STA 6166 UNIT 3 Section 3 Exercises
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# Unit 3 Section 3 Exercises

You can choose to work some or all of the problems listed below. We recommend that you at least work the problems listed in your major area of interest. Answers to these exercises can be found here. (ANSWERS).

General Questions.
1. The Binomial distribution has the form:
2. Using this equation, compute the following probabilities.

1. P(y=3) given n=20 and p=.5
2. P(y<4) given n=10 and p=.75
3. P(y>1) given n=5 and p=.1
3. Under what conditions can the formula be used to express a confidence interval for p.
4. Using the equations on page 472-473 for the confidence interval for a proportion, compute 95% confidence intervals for p for the following:
1. y=5, n=20 (that is, five successes in 20 trials)
2. y=1, n=10
3. y=0, n=75
4. y=35, n=35
5. Microsoft Corporation released its Windows XP operating system in November of 2001. You have been asked to design a survey to determine how many Windows 2000 users have already switched to Windows XP. Microsoft, being optimistic, suspects the proportion is as high as 20%. You want to be certain of the estimated proportion to within ±0.02 (i.e. ±2%). Determine the sample size needed for this survey using 0.2 as the guess for p. Redo the sample size determination using 0.5 (a worst case scenario for p). [equation on page 474].
6. The sensitivity, specificity and predictive power of a diagnostic test for a disease are defined as follows:
• Sensitivity is the probability that the test will give a positive result (indicating the presence of the disease) in a subject who has the disease.
• Specificity is the probability that the test will give a negative result (indicating the absence of the disease) in a subject who does not have the disease.
• Predictive power is the probability that the test will make the correct diagnosis.

Back in Exercise 4.28 we were presented with a table describing the results of a radiological determination test for presence of Appendicitis. Suppose this study had been redone with 220 patients suspected of having appendicitis being subjected to the radiological determination test (CAT scan-the experimental approach) as well as to a clinical assessment (the expensive but definitive answer). Only two posible choices are available for each method, a decision of definitely apendicitis (DA) or definitely not appendicitis (DNA). The result of this study is in the table below.

 Clinical Assessment Radiologic Determination Confirmed (DA) Ruled Out (DNA) Definitely Appendicitis (DA) 120 7 Def. Not Appendicitis (DNA) 10 83 Total 130 90

Using these data do the following:

1. Estimate the sensitivity, specificity and predictive power of the diagnostic test.
2. Construct 95% confidence intervals for the parameters estimated in (a) and interpret them.
3. Perform a hypothesis test to verify the claim that the radiologic determination will detect more than 85% of the cases who have the disease.
4. Construct a 95% lower confidence bound to the predictive power of the test and interpret it.
For students in agriculture and environmental fields.

1. An experiment is run to examine two different methods methods of preserving onions. The researcher proposes to compare Method A to Method B, by storing a large number of onions using each method and after 6 months to classifying each onion as 'good' or 'defective'. The result of the experiment is as below:

 Experiment Result Method A Method B Good 134 207 Defective 16 33 Total number 150 240

a. Use these data to test whether the proportion of GOOD onions in each method are different from each other. Use a=.05.

b. Calculate a 95% confidence interval for the difference tested above.

2. A second aspect of the study was to examine the effects of different fertilizer treatments on the incidence of blackleg (Bacterium phytotherum) on potato seedlings. Seedlings were randomly assigned to a fertilizer treatment and after three weeks classified as to whether it was contaminated by blackleg or free. The results for four treatment were as follows:

 Observed frequencies Blackleg Total 1.No fertilizer 16 101 2.Nitrogen only 10 95 3.Dung only 4 113 4.Nitrogen and dung 14 141

Use these data to determine if the presence of Blackleg is independent of the Fertilization method. Use a=.05.

For students in engineering fields.

1. A project manager for an engineering firm submitts a bid for engineering design for two projects. The firm formally evaluates the chances that each bid will be accepted by surveying its engineers. The following table summarizes the collective assessment from this survey.

 Results of bids Project A Project B Accepted 28 25 Refused 29 24 Total number of bits submitted 57 49

a. Test whether the chances that each bid was accepted are different. Use a=.05.

b. Calculate a 95% confidence interval for the difference of chances that each bid was accepted.

2. An study plans to assess the impact of several factors involving the heat treatment of leaf springs. In this process, a conveyor system transports leaf spring assemblies through a high-temperature furnace. After heating, a high-pressure press induces curvature in the medal. Once the spring leaves the press, an oil quench cools it to near ambient temperature. An important quality characteristic of this process is the resulting free height of the spring. The researchers are interested in examining four factors known to affect this free height. For each spring tested, the final classification is whether the spring is below the specified minimal spring height. Results of numerous tests are given in the following table.

 Factors Low level Total 1.High heat temp. 14 32 2.heating time 23 48 3.Transfer time 10 22 4.Hold down time 2 5

Use these data to run a Chi-square test to determine whether a finding of low level is independent of the factors used. Use a=.05.

For students in toxicology and health science fields.

1. In a study of children aged 0 to15, concern is focused on the presence or absence of the carrier for Streptococcus Pyogenesis and the relationship between the presence of the carrier and tonsil size.

 Tonsils present. Not enlarged Tonsils enlarged Presence of the carrier 15 19 Absence of the carrier 380 360 Total number 395 379

a. Is the presence of the carrier related to whether tonsils are enlarged or not. Test this hypothesis at the P(Type I error)=a=.05 level.

b. Calculate a 95% confidence interval for the difference between the Tonsils enlarged probabilities in the presence of the carrier versus that in the absence of the carrier.

2. In a controlled clinical trial to determine the efficiency of an experimental drug for treating migraine headache, patients were divided into four groups and were treated with four drugs: A, B, C and D. Each treatment lasted 12 weeks. At the end of the treatment period, the effect of the drug was classified into two categories:  effective and not effective. The migraine headache data obtained are as follows:

 Drug Effective Total A 26 45 B 14 28 C 10 17 D 22 90

Use these data to run a chi-square test to determine whether the effectiveness of all drugs are the same.This is equivalent to testing that all effectiveness proportions are the same.

For students in community development, education and social services fields. 1. A company uses a pre-employment test to screen applicants for sales jobs. We are interested in whether the screning test is effective. In an experiment, a random samples of applicants who pass the test and a second sample of those who do not pass the screening test are all employed. The number of employees who successfully completed the training program in these two samples was recorded as follows.

 Experiment Result Applicant who pass the pre-employment test Applicant who do not pass the pre-employment test Employees completed in the training program 67 15 Employees failed in the training program 28 22 Total number of Employees 95 37

a. Test whether the proportions of employee who successfully completed the training program is the same for the group who initially passed the pre-employment test to those who did not initially pass the pre-emplyment test. For this test use P(Type I error)=a=.05.

b.   Calculate a 95% confidence interval for the above difference.

2.A survey was made of 164 customers in a department store. These customers were divided into four groups with members of each group viewing only one of the four types of advertisements available for a certain product. The total numbers of customers who made purchases by viewing group are given below.