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APSC 3115 Final Exam Question 1

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APSC 3115 Final Exam
Question 1. Consider the iid random variables Y?, Y?, and Y?, with  mean μ and variance σ². Suppose we have the following estimators for μ:

 θ? = (8Y? + 5Y? + 9Y?)/22 
 θ? = (9Y? + 8Y? + 3Y?)/20
 
(a) (3 points) Verify if both estimators are unbiased for μ.
(b) (3 points) What is the variance of each estimator?
(c) (4 points) Which estimator is better and why?

Question 2. A manufacturer produces shafts for an airplane engine. The shafts wear after 1,000 hours on flight is of interest, because the wear can affect the airplane performance. A random sample of 11 shafts is tested, and the sample mean wear is found to be 2.34. It is known that σ = 0.7 and that the wear is normally distributed. To answer the questions below, use α = 0.01.

(a) (5 points) Test H?: μ = 2.0 versus H?: μ ≠ 2.0. 
(b) (5 points) What is the power of the test in part a if  μ = 2.15? 
(c) (5 points) What sample size would be required to detect a true mean of 2.55 if we wanted the power of the test to be at least 0.99?
Question 3. Consider a research that tried to evaluate whether the upper limit of the normal human body temperature (98.6 degrees Fahrenheit) was adequate. The research team collected data on body temperature, gender, and heart rate for a several subjects. The body temperature for 16.0 female subjects is as follows:
[ 98.6  98.5  98.9  98.   99.1  98.7  98.7  98.6  98.3  98.   98.4  98.2
  97.7  98.5  98.2  98.7].
Assume that the body temperature for female subjects is normally distributed.

(a) (5 points) Test the hypothesis H?: μ = 98.6 versus H?: μ ≠ 98.6, using α = 0.1.
(b) (5 points) Explain how the question in part (a) could be answered by  constructing a two-sided confidence interval on the mean female body temperature.

Question 4. A poll of the 2012 presidential election for the state of Georgia had 2000 respondents, of which 733.0 had a college degree. Among the college graduates, 440.0 voted for Barack Obama, and 293.0 voted for Mitt Romney. For all items below, use method 1 to estimate confidence intervals for the sample proportion.

(a) (5 points) Calculate a 90.0% confidence interval for the proportion of college graduates in Georgia that voted for Barack Obama.
(b) (5 points) Calculate a 95.0% upper confidence bound for the proportion of college graduates that voted for Mitt Romney.

Question 5. A cosmetic manufacturer is formulating a new shampoo and is interested in foam height (in millimeters). The foam height is approximately normally distributed, with standard deviation 20 millimeters. The company wishes to test  H?: μ = 180.0 versus H?: μ > 180.0, using the results of n = 18.0 samples.

(a) (5 points) Find the type I error probability α if the critical region lower bound is 189.0.
(b) (5 points) Given the critical region in part a, what is the probability of type II error if the true mean foam height is 189.0? 
(c) (5 points) Find β when the true mean is 189.0.

Question 6. The maximum compressive strength of a certain type of cement is being tested. The following 12.0 specimens were collected by a civil engineer:
[ 2274.  2294.  2201.  2227.  2277.  2284.  2241.  2245.  2224.  2260.
  2197.  2271.]
Assume that the maximum compressive strength is normally distributed.

(a) (5 points) Construct a 90.0% two-sided confidence interval on the mean strength.
(b) (5 points) Construct a 95.0% lower confidence bound on the mean strength. Compare this bound with the lower bound of the two-sided confidence interval  and explain why they are different.

Question 7. Consider the following data set for the stream flow (1000’s of acre-feet) in a river:
 [ 178.64  142.85  195.24  107.46  133.03  118.61  209.65  189.19  261.29
  195.59  186.27  171.08  174.52  132.67  196.05  156.28  291.65  275.99
  267.27  214.09  191.22  195.71  183.81  154.87  113.71  182.35  203.58
  135.72]
It is known that the stream flow is normally distributed, but the parameters are not known. 
 
(a) (5 points) Estimate the parameters of the distribution of the stream flow. 
(b) (5 points) Calculate an estimate of the value that separates the largest  10% of all values in the stream flow distribution from the remaining 90%. 
(c) (5 points) Estimate P(X< 253.11), that is, the proportion of all stream flow values that are less than 253.11.

Question 8. A group of researchers quantified the absorption of electromagnetic energy and the thermal effects from microwaves. The results were obtained from experiments conducted on chimpanzees. The blood pressure values (mmHg) for the control group (14.0 chimpanzees) and the test group (17.0 chimpanzees)  are as follows:
 
 group 1 (control group): sample mean = 110.0, sample standard deviation = 7.0 
 group 2 (test group): sample mean = 113.0, sample standard deviation = 10.0 
 
(a) (5 points) Is there evidence to support the claim that the test group has higher mean blood pressure? State the appropriate hypotheses, use α = 0.1, and assume that both populations are normally distributed, but the variances are not equal. What is the P-value for this test? 
(b) (5 points) Calculate a confidence interval to answer the question in part (a).
(c) (5 points) How could we use the confidence interval in part (b) to check the claim that the mean blood pressure from the test group is at least 2.0 higher than the control group? State the appropriate hypothesis and use the same assumptions from part (a).