Name:                      

                                                                                           FINAL EXAM

Use only the space provided to answer the following questions.  Whenever possible, show your work for potential partial credit.  You may use the extra sheets of paper for additional space.  NOTE:  When performing numerical calculations, keep at least 4 decimals.  (i.e., do NOT round .2265 to .227 or .23)

1.  In many universities, students evaluate their professors by means of answering a questionnaire.  Assume a questionnaire is distributed to a class of 45 students.  Students are asked to answer the following questions:

  • Sex (M,F)
  • Age
  • Number of Hours Completed
  • Grade Point Average
  • My instructor is a very effective teacher

1

2

3

4

5

Strongly Agree

Moderately Agree

Neutral

Moderately Disagree

Strongly Disagree

a.  How many elements are in the above data set?  (3 points)

 

 

b.  How many variables are in this data set? (3 points)

 

 

c.  Which variables are qualitative and which are quantitative?  How can you tell? (5 points)

 

 

 

 

 

2.  Distinguish between descriptive statistics and statistical inference.  Use an example to show how the two concepts may differ.  (16 points)

 

 

 

 

 

 

3.  A sample of employees is taken from a large manufacturing company.  The data below represents the number of hours of sick leave that these employees have taken in the first quarter of the year (rounded to the nearest hour).  Use this data to develop a frequency and relative frequency distribution.  Use statistical inference to make two comments on sick leave for this firm.  (15 points)

19

22

27

24

28

12

23

47

11

55

25

42

36

25

34

16

45

49

12

20

28

29

21

10

59

39

48

32

40

31

 

4.  The price of gold at the end of each month of the year 2000 is shown below.

Month

Price per Ounce

January

a.  Determine the mean price of gold for the year.  Fully explain its meaning.  (5 points)
 
$225

February

230

March

225

April

236

May

270

June

382

July

322

August

324

September

320

October

310

November

368

December

388

 

b.   Determine the median price of gold for the year.  Fully explain its meaning. (5 points)

 

 

 

 

 

 

c.  Determine the mode price of gold for the year.  Fully explain its meaning. (4 points)

 

 

 

 

 

d.  Compute the 30th and the 80th percentiles. Fully explain their meanings. (8 points)

 

 

 

 

e.  The sample standard deviation of these gold prices is $64.64.  Interpret this value. (5 points)

 

 

 

 

 

 

 

5.  Explain the difference between the covariance and the correlation between two variables.  What does each of these measures tell us about the relationship between two variables?  (15 points)

6.  You have just applied to two universities (A and B) to pursue your graduate work.  In the past, students with your credentials were accepted at University A; while University B accepts 35% of applicants with your credentials. 

a.  Assume that acceptance at the two universities are independent events.  Explain the meaning of this assumption.  (8 points)

 

 

 

 

 

 

 

b.  What is the probability that you will be accepted in both universities?  (4 points)

 

 

 

c.  What is the probability that you will be accepted to at least one graduate program?  (5 points)

 

 

 

 

7.  Betting on the color red in a standard game of roulette results in an 18/38 chance of winning.  Suppose your high roller grandma decides to play 5 consecutive games of roulette, each time putting her money ($10) on red. 

a.  What is the probability that she will win exactly one time?  (4 points)

 

 

 

b.  What is the probability that she will win at least twice?  (5 points)

 

 

 

c.  If the casino pays 3 times the bet to all winners, how much money will grandma expect to win or lose after 5 games? What’s your recommendation to grandma?  (11 points)

 

 

 

 

 

 

8.  A statistics professor notes that the grades of his students were normally distributed with a mean of 74 and a standard deviation of 10.  The professor has informed us that 6.3% of his students received A’s while only 2.5% of his students failed the course and received F’s.

a.  What is the minimum score needed to make an A?  (4 points)

 

 

 

 

b.  What is the maximum score among those who received an F?  (4 points)

 

 

9.  A department store has determined that that 25% of their sales are credit sales.  A random sample of 60 sales is selected.

a.  Describe the sampling distribution of the sample proportion.  (6 points)

 

 

 

 

 

 

b.  What is the probability that the sample proportion will be between .20 to .30?  (5 points)

 

 

 

 

 

 

 

10.  A sample of 116 college students are asked, “In a typical week, how many hours do you study outside of class?”  The sample mean is 18.319 hours with a sample standard deviation of 8.169 hours. 

a.  Conduct a 99% confidence interval for the mean number of study hours for college students.  Interpret this interval.  (8 points)

 

 

 

 

 

b.  How many students would you need to survey to cut the margin of error in half?  Would you suggest that the surveyors do this?  Explain.  (10 points)

 

 

 

 

 

 

11.  You are tired of hearing that the Dean of Admissions at Hanover College has been successful at significantly increasing the average SAT scores of incoming first year students.  You randomly sample students from two of the most recent incoming classes and calculate point estimators in the table below.  Have incoming SAT scores significantly increased?  Set up and conduct the hypothesis test and then explain your results to the Dean of Admissions.  (12 points)

Class of 2003

Class of 2006

Sample size = 16

Sample size = 43

Sample mean SAT = 1171.25

Sample mean SAT = 1201.7442

Sample variance SAT = 11,425

Sample variance SAT = 12,593

 

 

 

 

 

 

 

 

 

12.  The percentage of students who enroll at a college or university and actually graduate is an important statistic for university administrators.  Some of the factors related to the graduation rate include the percentage of classes with fewer than 20 students, the percentage of classes with more than 50 students, the student-faculty ratio, the percentage of students who apply to the university and are admitted, the percentage of first-year students in the top 10% of their high school class, and the academic reputation of the university.  Data for 48 national universities was collected.

 

  • Graduation rate is your dependent variable.
  • % of Classes Under 20:  the percentage of classes with fewer than 20 students.
  • % of Classes of 50 or more: the percentage of classes with more than 50 students.
  • Student-Faculty Ratio:  The ratio of the number of students enrolled divided by the total number of faculty.
  • Acceptance Rate:  The percentage of students who apply and are accepted.
  • 1st-Year Students in Top:  The percentage of students admitted who were in the top 10% of their high school class.
  • Academic Reputation Score:  A measure of the school’s reputation determined by surveying administrators at other universities.  Measured on a scale from 1 (marginal) to 5 (distinguished)

 

a.  Interpret each of the estimated coefficients.  Do they make economic sense?  Explain.  (10 points)

b.  Discuss how well the model fits the data.  (10 points)

c.  Discuss statistical significance, being sure to explain the appropriate hypothesis tests. (10 points)

SUMMARY OUTPUT

 

 

 

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

Multiple R

0.874554829

 

 

 

 

R Square

0.764846149

 

 

 

 

Adjusted R Square

0.730433391

 

 

 

 

Standard Error

4.468824563

 

 

 

 

Observations

48

 

 

 

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

df

SS

MS

F

Significance F

Regression

6

2663.130555

443.8550924

22.2256564

1.88742E-11

Residual

41

818.786112

19.97039298

 

 

Total

47

3481.916667

 

 

 

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

 

Intercept

84.9080963

11.67611487

7.271947665

6.82689E-09

 

% of Classes Under 20

-0.169549258

0.096720653

-1.752978838

0.087080638

 

% of Classes of 50 or More

-0.504289007

0.19611488

-2.571395942

0.013853439

 

Student/Faculty Ratio

-0.264338381

0.260773307

-1.013671163

0.316684596

 

Acceptance Rate

-0.20872764

0.055833427

-3.738399257

0.000566225

 

1st-Year Students in Top 10%

of HS Class

0.129942773

0.058095898

2.236694448

0.030807058

 

Academic Reputation Score

3.936719582

1.92178851

2.048466604

0.046953102