Assignment 3 | Complete Solution

Assignment 3

1. The average expenditure on Valentine’s Day was expected to be $100.89 (USA Today, February 13, 2006). Do male and female consumers differ in the amounts they spend? The average expenditure in a sample survey of 40 male consumers was $135.67, and the average expenditure in a sample survey of 30 female consumers was $68.64. Based on past surveys, the population standard deviation for male consumers is assumed to be $35, and the population standard deviation for female consumers is assumed to be $20. 
a. What is the point estimate of the difference between the population mean expenditure for males and the population mean expenditure for females? 
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
b.    At 99% confidence, what is the margin of error? 

__________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________
c.    Develop a 99% confidence interval for the difference between the two population means. 

__________________________________________________________________________________________________________________________________________________________________________
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

2. During the 2003 season, Major League Baseball took steps to speed up the play of baseball games in order to maintain fan interest (CNN Headline News, September 30, 2003). The following results come from a sample of 60 games played during the summer of 2002 and a sample of 50 games played during the summer of 2003. The sample mean shows the mean duration of the games included in each sample. 
2002 Season    2003 Season
n1=60    n2=50
 2 hours, 52 minutes
 2 hours, 46 minutes

a. A research hypothesis was that the steps taken during the 2003 season would reduce the population mean duration of baseball games. Formulate the null and alternative hypotheses. (fill in the blanks below and define the notations of parameters you use for the two years). 

H0 = ____________
Ha = ____________
Definition of notations:

__________________________________________________________________________________________________________________________________________________________________________
b. What is the point estimate of the reduction in the mean duration of games during the 2003 season? 

__________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________
c. Historical data indicate a population standard deviation of 12 minutes is a reasonable assumption for both years. Conduct the hypothesis test to determine whether the steps taken during the 2003 season would reduce the population mean duration of baseball games. What is the test statistic? 

(1) Write down the formula calculating the test statistic: _______________________________________
(2) Start your calculation here: ____________________________________________________________
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
d. What is the p-value of the test? (If you use Excel, provide the function you use) 
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
e. What is the critical value based on α = 0.05? (If you use Excel, provide the function you use) 
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
f. Use α = 0.05.  Do you reject or not reject H0? What is the rejection rule you use? What is your conclusion. 
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
_____________________________________________________________________________________
g.    Develop a 95% confidence interval for the difference between the two population mean duration of the games. 

_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
_____________________________________________________________________________________
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
_____________________________________________________________________________________

 
3. FedEx and United Parcel Service (UPS) are the world’s two leading cargo carriers by volume and revenue (The Wall Street Journal, January 27, 2004). According to the Airports Council International, the Memphis International Airport (FedEx) and the Louisville International Airport (UPS) are 2 of the 10 largest cargo airports in the world. The following random samples show the tons of cargo per day handled by these airports. Data are in thousands of tons. 
Memphis    Louisville
9.1    4.7
15.1    5.0
8.8    4.2
10.0    3.3
7.5    5.5
10.5    2.2
8.3    4.1
9.1    2.6
6.0    3.4
5.8    7.0
12.1    
9.3    
a. Formulate the null and alternative hypotheses that can be used to determine whether the tons of cargo per day handled by these airports are the same. (fill in the blanks below). 

H0 = ____________
Ha = ____________
b. What is the test statistic? 
(1) Write down the formula calculating the test statistic: _______________________________________
(2) Start your calculation here: ____________________________________________________________
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
c. What is the p-value? (If you use Excel, provide the function you use) 
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

d. What is the critical value based on α = 0.05? (If you use Excel, provide the function you use) 
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
e. Use α = 0.05.  Do you reject or not reject H0? What is the rejection rule you use? What is your conclusion. 
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

4. A market research firm used a sample of individuals to rate the purchase potential of a particular product before and after the individuals saw a new television commercial about the product. The purchase potential ratings were based on a 0 to 10 scale, with higher values indicating a higher purchase potential. You are asked to test whether the commercial improved the mean purchase potential rating (the mean rating “after” would be greater than the mean rating “before.”) Use .05 as the level of significance and the following data to test the hypothesis. 
    Purchase Rating
Individual    After    Before
1    6    5
2    6    4
3    7    7
4    4    3
5    3    5
6    9    8
7    7    5
8    6    6

a. Formulate the null and alternative hypotheses that can be used to determine whether the commercial improved the mean purchase potential rating. (fill in the blanks below). 

H0 = ____________
Ha = ____________

b. What is the test statistic? 
(1) Write down the formula calculating the test statistic: _______________________________________
(2) Start your calculation here: ____________________________________________________________
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
c. What is the p-value? (If you use Excel, provide the function you use) 
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
d. What is the critical value based on α = 0.05? (If you use Excel, provide the function you use) 
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
e. Use α = 0.05.  Do you reject or not reject H0? What is the rejection rule you use? What is your conclusion. 
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

 
5. To study the effect of temperature on yield in a chemical process, five batches were produced at each of three temperature levels. The results follow. Construct an analysis of variance table. Use α = .05 level of significance to test whether the temperature level has an effect on the mean yield of the process. 
50°C    60°C    70°C
34    30    23
24    31    28
36    34    28
39    23    30
32    27    31

a.  Formulate the null and alternative hypotheses that can be used to determine whether the temperature level has an effect on the mean yield of the process. (fill in the blanks below). 

H0 = ____________
Ha = ____________

b. What is the test statistic? 
(1) Write down the formula calculating the test statistic: _______________________________________
(2) Start your calculation here (if you use Excel, provide the steps you follow): ____________________
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
__________________________________________________________________________________________________________________________________________________________________________
c. Complete the following ANOVA table. 
Source of Variation    Sum of Squares    Degrees of Freedom    Mean Square    F
Treatments                
Error                ---
Total            ---    ---

d. What is the p-value? (If you use Excel, provide the function you use) 
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

e. Use α = 0.05.  Do you reject or not reject H0? What is your conclusion? 
(1) Write down the rejection rule: _________________________________________________________
(2) Write down your decision: ____________________________________________________________
(3) Interpret your result: _________________________________________________________________
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
_____________________________________________________________________________________

 
Formulas:
•     
•     
•     
•     
•     
•     
•     
•    F = MSTR /MSE
•     
•    MSTR = SSTR /(k - 1)
•     
•    MSE = SSE /(nT - k)
•    SST=SSTR+SSE
•     , nT = n1 + n2 + … + nk
Excel Functions:
•    NORMSDIST
•    NORMSINV
•    NORMDIST
•    NORMINV
•    TDIST
•    TINV
•    FDIST
•    FINV

Useful Values:
Confidence Level    α    α/2    zα/2
90%    0.1    0.05    1.645
95%    0.05    0.025    1.960
99%    0.01    0.005    2.576

ANOVA Table
Source of Variation    Sum of Squares    Degrees of Freedom    Mean Square    F
Treatments    SSTR    k - 1    MSTR    MSTR/MSE
Error    SSE    nT  - k    MSE    ---
Total    SST    nT - 1    ---    ---