GM 533 Statistics - Final Examination SU15

        
Data Set  Ford Mustang Automobiles        
PRICE CONVERT AGE MILES TRANS  Variables:  
12688 0 1 8200 0  PRICE = Asking price in dollars  
11500 0 1 27000 1  CONVERT = 1 if convertible and 0 if not  
11650 0 1 3000 0  AGE = age of car in years  
10995 1 2 27000 1  MILES = odometer reading in miles  
6200 0 3 55000 1  TRANS = 1 if automatic and 0 if manual  
9500 0 3 22000 0    
8700 0 3 25000 1  Source:  Practical Data Analysis 2nd, Vol I, II  
12500 0 2 23800 0  Bryant-Smith McGraw-Hill ISBN -13:978-0-69-778527-5  
10488 1 4 20000 0  For Applied Managerial Statistics GM533 DeVry University  
5988 0 4 37000 0    
7288 1 5 84000 1    
3590 0 5 55000 0    
2800 0 6 62000 0    
6995 1 7 78000 1    
4900 0 7 60000 0    
2488 0 7 67000 1    
4200 1 8 65000 1    
12990 1 1 13300 1    
14500 1 3 31000 1    
10900 1 3 4000 1    
13500 1 3 23000 0    
12000 0 3 89000 0    
8500 0 3 34000 1    
4388 0 4 80000 0    
3000 0 7 95000 0    
3900 0 8 90000 0    
3350 0 8 91000 0    
10500 0 3 17000 0    
12988 0 1 21000 1    
6500 0 3 73000 0    
12900 1 2 19000 0    
9000 0 2 32000 0    
7000 0 6 56000 0    
5900 0 4 110000 1    
2500 0 6 92000 0    
        
        
1) Use descriptive statistics to describe the price of the autos in the data set        TCO A
    assuming it is a sample.       5 points 
2) Draw a histogram of the Price of autos in the data set.        TCO A
        
3)  Compare the average price of the five autos with the lowest mileage         TCO A
to the average price of the five autos with the highest mileage.       5 points 
Use a column graph to illustrate your findings.        

4)   Stanford-Binet IQ Test scores are normally distributed with a mean score of 100        TCO B
 and a standard deviation of 16.        
       25 points  
a) Find the probability that a randomly selected person has an IQ score over 133         
         
         
b) Find the probability that a randomly selected person has an IQ score under 86         
         
c) Find the probability that a randomly selected person has an IQ score between 72 and 128         
         
d) Find the probability a randomly selected person has an IQ within 1.5 standard deviations of the mean.         
         
e) Find the lowest IQ score for a person in the upper 6% of the Stanford-Binet IQ Test score distribution.         

Use the following Data Set A to answer questions 5, 6, and 7.          
          
 Data Set A         
 5         
 8         
 12         
 3         
 14         
 2         
 4         
 7         
          
5)  What is the mean of the data in set A?          
 a) 1.52      5 points  TCO A 
 b) 6         
 c) 6.87         
 d) 55         
          
          
          
          
6)  What is the range of the data in Data Set A?       5 points  TCO A 
 a) The distance to the target         
 b) The difference of the maximum and minimum observations         
 c) The average squared deviation from the mean         
 d) kurtosis divided by skewness         
          
7)  Why can’t the Mode be calculated for Data Set A?       5 points  TCO A 
 a) No observation occurs more than once         
 b) The Mean is greater than the Median         
 c) The number of observations is not 30 or more         
 d) EXCEL doesn’t work on small data sets.         
          
8) Which graph best shows the relationship between the Year and the Price Index (CPI)?          TCO A
 Choose the best answer  a), b), c), d)        5 points   
          
 a)    b)    
          
 c)    d)     
          
9)  Which of the following is a probability distribution?       5 points  TCO B 
          
a)   b)       
X P(X)  X P(X)      
Green 0.5  5 0.1      
Red 0.3  12.5 0.5      
Blue 0.1  26.2 0.4      
          
c)   d)       
X P(X)  X P(X)      
Sam 0.2  0.4 0.1      
Ned 0.5  0.4 0.5      
Sally 0.4  0.2 0.3 

10)   Tires made by the Schlock Company have mean life of 50,000 miles, Standard  Deviation        
 of 4,200 miles.  What mileage should the company guarantee so that no more        
 than 3% fail to make the guaranteed mileage if the distribution is normal?       10 points

11) Using the "Ford Mustang Data Set" to represent a random sample, determine a 95%          TCO C
 confidence interval for the mean price of Ford Mustangs.       10 points  
          
12) Using the "Ford Mustang Data Set" to represent a random sample, determine a 90%          TCO C
 confidence interval for the mean value of odometer mileage.(MILES)       10 points  
     
13) The Internal Revenue Service has stated that historically 85% of federal tax returns         TCO B
 filed are free of arithmetic errors.  A random sample of 25 returns is selected and          
 checked carefully for arithmetic errors.  Assuming independence, find the probability that:         
  hint:  Binomial problem      20 points  
          
 a. all 25 returns are free of arithmetic errors.         
 b. at most 23 returns are free of arithmetic errors.         
 c. more than 17 are free of arithmetic errors.         
14) A new brand of laptop battery is produced by ZELCON. The company claims that the          TCO B
 battery will last for an extended period of time before a recharge is necessary. A sample          
 of 49 batteries is tested for the length of usage time to recharge. The sample results are          
 as follows:       20 points  
          
 Sample Size = 49         
 Sample Mean = 6.4 hrs.         
 Sample Standard Deviation = 1.3 hrs.          
          
 a. Construct a 95% confidence interval for the average length of usage time to recharge.         TCO C
 b. Interpret this interval.         
 c. How large a sample size will need to be selected if we wish to have a 95% confidence          
     interval with a margin for error of .25 hours?          

15) A marketing research firm wishes to estimate the proportion of adults who          TCO C
 are planning to buy a new car in the next 6 months. A simple random sample of 100          
 adults led to 25 who were planning to buy a new car in the next 6 months.       15 points  
          
 a. Compute the 95% confidence interval for the proportion of adults who are planning to          
 buy a new car in the next 6 months.         
          
 b. Interpret this confidence interval.         
          
 c. How large a sample size will need to be selected if we wish to have a 95% confidence         
  interval that is accurate to within 2%.          

16. Bill Smith is the Worthington Township manager. When citizens request a traffic light, the           
staff assesses the traffic flow at the requested intersection. Township policy requires the installation of           
a traffic light when an intersection averages more than 150 vehicles per hour. A random sample of 50          
 vehicle counts is done. The results are as follows:        40 points  
          
 Sample Size = 50         
 Sample Mean = 159.3 vehicles/hr.         
 Sample Standard Deviation = 27.5 vehicles/hr.         
          
Does the sample data provide evidence to conclude that the installation of the traffic light is warranted          
 (using α= .10)?  Use the hypothesis testing procedure outlined below.          
          
a. Formulate the null and alternative hypotheses.          TCO D
b. State the level of significance.          
c. Find the critical value (or values), and clearly show the rejection and nonrejection regions.          
d. Compute the test statistic.          
e. Decide whether you can reject Ho and accept Ha or not.          
f. Explain and interpret your conclusion in part e. What does this mean?          
g. Find the observed p-value for the hypothesis test and interpret this value. What does this mean?          
h. Does this sample data provide evidence (with α = 0.10), that the installation of the traffic light           
    is warranted?          

17) Given the following regression output:       
 SUMMARY OUTPUT       15 points
        
 Regression Statistics       TCO E
 Multiple R  0.835     
 R Square  0.697     
 Adjusted R Square  0.674     
 Standard Error  0.972     
 Observations  15     
        
 ANOVA       
  df SS MS F Significance F  
 Regression 1 28.287 28.287 29.970 0.000  
 Residual 13 12.270 0.944    
 Total 14 40.557     
        
   Standard   Lower Upper 
  Coefficients Error t Stat P-value 95% 95% 
 Intercept -0.309 3.109 -0.099 0.922 -7.025 6.407 
 X 0.212 0.039 5.475 0.000 0.128 0.296 
        
A) Approximately what percentage of the variation in the Y variable        
 is explained by the regression?       
        
a) 83.50%       
b) 97.20%       
c) 69.70%       
d) 29.97%       
        
B) What is the equation of the regression line?       
        
a) Y = .212 - 0.309X       
b) Y = -309 + 0.212X       
c) Y = 0.835 + 0.697X       
d) We fail to reject the null hypothesis       
        
C) State a 95% confidence interval for the coefficient of X       
        
a) P(-7.025 < b1 < 6.407) = 95%       
b) P(0.128< b1 < 0.296) = 95%       
c) P(28.287< b1 < 0.944) = 95%       
d) P( -0.309 < b1 < 0.212) = 95%       

18)  Using the "Ford Mustang Data Set" and alpha = .05        25 points  TCO E
 Hint - Use Megastat to calculate question #18 and #19 for ease of calculation of the confidence intervals requested         
          You may also use the CI techniques at the end of my spreadsheet "Regression Explanation with CI graphics" for #19         
          
 a)  Calculate a multiple regression to explain "Price" as the Y variable.         
 b)  state the regression equation         
 c)  Use the regression equation to estimate the average price for a          
  Convertible, 3 years old, 23,000 miles, standard transmission        
 d) State a 95 % confidence interval for the average price of the auto in "c" above          
          
 e)  Is the coefficient of "Age" significantly different from zero?  How can you tell?         
 f)  Which variables are least trustworthy in the multiple regression you calculated?         
      How can you tell?         
          
19)   a)  Calculate another regression using only the age variable to estimate         25 points  TCO E
  the price of Mustang Automobiles.        
 b) Compare the R square from this calculation to the previous regression         
  In common sense terms, explain which of these calculations you would recommend        
  to estimate the prices of Ford Mustangs and why?        
 c) From the regression equation, calculate the expected price for a 5 year old Mustang.         
          
 d) State a 95% confidence interval for the average price of a 5 year old Mustang