Final Exam; Stat. 4033

For each problem on this exam clearly mark your final answers. If you wish to receive partial credit, show all your work, give all relevant formulas, and justify your answers completely. Also for all tests (unless otherwise specified) state the appropriate hypotheses, give the test statistic (if possible), state the p-value, and state the conclusion in terms of the original problem. Unless otherwise stated use α = 0.05 for all tests.

1.  We want to predict the per capita tax for a state from its per capita debt. A random sample of states is taken and the data are:

It may be helpful to know that SSXX  = 1067630, SSYY  = 185866.8, SSXY  = 230565, Σxi  = 6545, and Σyi  = 8416

a)  (7 pts.) Give the least squares line and interpret the coefficients in terms of the original problem.

b)  (5 pts.) Calculate the coefficient of determination and interpret its value in terms of the original problem.

c)  (2 pts.) Predict the per capita tax for a state with a per capita debt of 1200.

2. The amount of laundry a family of four washes in a year is normally distributed with a mean of 2000 lbs. and a standard deviation of 187.5 lbs.

a)  (7 pts.) Find the probability a family of four washes between 1800 and 2300 pounds in a year.

b)  (8 pts.) A random sample of 50 families is taken. What is the probability the average amount of laundry washed in a year is greater than 1985 pounds?

3.  ( 10 pts.)  A random sample of 20 homes in Whiting, Indiana is taken and the mean age of the homes is 62.1 years with a standard deviation of 5.4 years.  A random sample of 20 homes in Franklin, Pennsylvania is taken and the mean age of the homes is 55.6 years with a standard deviation of 3.9 years.  The age of the homes is approximately normally distributed for both locations and the variances are approximately equal.  Find a 90% confidence interval for the mean difference of the age of the homes at the two locations.   Also interpret your results in terms of the original problem.

4.    (8 pts.) We want to estimate the proportion of homes that have a direct satellite television receiver to within 3 percentage points with a 95% confidence interval. How large a sample is needed?

5.  We want to prove the proportion of women with high blood pressure is greater than the proportion of men who have high blood pressure. A sample of 150 women had 52  with high blood pressure, and a random sample of 150 men had 43 with high blood pressure. Use a = 0.05.

a)  (3 pts.) State the appropriate hypotheses.

b)  (4 pts.) State and calculate the appropriate test statistic.

c)  (4 pts.) Find or bound the p-value.

d)  (5 pts.) State the conclusion in terms of the original problem.

6.  (8 pts.) A random sample of 24 students from Oak Park College gives a standard deviation for the age of the students of 2.3. The ages of the students are approximately normally distributed. Find a 98% confidence interval for the variance of the student ages.

7.  ( 10 pts.) We want to estimate the difference in the mean test scores for students that  watched a DVD based on a novel and those that read the novel. A random sample of 7   students was taken and they each took a test over one novel where they read the text and a separate one where they watched the DVD.  The test scores are approximately normal for those that watched DVDs and for those that read the novel. The data is given below. Find a 95% confidence interval for the difference in the mean test score for those that    watched the DVD and those that read the novel. Interpret your results in terms of the original problem.

8.  We want to predict the height of a building from the number of stories of the building. Utilize the attached Excel output, which contains the data, to answer the following questions.

a)  (8 pts.) Give the least squares line and interpret the coefficients in terms of the original problem.

b)  (10 pts.) Perform a test to see if the height of the building increases as the number of stories increases by one. Use a = 0.05.

c)  (6 pts.) Find a 90% confidence interval for the change in the average height of the building when the number of stories increases by 1.

9.  The U.S. Bureau of Labor and Statistics states that individuals between 18 and 34 have had an average of 9.2 jobs. We want to prove that the Bureau is wrong. The number ofjobs for individuals between 18 and 34 is approximately normally distributed. A          random sample of individuals between 18 and 34 is taken and the number ofjobs is recorded. The data are: 8, 12, 15, 6, 1, 9, 13, 2. Use a = 0.01.

a)  ( 15 pts.) Calculate the sample mean and sample standard deviation for the data above.

b)  (3 pts.) State the appropriate hypotheses.

c)  (4 pts.) State and calculate the appropriate test statistic.

d)  (4 pts.) State the rejection region.

e)  (5 pts.) State the conclusion in terms of the original problem.

10. The proportion of people who respond to a particular mail-order solicitation has the following distribution

fX (x) = | , 0 ≤ x ≤ 1

b) (6 pts.)  Find the mean percentage of people who respond.

c)  (8 pts.)  Find the variance of the percentage of people that respond.

11.  We are interested in predicting a college freshman’s GPA for their first semester from their overall ACT test score using a linear relationship.

a)   (3 pts.) Do we need to consider anything before we find the least squares line? If so, what and why? If not, why not?

b)  (3 pts.) Once we have calculated the least squares line we want to utilize it to predict a student’s first semester GPA. Are there any concerns with utilizing this line for prediction or can we utilize this line for any values of ACT test scores? Explain your reasoning completely.

c)   (4 pts.) Now suppose we wish to estimate the slope of the least squares line utilizing a 95% confidence interval. Are any assumptions necessary? If so, what?  If not, why not?

12.  Suppose that 1 out of every 5 marriages began as an online relationship.

(a) (5 pts.) Find the probability that at least 2 of 8 marriages began as an online relationship.

(b) (6 pts.) Find the probability that exactly 6 out of 30 marriages began as an online relationship.

(c) (7 pts.) Find the mean and variance of the number of marriages out of 15 that began as online relationships.