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 23 students.  Students are asked to answer the following questions:

  • How much did you learn in this course?

Exceptional amount _____                 A lot ______        Very little _____

  • Age ____
  • Class Standing (Fr, So, Jr, Sr)
  • Grade Point Average _____
  • The textbook is very effective

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.  The age of 16 employees are shown below.  Use this data to develop a frequency and relative frequency distribution.  Use statistical inference to make two comments on employee age for this firm.  (15 points)

22

40

34

36

35

27

30

32

39

46

32

48

45

36

41

41

4.  The prices of two grades of unleaded gasoline at the end of each month in 2002 are shown below.

 
Month
 
Unleaded Regular
 
Unleaded Premium
 
January
 
113.9          
132.3          
February
 
113.0          
133.0          
March
 
124.1          
145.0          
April
 
140.7          
162.2          
May
 
142.1          
162.5          
June
 
140.4          
160.6          
July
 
141.2          
160.7          
August
 
142.3          
162.0          
September
 
142.2          
161.9          
October
 
144.9          
164.3          
November
 
144.8          
164.3         

December

139.4         

158.9 

a.   Determine the mean and median price of both regular and premium unleaded gasoline for the year.  Fully explain the meaning of each. (6 points)

 

 

 

 

 

 

b.  Determine the standard deviation of both grades for the year.  Fully explain their meaning. (8 points)

 

 

 

 

 

c.  If unleaded gasoline prices were distributed normally, at what prices would you consider “outliers” for each grade?  Explain your reasoning. (6 points)

 

 

 

 

 

 

d The sample covariance between these grades of gasoline is 127.2679.  Interpret this value.  Calculate the sample correlation coefficient between these grades of gasoline.  Interpret this value.  (9 points)

 

 

 

 

 

 

6.  You have just applied to two universities (A and B) to pursue your graduate work.  In the past, 90% of students with your credentials were accepted at University A; while University B accepts 35% of applicants with your credentials. 

a.  Your advisor says that she is assuming your acceptance at the two universities are independent events.  Explain the meaning of this assumption.  (6 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?  (4 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 4 consecutive games of roulette, each time putting her money 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, and grandma always places $10 bets, how much money will grandma expect to win or lose after 4 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 population mean of 72.5 and a population standard deviation of 8.  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 30% of their sales are credit sales.  A random sample of 50 sales is selected.

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

 

 

 

 

 

 

b.  What is the probability that the sample proportion will be between .20 to .35?  (5 points)

 

 

 

 

 

 

 

10.  A sample of 121 college students are asked, “In a typical week, how many hours do you study outside of class?”  The sample mean is 19.0744 hours with a sample standard deviation of 11.5119 hours. 

a.  Conduct a 95% 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.  (8 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, but you are skeptical.  You randomly sample students from the current senior class and the current sophomore class.  You then calculate point estimators in the table below.  Have incoming SAT scores significantly changed?  Set up and conduct the hypothesis test and then explain your results to the Dean of Admissions.  (12 points)

Class of 2003

Class of 2005

Sample size = 44

Sample size = 24

Sample mean SAT = 1235

Sample mean SAT = 1211.542

Sample variance SAT = 11,987.32

Sample variance SAT = 15,707.48

 

 

 

 

 

 

 

 

 

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 (how selective the university is), 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)

 

In your analysis, please clearly separate your responses into these parts.

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)

d. What is muticollinearity?  In this model of graduation rates, do you see the potential for this problem?  How would you check for it and how might you adjust your model accordingly?  (10 points)
NOTE:  You may need to complete some missing information. (8 points)

SUMMARY OUTPUT

 

 

 

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

Multiple R

0.874554829

 

 

 

 

R Square

 

 

 

 

 

Adjusted R Square

 

 

 

 

 

Standard Error

4.468824563

 

 

 

 

Observations

48

 

 

 

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

df

SS

MS

F

Significance F

Regression

6

 

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

 

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

 

0.19611488

-2.571395942

0.013853439

 

Student/Faculty Ratio

-0.264338381

0.260773307

 

 

 

Acceptance Rate

 

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

 

1.92178851

2.048466604

0.046953102