QUESTION ONE
(a) A factory manager claims that workers at plant A are faster than those at plant B. To test the claim, a random sample of times (in minutes) taken to complete a given task was taken from each of the plants. The following results were obtained:
Plant A: n1 = 10, x1¯ = 74 , S12 = 60
Plant B: n2 = 13, x2¯ = 81 , S22 = 40
Assume that the populations are normal.
(i) Is the claim valid? Assume the population variances are equal. Use a 5% level of significance.
(ii) Explain how one can commit a type I error.
(b) A medical doctor suspected that the amount of fat in the blood in females was 5g more than males, on average in every litre of blood. A study was conducted and the following results were obtained:
Males: n1 = 10 , x1¯ = 39.6 s1 = 6.8
Females: n2 = 10 x2¯ = 48.1 s2 = 4.4
Assuming independent samples from approximately normal populations complete the test at a 5% level of significance.
QUESTION TWO
Four salesmen in XYZ Company are competing for the title ‘Salesman of the year'. Each has the task of selling milk in three different types of location. Their resulting sales, in K' 000, were as follows:
|
Area
|
A
|
Salesmen
B C
|
D
|
|
1
|
52.8
|
49.4
|
58.6
|
42.9
|
|
2
|
60.1
|
48.1
|
61.0
|
50.3
|
|
3
|
62.0
|
56.4
|
63.3
|
61.2
|
(a) Write down a model for the above design. Explain each term in the model in the context of the given information.
(b) Consider the observation on the salesman B and location 2 (x22 = 48.1) .
i. Estimate μ
ii. Estimate and interpret β2
iii. Estimate and interpret τ1
iv. Estimate ε21
(c) What is the blocking variable and what is the treatment?
(d) What is the purpose of the blocks in this experiment?
(e) Prepare a two -way analysis of variance table
(f) Test at 5% level of significance the null hypothesis that the population mean sales are identical for all the four salesmen.
(g) Test at 5% level of significance the null hypothesis that the population mean sales are the same for all three locations.
(h) Which salesman won? Did he do significantly better than his nearest rival?
QUESTION THREE
A certain market has both an express checkout line and a super express checkout line. Let X denote the number of customers in line at the express checkout line at a particular time of day and let Y denote the number of customers in line at the super express checkout at the same time. Suppose the joint probability distribution of X and Y is as given in the following table:
|
C
|
Y
|
|
0
|
1
|
2
|
3
|
|
0
|
0.08
|
0.07
|
0.04
|
0.00
|
|
1
|
0.06
|
0.15
|
0.05
|
0.04
|
|
2
|
0.05
|
0.04
|
0.10
|
0.06
|
|
3
|
0.00
|
0.03
|
0.04
|
0.07
|
|
4
|
0.00
|
0.01
|
0.05
|
0.06
|
(a) What is P (X = 1,Y = 1), line?
that is, the probability that there is exactly one customer in each
(b) What is P (X = Y ), that is, the probability that the number of customers in the two lines are identical?
(c) Let A denote the event that there are at least two more customers in one line than in the other line. Express A in terms of X and Y and calculate the probability of this event.
(d) What is the probability that the total number of customers in the two lines is
(i) Exactly four?
(ii) At least four?
(e) Determine the marginal probability distribution of X and then calculate the expected number of customers in line at the express checkout.
(f) Determine the marginal probability distribution of Y and then calculate the expected number of customers in line at the super express checkout.
(g) By inspection of the probabilities P (X = 4), P (Y = 0), and P (X = 4,Y = 0), are X and Y
independent random variables? Explain your answer.
(h) Compute the covariance for X and Y
(i) Compute the correlation for X and Y
QUESTION FOUR
A random sample of 50 ZAOU students was obtained to test the claim that gender and political affiliation were independent. Use the chi-square to test the claim at a 10% level of significance. The following data were obtained:
|
|
PF
|
MMD
|
UPND
|
|
Females
|
6
|
12
|
8
|
|
Males
|
10
|
6
|
8
|
QUESTION FIVE
A state official was investigating the relationship between salary (x) and the number of absences (y) for state employees. The variable y in the following table represents the average number of absences per year for employees at that salary.
|
Salary in K Thousands (x)
|
20.0
|
22.5
|
25.0
|
27.5
|
30.0
|
32.5
|
35.0
|
37.5
|
40.0
|
|
Number of absences ( y )
|
2.3
|
2.0
|
2.0
|
1.8
|
2.2
|
1.5
|
1.2
|
1.3
|
0.6
|
(a) Draw a scatter diagram. Does a linear relationship between x and y seem appropriate?
(b) Estimate the simple linear regression line. Interpret the parameters in the model.
(c) Estimate the average number of absences for employees earning K29, 000.
(d) At the 5 % level of significance would you conclude that there is a linear relationship between x and y ?
(e) Construct a 95 % confidence interval for the slope parameter β .
(f) Find the coefficient of determination and interpret the value.
(g) Find the correlation coefficient.