Question

An Investment Bank compared the profiles of a sample of 25 firms that merged during 1985 to 2005 with those of a sample of 25 firms that did not merge. The table displays the information obtained on the firms' price earnings ratio.

Price-Earnings Ratio
	Merged Firms	Non-Merged Firms
Sample Mean	7.3	14.7
Sample Standard Deviation	11.0	16.0

1. a) At the 5% significance level, carry out a hypothesis test to investigate if the two population variances are equal. [2 marks]

2. b) At the 1% significance level, is there evidence to indicate that merged firms generally have smaller price-earnings ratios? [6 marks]

3. c) Is it appropriate to use a pooled variance t–test to test for equality of the mean price- earnings ratio for the two firms [merged and non-merged firms]? What additional assumption is necessary for you to use the t-test?

Answer 1

(a)

Let the population variance of the price-earning ratio for merged firms be and that for non-merged firms be

Here we are to test

The given data are summarized as

Statistic	Merged Firms	Non-Merged firms
Sample size	n1=25	n2=25
Sample mean	7.3	14.7
Sample SD	s1=11	s2=16

The test statistic is given by

The test statistic under the null hypothesis follows F distribution with df 24,24

The critical value is obtained from the Biometrika table as

As both the observed values are less than the critical value, we fail to reject the null hypothesis at 5% level of significance and thus we can conclude that their variances are equal.

(b)

Let the population mean of the price-earning ratio for merged firms be and that for non-merged firms be

Here we are to test

The given data are summarized as

Statistic	Merged Firms	Non-Merged firms
Sample size	n1=25	n2=25
Sample mean	7.3	14.7
Sample SD	s1=11	s2=16

The test statistic is given by

where s' is given by

The test statistic under the null hypothesis follows t distribution with df 48

The critical value is obtained from the Biometrika table as

we fail to reject the null hypothesis at 1% level of significance and thus we can conclude that the mean ratios are equal.

(c)

As the variances are qeual by the result of (a) , it is appropriate to use pooled t test.

The other assumptions made are as follows:

(i) The sample are drawn randomly i.e. they are independent.

(ii) They are drawn from samples following Normal distribution.

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