Question

NB:

1. Questions: 3.1 - 3.3 are related. Make use of the information provided in 3.1 to answer 3.2 & 3.3.

2. Questions: 4.1 - 4.4 are related. Make use of the information provided in 4.1 to answer 4.2, 4.3 & 4.4.

3. Questions: 5.1 - 5.3 are related. Make use of the information provided in 5.1 to answer 5.2 & 5.3.

3.1 An IQ test was given to five MBA students before and after they completed the MBA degree. Test whether there is any improvement (increase) in the IQ of the same students after completing MBA degree. Note: μ1 = population mean IQ before; μ2 = population mean IQ after. Hint: If the mean IQ has improved, then the mean IQ difference ∂ < 0; otherwise, the mean IQ difference ∂ > 0. Question: Formulate the Null and Alternative Hypothesis for this problem.

2 points

Students 1 2 3 4 5

IQ test scores before MBA 110 120 123 132 125

IQ test score After MBA 120 118 125 136 121

use above information

a) H0: IQ mean difference ∂≥0 vs H1: IQ mean difference ∂>0

b) H0: IQ mean difference ∂≤0 vs H1: IQ mean difference ∂>0

c) H0: IQ mean difference ∂=0 vs H1: IQ mean difference ∂≠0

d) H0: IQ mean difference ∂≥0 vs H1: IQ mean difference ∂<0

3.2 Make use of the information provided in the previous question calculate the t-statistic using T-test and tick the correct answer below.

10 points

a) t-statistic = -2.319

b) t-statistic = 0.1028

c) t-statistic = -1.028

d) t-statistic = -0.816

3.3 Based on your empirical evidence in the previous question make your statistical conclusion at the 5% level of significance whether there is any improvement in IQ of the same students after completing MBA degree. Tick the correct answer below.

5 points

a) None of these answers is correct.

b) Fail to reject the Null hypothesis, the Null is probably true that IQ remains the same has not improved after completing the MBA degree.

c) Reject the Null hypothesis. The alternative is probably true that IQ has improved after completing the MBA degree.

d) Accept the Null hypothesis because the t-statistic is very close to zero.

4.1 A company that manufacturers wooden products (e.g. garden furniture, ladders, benches) regularly maintains its lathe machines, which are used for cutting and shaping components. The manager would like to know whether the cost of machine maintenance is related to the age of the machines. For a random sample of 8 lathe machines in the company's factory, the annual maintenance cost (in N$100s) and age of each machine was recorded. Question: Identify the independent variable and the dependent variable. Tick the correct answer below.

2 points

Machine 1 2 3 4 5 6 7 8

Age 4 2 3 8 6 7 1 2

Annual Cost 45 20 39 66 58 50 14 18

Table 2. Maintenance costs analysis

a) Machine =Independent variable & Age =Dependent variable

b) Machine & Age =Independent variables, Annual Cost =Dependent variable

c) Annual cost=Dependent variable & Age =Independent variable

d) Age =Independent variable & Machine =Dependent variable

4.2 Use the information given in the previous question and using the method of least squares, which of the equation listed below represent the best fitting regression line between the age of lathe machines and their annual maintenance costs? Tick the correct answer below.

10 points

a) Machine & Age =7.1306 + 9.336Annual costs

b) Annual cost = 7.336 +6.1306Age

c) Annual cost = 9.336 + 7.1306Age

d) Age = 7.1306 + 9.431Annual cost

4.3 Calculate the sample correlation coefficient (r) between the annual maintenance cost and age of each machine. Question: Which of the options below is the correct answer?

5 points

a) r = -0.91

b) r = 0.89

c) r = 0.87

d) r = 0.94

4.4 Question: What is the expected average maintenance cost of a lathe machine that is five years old? Tick the correct answer below.

4 points

a) 45.04

b) 40.67

c) 42.77

d) 41.77

5.1 In May 2010, the Snap poll asked British adults their opinion on whether they are in favour of or opposed to using profiling to identify potential terrorists at airports, a practice used routinely in Israel, but not in the UK. Does opinion depend on age? Or are opinion and age independent? Table below show some numbers from Snap Poll. Question: Formulate the Null and Alternative Hypothesis for this problem.

2 points

Age 18-29 30-49 50-64 65+

Favour 57 66 77 87

Oppose 43 34 23 13

Table 3. Snap Poll Results

a) The null hypothesis is that Opinion and Age are the variables.

b) The null hypothesis is that Opinion and Age are associated vs. The alternative hypothesis is that Opinion and Age are not associated.

c) The null hypothesis is that Opinion and Age are independent vs. The alternative hypothesis is that Opinion and Age are dependent.

d) The alternative hypothesis is that Opinion and Age are independent vs. The null hypothesis is that Opinion and Age are dependent.

5.2 Does opinion depend on age? Or are opinion and age independent? Using the information in the previous question calculate the Chi-square-statistic using Chi-square-test. Tick the correct answer below.

10 points

a) Chi-squares-stat = 25.20

b) Chi-squares-stat = 14.76

c) Chi-squares-stat = 23.20

d) Chi-squares-stat = 21.19

5.3 Based on your empirical evidence in the previous question, and using a 5% level of significance, make your statistical conclusion about the association between opinion and age. Tick the correct answer below.

5 points

a) Reject the Null hypothesis and conclude that Age and Opinion about Profiling are not independent. The alternative is probably true.

b) Accept the alternative and conclude that Age and Opinion about Profiling are independent.

c) Fail to reject the Null hypothesis, the alternative is probably false.

d) None of these answers is correct.

Answer 1

3.1: Option: d) H₀: IQ mean difference ∂≥0 vs H₁: IQ mean difference ∂<0.

3.2: Option: d) t-statistic = -0.816.

3.3: Option: b) Fail to reject the Null hypothesis, the Null is probably true that IQ remains the same has not improved after completing the MBA degree.

(since P-value=0.2300>0.05).

t-Test: Paired Two Sample for Means
	Before MBA	After MBA
Mean	122	124
Variance	64.5	51.5
Observations	5	5
Pearson Correlation	0.74607929
Hypothesized Mean Difference	0
df	4
t Stat	-0.8164966
P(T<=t) one-tail	0.23002538
t Critical one-tail	2.13184679
P(T<=t) two-tail	0.46005075
t Critical two-tail	2.77644511

4.1: Option: c) Annual cost=Dependent variable & Age =Independent variable

4.2: Option: c) Annual cost = 9.336 + 7.1306Age

4.3: Option: d) r = 0.94.

4.4: Option: a) 45.04

(Expected Annual cost = 9.336 + 7.1306*5=44.989 (which is closed to 45.04)).

R code:

Age=c(4, 2, 3, 8, 6, 7, 1, 2)
Annual_Cost=c(45, 20, 39, 66, 58, 50, 14, 18)
m=lm(Annual_Cost~Age)
round(m$coefficients,4)
round(cor(Age,Annual_Cost),2)

5.1: Option: c) The null hypothesis is that Opinion and Age are independent vs. The alternative hypothesis is that Opinion and Age are dependent.

5.2: Observed frequencies:

Age	18-29	30-49	50-64	65+	Totals
Favour	57=O11	66=O12	77=O13	87=O14	287=O10
Oppose	43=O21	34=O22	23=O23	13=O24	113=O20
Totals	100=O01	100=O02	100=O03	100=O04	400=N

Expected frequencies:

Age	18-29	30-49	50-64	65+
Favour	71.75=E11	71.75=E12	71.75=E13	71.75=E14
Oppose	28.25=E21	28.25=E22	28.25=E23	28.25=E24

Option: a) Chi-squares-stat = 25.20

5.3:

Option: a) Reject the Null hypothesis and conclude that Age and Opinion about Profiling are not independent. The alternative is probably true.