STT

homework_4_questions.pdf

STT 200 Due on 04 / 28 / 2015

Name PID

Homework 4

1. A poll by the Gallup Organization sponsored by Philadelphia-based CIGNA Integrated Care found that about 40% of employees have missed work due to a musculoskeletal (back) injury of some kind (National Underwriter, Apr. 5, 1999). A random sample of 100 workers is to be drawn from a particular manufacturing plant. Let p̂ be the proportion of workers who missed work due to back injuries in the sample.

(a) Does p̂ follow a normal distribution ? Answer by checking whether all the necessary conditions are met.

(b) Find the mean and standard deviation of p̂.

(c) Find the probability that between 37.5% and 45% of the employees in the sample have missed work due to a back injury. Draw the graph of the distribution of p̂ and shade the appropriate region on the graph.

2. According to a National Business Travel Association (NBTA) 2010 survey, the average salary of a travel management professional is $96850. Assume that the standard deviation of such salaries is $30000.

(a)Consider a random sample of 100 travel management professionals. Let X̄ represent the mean salary for the sample. Does X̄ follow a normal distribution ? Answer by checking whether all the necessary conditions are met.

(b) Find the expected value and the standard deviation of X̄.

(c) What is the probability that the average salary in the sample exceeds $100000 ? Draw the graph of the distribution of X̄ and shade the appropriate region on the graph.

(d) What is the probability that the average salary in the sample is below $90000 ? Draw the graph of the distribution of X̄ and shade the appropriate region on the graph.

(e) Now consider a random sample of 400 travel management professionals and let Ȳ represent the mean salary for the sample. Find the expected value and standard deviation of Ȳ .

(f) Based on your answers to parts (b) and (e), circle the correct answer in the following sentence:

The standard deviation of the sampling distribution for a sample mean based on samples of size 100 is

twice as large as / one-fourth as large as / four times as large as / one-half as large as

the standard deviation of the sampling distribution of a sample mean based on samples of size 400.

(g) In general, what happens to the standard deviation of the sample mean as the sample size increases ?

For Problems 3 and 4, you may use your calculator to double check your work, but for full credit, you must clearly write down the formula for the confidence interval and show how you found the z-value (critical value) by drawing a graph and shading the appropriate region.

3. Recently, a case of salmonella (bacterial) poisoning was traced to a particular brand of ice cream bar, and the manufacturer removed the bars from the market. Despite their response, many consumers refused to purchase any brand of ice cream bars for some period of time after the event (McClave, personal consulting). One manufacturer conducted a survey of consumers 6 months after the outbreak. A sample of 244 ice cream bar consumers was contacted, and 23 respondents indicated that they would not purchase ice cream bars because of the potential for food poisoning.

(a) What is the point estimate of the true fraction of the entire market who refuse to purchase bars 6 months after the outbreak ?

(b) Is the sample large enough to use the normal approximation for the sampling distribution of p̂ ? Justify your response.

(d) Construct a 95% confidence interval for the true proportion of the market who still refuses to purchase ice cream bars 6 months after the event.

(e) Construct a 99% confidence interval for the true proportion of the market who still refuses to purchase ice cream bars 6 months after the event.

(f) How does the width of the 95% confidence interval compare to that of the 99% confidence interval ? Explain why this result agrees with your intuition.

4. A company is interested in estimating µ, the mean number of days of sick leave taken by all its employees. The firm’s statistician selects at random 101 personnel files and notes the number of sick days taken by each employee. The following sample statistics are computed: x̄ = 12.2 days, s = 10 days.

(a) Are the necessary conditions for a valid confidence interval for µ satisfied in this problem ?

(b) Find the 90% confidence interval for µ. Give your final answer in the form ”(lower bound, upper bound)”; not in the form ”sample mean ± margin of error”.

(c) Find the 98% confidence interval for µ. Give your final answer in the form ”(lower bound, upper bound)”; not in the form ”sample mean ± margin of error”.

5. During routine screening, a doctor notices that 22% of her adult patients show higher than normal levels of glucose in their blood a possible warning signal for diabetes. Hearing this, some medical researchers decide to conduct a large-scale study, hoping to estimate the proportion to within 4% with 98% confidence. How many randomly selected adults must they test?

For Problems 6 and 7, simply circle the correct answer choice. Do not show any work.

6. Which of the following statements are true ?

(i) if we increase the sample size (while keeping the margin of error constant), the confidence level in- creases (ii) if we increase the sample size (while keeping the margin of error constant), the confidence level decreases (iii) if we increase the sample size (while keeping the confidence level constant), the width of the confidence interval increases (iv) if we increase the sample size (while keeping the confidence level constant), the width of the confidence interval decreases

(A) (ii) and (iv) (B) (i) and (iii)

7. Explain what the phrase ”99% confident” means when we interpret a 99% confidence interval for µ.

(A) The probability that the population mean falls in the calculated interval is 0.99.

(B) 99% of the observations in the population fall within the bounds of the calculated interval.

(D) The probability that the given interval captures the sample mean is 0.99.

8. A pharmaceutical company conducts a clinical trial to see if more patients who take a new drug experience headache relief than the 40% who claimed relief after taking the standard drug. Let p be the proportion of all people who would feel headache relief after taking the new drug.

(a) What are the hypotheses ? Write them in terms of p and also in words (sentences).

H0 : p Ha : p

The null hypothesis is that

The alternative hypothesis is that

(b) In this context, what is a Type I error ?

For Problems 9 and 10, use the calculator only as a computational tool. In other words DO NOT use the 1-PropZTest or Z-Test functions in the calculator to solve the entire problem (you can, of course, use them to double check your work). Any time you have to compute a test statistic, write the formula that you are using with the appropriate values from the problem. Any time you have to find a p-value, draw a graph and shade the appropriate region.

9. The placebo effect describes the phenomenon of improvement in the condition of a patient taking a placebo – a pill that looks and tastes real but contains no medically active chemicals. Physicians at a clinic in La Jolla, California, gave what they thought were drugs to a random sample of 1, 000 asthma, ulcer, and herpes patients. Although the doctors later learned that the drugs were really placebos, 55% of the patients reported an improved condition. If the placebo was ineffective, the probability of a patient’s condition im- proving would be 0.5. We would like to determine if there is enough evidence to conclude that the placebo is effective. We want to test H0 : p = 0.5 against Ha : p > 0.5.

(a) Are the necessary conditions for a valid test of hypothesis for p satisfied in this problem ? Explain.

(b) Find the value of the test statistic.

(d) State the appropriate conclusion at 5% level of significance.

(e) Suppose that in fact, the doctors in that clinic were simply lucky and that in reality only 50% of patients who take a placebo benefit from it. What can you say about the decision you made in part (d) ? In other words, was it a correct decision, a Type I error or a Type II error ?

10. The Lincoln Tunnel (under the Hudson River) connects suburban New Jersey to midtown Manhattan. On Mondays at 8 : 30 A.M., the mean number of cars waiting in line to pay the Lincoln Tunnel toll is 1, 220. Because of the substantial wait during rush hour, the Port Authority of New York and New Jersey is considering raising the amount of the toll between 7 : 30 and 8 : 30 A.M. to encourage more drivers to use the tunnel at an earlier or later time. Suppose the Port Authority experiments with peak-hour pricing for a year, increasing the toll from $4 to $7 during the rush hour peak. On 81 randomly selected different workdays (during the period with the higher toll) at 8 : 30 A.M. aerial photographs of the tunnel queues are taken and the number of vehicles counted. The average and the standard deviation of the number of vehicles in line for those 81 days are 1184 and 160.68 respectively. We would like to determine whether peak-hour pricing succeeded in reducing the average number of vehicles attempting to use the Lincoln Tunnel. We want to test H0 : µ = 1220 against Ha : µ < 1220.

(a) Are the necessary conditions for a valid test of hypothesis for µ satisfied in this problem ? Explain.

(b) Find the value of the test statistic.

(d) State the appropriate conclusion at 1% level of significance.

(e) Suppose that it turns out the true mean is actually µ = 1200 (So the number of cars did decrease). What can you say about the decision you made in part (d) ? In other words, was it a correct decision, a Type I error or a Type II error ?

11. Which of the following statements are true ? In a testing problem if the null hypothesis is

(i) rejected at 5% significance level, then it will also be rejected at 1% significance level. (ii) rejected at 1% significance level, then it will also be rejected at 5% significance level. (iii )accepted at 5% significance level, then it will also be accepted 1% significance level. (iv) accepted at 1% significance level, then it will also be accepted at 5% significance level.

(A) (i) and (iii) (B) (i) and (iv) (C) (ii) and (iii) (D) (ii) and (iv)

12. The Pew Research Center recently polled 1048 U.S. drivers and found that 728 enjoyed driving their automobiles.

(a) What is the sample proportion of drivers who enjoy driving their automobiles ?

(b) What is the standard error of this proportion ?

(c) Based on this sample, calculate a 98% confidence interval for the proportion of drivers who enjoyed driving their automobiles. Clearly write down the formula for the confidence interval and show how you found the z-value by drawing a graph and shading the appropriate region.

(d) Next test the validity of a television ad that was placed by an automobile manufacturer: it claimed that 85% of the people owning that manufacturer’s automobiles enjoyed driving them. What are the appropriate hypotheses ?

(e) What is the value of the test statistic ?

(f) What is the p-value ? Draw a graph and shade the area of interest to answer this question.

(g) What is your conclusion ?