MATH 533 week 8 final exam

1. (TCO A)Consider the following raw data, which is the result of selecting a random sample of 20 Bank Common Stocks and noting the dividend yields (as a %).
           

3.1                    4.2                    2.0                    3.5                    2.6
5.3                    3.5                    3.1                    2.6                    3.3
4.7                    3.7                    3.0                    2.6                    4.0
3.8                    4.4                    3.2                    3.2                    3.8

 
a. Compute the mean, median, mode, and standard deviation, Q₁, Q₃, Min, and Max for the above sample data on dividend yield.
b. In the context of this situation, interpret the Median, Q₁, and Q₃. (Points : 33)

2. (TCO B) Consider the following data on newly hired employees in relation to which part of the country they were born and their highest degree attained.
 

 
If you choose one person at random, then find the probability that the person

a. has a PHD.
b. is from the East and has a BS as the highest degree attained.
c. has only a HS degree, given that person is from the West. (Points : 18)

5. (TCO C) A large-sized bag of Neato Chips should contain 16 oz of potato chips. A sample of 50 large-sized bags is selected with the following results.
 
Sample Size = 50
Sample Mean = 15.85 oz
Sample Standard Deviation = 1.53 oz

a. Construct a 95% confidence interval for the mean amount of contents per bag. .
b. Interpret this interval.
c. How large a sample size will need to be selected if we wish to have a 99% confidence interval with a margin for error of .10 oz? (Points : 18)

7. (TCO D) A Ford Motor Company quality improvement team believes that its recently implemented defect reduction program has reduced the proportion of paint defects. Prior to the implementation of the program, the proportion of paint defects was .03 and had been stationary for the past 6 months. Ford selects a random sample of 2,000 cars built after the implementation of the defect reduction program. There were 45 cars with paint defects in that sample. Does the sample data provide evidence to conclude that the proportion of paint defects is now less than .03 (with a = .01)? Use the hypothesis testing procedure outlined below.
 
a. Formulate the null and alternative hypotheses.
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. Determine the observed p-value for the hypothesis test and interpret this value. What does this mean?
h. Does the sample data provide evidence to conclude that the proportion of paint defects is now less than .03 (with a = .01)? (Points : 24)

8. (TCO D) The duration of worker’s unemployment seems to be increasing. Suppose that special federal funds are available for your state when the mean period of unemployment exceeds 40 weeks. As an economist with your state’s department of labor, you want to test whether the mean period of unemployment is more than 40 weeks. A random sample of 60 unemployed persons reveals the following.
 
Sample Size = 60
Sample Mean = 41.7 weeks
Sample Standard Deviation = 6.1 weeks
 
Does the sample data provide sufficient evidence to conclude that the population mean period of unemployment is greater than 40 weeks (using a = .05)? Use the hypothesis testing procedure outlined below.
 
a. Formulate the null and alternative hypotheses.
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. Determine the observed p-value for the hypothesis test and interpret this value. What does this mean?
h. Does the sample data provide sufficient evidence to conclude that the population mean period of unemployment is greater than 40 weeks (using a = .05)? (Points : 24)

9 (TCO E) Comp-U-Systems manufactures, sells, and services its own brand of computer. As part of the standard purchase contract, Comp-U-Systems agrees to perform regular service on its computers. To better schedule service calls, Comp-U-Systems is interested in the relationship between the total service time required for a service call (TIME, Y in minutes) and the number of computers serviced (NUMBER, X). A random sample of 11 service calls is selected, yielding the data found below.
 

 
 
Correlations: NUMBER, TIME
 
Pearson correlation of NUMBER and TIME = 0.995
P-Value = 0.000Regression Analysis: TIME versus NUMBER
 
The regression equation is
TIME = 11.5 + 24.6 NUMBER
 
 
Predictor     Coef  SE Coef       T      P
Constant    11.464    3.439    3.33  0.009
NUMBER     24.6022   0.8045   30.58  0.000
 
 
S = 4.61521   R-Sq = 99.0%   R-Sq(adj) = 98.9%
 
 
Analysis of Variance
 
Source          DF      SS      MS        F      P
Regression       1   19919   19919  935.15  0.000
Residual Error   9     192      21
Total           10   20111
 
 
Predicted Values for New Observations
 
New Obs     Fit  SE Fit       95% CI            95% PI
      1  134.48    1.65  (130.75, 138.20)  (123.39, 145.56)
      2  503.51   13.02  (474.06, 532.96)  (472.26, 534.76)XX
 
XX denotes a point that is an extreme outlier in the predictors.
 
 
Values of Predictors for New Observations
 
New Obs  NUMBER
      1     5.0
      2    20.0
 
a. Analyze the above output to determine the regression equation.
b. Find and interpret in the context of this problem.
c. Find and interpret the coefficient of determination (r-squared).
d. Find and interpret coefficient of correlation.
e. Does the data provide significant evidence (a = .05) that the number of computers to be serviced can be used to predict the total service time? Test the utility of this model using a two-tailed test. Find the observed p-value and interpret.
f. Find the 95% confidence interval for the mean service time for all occurrences of having five computers to be serviced. Interpret this interval.
g. Find the 95% prediction interval for the service time for one occurrence of having five computers to be serviced. Interpret this interval.
h. What can we say about the service time when we had 20 computers to service? (Points : 48)

 10. (TCO E) Holding goods in inventory is costly because inventoried goods are susceptible to breakage and other forms of physical damage. Typically, the amount of damage increases with the level of inventory, but some of the damage is unrelated to the amount of inventory. In addition, the seasonality may make a difference. A random sample of 10 observations is selected with the variables INVTRY (X1, inventory in $1,000,000s), SEASON (X2, with spring and summer being 0 and fall and winter being 1), and DAMAGE (Y, in $10,000s). The results are found below.
 

 
 
Correlations: INVTRY, SEASON, DAMAGE
 
        INVTRY   SEASON
SEASON   0.349
         0.242
 
DAMAGE   0.798    0.578
         0.001    0.038
 
 
Cell Contents: Pearson correlation
               P-Value
 
 
Regression Analysis: DAMAGE versus INVTRY, SEASON
 
The regression equation is
DAMAGE = 21.2 + 4.49 INVTRY + 11.7 SEASON.
 
 
Predictor    Coef  SE Coef     T      P
Constant    21.16    12.44  1.70  0.120
INVTRY      4.485    1.138  3.94  0.003
SEASON     11.670    5.896  1.98  0.076
 
 
S = 9.93147   R-Sq = 73.9%   R-Sq(adj) = 68.7%
 
 
Analysis of Variance
 
Source          DF      SS      MS      F      P
Regression       2  2797.4  1398.7  14.18  0.001
Residual Error  10   986.3    98.6
Total           12  3783.7
 
 
Predicted Values for New Observations
 
New Obs    Fit  SE Fit      95% CI           95% PI
      1  86.65    3.76  (78.27, 95.02)  (62.99, 110.30)
 
 
Values of Predictors for New Observations
 
New Obs   INVTRY  SEASON
      1     12.0    1.00
 
 
a. Analyze the above output to determine the multiple regression equation.
b. Find and interpret the multiple index of determination (R-Sq).
c. Perform the multiple regression t-tests on βˆ1, βˆ2 (use two tailed test with (a = .10). Interpret your results.
d. Predict the damage for a single case in the spring or summer with an inventory of $12,000,000. Use both a point estimate and the appropriate interval estimate.