatCrunch into the word file Assignment4B under the subheading Problem 1a. [1 mark] n. Use StatCrunch to display the graph and compute the mean and standard deviation for the population distribution in problem 1a. Copy and paste the graph and mean and standard deviation from StatCrunch to the word file Assignment4B under the subheading Problem 1b. [1 mark] o. With the population probability distribution that you created in 1b displayed in the data table, use StatCrunch to generate 10,000 repetitions of the following sampling experiment: Drawing on the population values 6, 8, 10, 12, 14, randomly select a sample of 2 values, with replacement, and observe the sample mean. StatCrunch will simulate this experiment 10,000 times, so that 10,000 sample means are created in one column of the data table. Copy and paste the first 5 sample means (along with the variable name) from StatCrunch to the word file Assignment4B under the subheading Problem 1c. [1 mark] p. Use StatCrunch to compute the mean and standard deviation of the 10,000 sample means you just generated in 1c. Copy and paste the Summary Statistics window that displays the mean and standard deviation of the sampling distribution to the word file Assignment4B under the subheading Problem 1d. [1 mark] q. Based on the Central Limit Theorem, the mean and standard deviation of the sampling distribution in problem 1d should approximate what values? Type your answer into the word file Assignment4B under the subheading Problem 1e. [1 mark] 4 r. Use StatCrunch to create a relative frequency histogram for the 10,000 sample means you generated in Problem 1c. Copy and paste the relative frequency histogram from StatCrunch to the word file Assignment4B under the subheading Problem 1f. [1 mark] s. How does the shape of the relative frequency histogram created in Problem 1f compare to what one would expect to see based on the Central Limit Theorem? Type your answer into the word file Assignment4B under the subheading Problem 1g. [1 mark] Problem 2. Funland questionnaire Funland, an indoor amusement park located in a large mall, offers midway rides, games, fast foods, and beverages. The owners of Funland distributed the survey questionnaire (Figure 2) to its regular customers. Twenty-five regular customers responded to this survey, and their responses are stored in the StatCrunch file Funland in the Math 216 group folder on StatCrunch. Funland Questionnaire For questions 1-4, please select the responses that apply to you. For questions 5-8, please write the number that applies. Variable 1. Please indicate your marital status. Marital single 1 married 2 2.Please indicate your gender. Gender Code female 1 male 2 3.Do you frequently bring children to Funland? Child Code yes 1 no 2 4.Do you purchase a monthly pass? 5 Text Code Pass Code yes 1 no 2 5.On a typical visit, please estimate the amount that you spend at Funland. Spend $ _______ 6.How many times per month do you typically visit Funland? Visits _______ 7.Please indicate your monthly family income before taxes. Income $ _______ 8.Please indicate your age. Age _______ a Open the StatCrunch data file Funland and answer the following. Use StatCrunch to construct a 99% confidence interval for the mean amount spent, per visit, by the population of regular customers of Funland. Copy and paste the StatCrunch Confidence Interval Table created to the word file Assignment4B under the subheading Problem 2a. [1 mark] t. Use StatCrunch to construct a 90% confidence interval for the mean amount spent, per visit, by the population of regular customers of Funland. Copy and paste the StatCrunch Confidence Interval Table created to the word file Assignment4B under the subheading Problem 2b. [1 mark] u. What happens to the width of the confidence interval, when the level of confidence decreases. Type your answer in the word file Assignment4B under the subheading Problem 2c. [1 mark] v. In constructing the confidence intervals in Problem 2a., what assumption are you making regarding the population of monthly expenditures made by all regular customers of Funland? Type your answer in the word file Assignment4B under the subheading Problem 2d. [1 mark] w. Suppose you wish to estimate, with 95% confidence, the mean amount spent, per visit, by the population of regular customers of Funland so that your estimate is within $10 of the true mean amount spent. Assume that the population standard deviations is $50. Find the minimum required sample size. Copy and paste the StatCrunch Confidence Interval Width Window to the word file Assignment4B under the subheading Problem 2e. [1 mark] Problem 3. Funland: Marital status Open the StatCrunch data file Funland and answer the following a Use StatCrunch to construct a 90% confidence interval for the population proportion of Funland regular customers who are married. Copy and paste the StatCrunch Confidence Interval Table created to the word file Assignment4B under the subheading Problem 3a. [1 mark] x. Suppose you wish to estimate, with 95% confidence, the population proportion of Funland regular customers who use the monthly pass. Your estimate must be within 5% of the population proportion. Find the minimum sample size required, when no preliminary estimate is available. Copy and paste the StatCrunch Confidence Interval Width Window to the word file Assignment4B under the subheading Problem 3b. [1 mark] y. 6 Suppose you wish to estimate, with 95% confidence, the population proportion of Funland regular customers who use the monthly pass. Your estimate must be within 5% of the population proportion. Find the minimum sample size required if a preliminary study found that 65% of customers use the monthly pass. Copy and paste the StatCrunch Confidence Interval Width Window to the word file Assignment4B under the subheading Problem 3c. [1 mark] Problem 4. Funland: Ages Open the StatCrunch data file Funland and answer the following. a Use StatCrunch to test the hypothesis that the mean age of the population of Funland regular customers exceeds 40 years, at a 5% level of significance. Use the four-step P-value approach. Copy and paste the Hypothesis Test Results window from StatCrunch to the word file Assignment4B under the subheading Problem 4a under Step 2 of the test. [1 mark] z. What assumption did you make in conducting the hypothesis test in 4a? Type your answer in the word file Assignment4B under the subheading Problem 4b. [1 mark] aa. Conduct the appropriate hypothesis test to determine if the sample of customer ages comes from a normal population. Use the four-step P-value approach. Copy and paste the Hypothesis Test Results window from StatCrunch to the word file Assignment4B under the subheading Problem 4c under Step 2 of the test. [1 mark] Problem 5. Funland: Monthly passes Open the StatCrunch data file Funland. Use StatCrunch to test the hypothesis that the population proportion of Funland regular customers who use the monthly pass is at least 70%. Use the four-step P-value approach. Copy and paste the Hypothesis Test Results window from StatCrunch to the word file Assignment4B under the subheading Problem 5 under Step 2 of the test. [2 marks] 7