An investment house is looking at the potential growth in shares from different business sectors. The price of a sample of nine shares from the manufacturing sector are recorded at the start of the year and again at the end of the year. The two prices (in pence) for each share are given below. Share No. 1 2 3 4 5 6 7 8 9 Start of year 42 39 45 50 36 61 36 44 37 End of year 74 66 60 63 59 48 59 86 63 Examine the data to test the hypothesis that there is no change in the expected prices of shares from the manufacturing sector from the start to the end of the year.

Solution:

Null Hypothesis (Ho): There is no difference in the population average price in the start and end of the year for manufacturing sector. (µ1-µ2 = 0)

Alternative Hypothesis (Ha): There is a difference in the population average price in the start and end of the year for manufacturing sector.(µ1-µ2 ≠ 0)

Level of significance = 0.05

Test statistics is paired sample t test

Paired t Test
Data
Hypothesized Mean Difference / 0
Level of significance / 0.05
Intermediate Calculations
Sample Size / 9
DBar / -20.8889
Degrees of Freedom / 8
SD / 15.3578
Standard Error / 5.1193
t Test Statistic / -4.0805
Two-Tail Test
Lower Critical Value / -2.3060
Upper Critical Value / 2.3060
p-Value / 0.0035
Reject the null hypothesis

Conclusion:

Since the p-value is smaller than the 5% level of significance so we will reject the null hypothesis and conclude that There is a difference in the population average price in the start and end of the year for manufacturing sector. (µ1-µ2 ≠ 0).

The price of a sample of seven shares from the banking sector are valued at the start of the year and again at the end of the year. The two prices (in pence) for each share are given Share No. 1 2 3 4 5 6 7 Start of year 48 65 67 58 50 53 43 End of year 69 56 61 77 71 78 73 Examine the data to test the hypothesis that there is no change in the expected prices of shares from the banking sector from the start to the end of the year.

Solution:

Null Hypothesis (Ho): There is no difference in the population average price in the start and end of the year for banking sector. (µ1-µ2 = 0)

Alternative Hypothesis (Ha): There is a difference in the population average price in the start and end of the year for banking sector. (µ1-µ2 ≠ 0)

Level of significance = 0.05

Test statistics is paired sample t test

Paired t Test
Data
Hypothesized Mean Difference / 0
Level of significance / 0.05
Intermediate Calculations
Sample Size / 7
DBar / -14.4286
Degrees of Freedom / 6
SD / 15.4257
Standard Error / 5.8304
t Test Statistic / -2.4747
Two-Tail Test
Lower Critical Value / -2.4469
Upper Critical Value / 2.4469
p-Value / 0.0481
Reject the null hypothesis

Conclusion:

Since the p-value is smaller than the 5% level of significance so we will reject the null hypothesis and conclude that there is a difference in the population average price in the start and end of the year for banking sector. (µ1-µ2 ≠ 0).

Test the hypothesis that the two populations of changes in share price from the manufacturing and banking sectors have equal variances. Hence test whether there is any evidence that sector has an effect on the expected change in share prices. Also give a 95% confidence interval for the difference in expected change in share prices for the manufacturing sector compared with the banking sector.

Solution:

Null Hypothesis (Ho): There is no difference in the population variance betweenmanufacturing andbanking sector. (σ21 =σ22 )

Alternative Hypothesis (Ha): There is a difference in the population variance between manufacturing and banking sector. (σ21≠σ22 )

Level of significance = 0.05

Test statistics is F test

F Test for Differences in Two Variances
Data
Level of Significance / 0.05
Larger-Variance Sample
Sample Size / 7
Sample Variance / 237.952381
Smaller-Variance Sample
Sample Size / 9
Sample Variance / 235.8611111
Intermediate Calculations
F Test Statistic / 1.0089
Population 1 Sample Degrees of Freedom / 6
Population 2 Sample Degrees of Freedom / 8
Two-Tail Test
Upper Critical Value / 4.6517
p-Value / 0.9616
Do not reject the null hypothesis

Conclusion:

Since the p-value is bigger than the 5% level of significance so we will not be able to reject the null hypothesis and conclude that there is no difference in the population variance between manufacturing and banking sector.

95% confidence interval for the difference in expected change in share prices for the manufacturing sector compared with the banking sector is given below:

Confidence Interval Estimate
for the Difference Between Two Means
Data
Confidence Level / 95%
Intermediate Calculations
Degrees of Freedom / 14
t Value / 2.1448
Interval Half Width / 16.6313
Confidence Interval
Interval Lower Limit / -10.1710
Interval Upper Limit / 23.0916