Question

One measure of the value of a stock is its price to earnings ratio (or P/E ratio). It is the ratio of the price of a stock per share to the earnings per share and can be thought of as the price an investor is willing to pay for $1 of earnings in a company. A stock analyst wants to know whether the P/E ratios for three industry categories differ significantly. The following data represent simple random samples of companies from three categories: (1) financial, (2) food, and (3) leisure goods.

Financial	Food	Leisure Goods
8.83	19.75	14.1
12.75	17.87	10.12
13.48	15.18	15.57
14.42	22.84	13.48
10.06	15.6	11.27

1. Test the null hypothesis that the mean P/E ratio for each category is the same at the alpha = 0.05 level of significance with the following steps:

• Formulate the null hypothesis and alternative hypothesis.

H₀: ______________________________ versus   H₁: _________________________________________.

• Check the assumptions and find MST, MSE, and the test statistic of fo with degrees of freedom of numerator and denominator.
• Make a decision and draw a conclusion with alpha = 0.05 level of significance.

2. If the null hypothesis is rejected in part (a), use Tukey’s test to determine which pairwise means differ using a familywise error rate of alpha = 0.05, where qa,v,k = q0.05,12,3 = 3.773 from the table of critical values for Tukey’s Test.

H₀: __________________ versus   H₁: ____________________.

q0=

Decision:

H₀: __________________ versus   H₁: ____________________.

q0=    

Decision:

H₀: __________________ versus   H₁: ____________________.

q0=

Decision:

• Use lines to indicate which population means are not significantly different.

Answer 1

From above probability plot we observed that the observations come from normal distribution.

Test for Equal Variances: P/E ratio versus categories

95% Bonferroni confidence intervals for standard deviations

   categories     N    Lower    StDev    Upper
    Financial      5 1.27831 2.36535   9.1040
         Food       5 1.70707 3.15871 12.1576
Leisure Goods 5 1.18681 2.19604   8.4523

Bartlett's Test (Normal Distribution)
Test statistic = 0.56, p-value = 0.757

Levene's Test (Any Continuous Distribution)
Test statistic = 0.28, p-value = 0.760

Since p-value>0.05 so we can assume population variances for three categories are same.

Now we construct ANOVA table:

ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	116.185333	2	58.09267	8.545171	0.004927135	3.885293835
Within Groups	81.57964	12	6.798303
Total	197.764973	14

MST=mean square due to treatments/groups=58.0927

MSE=mean square due to error=6.7983

Test statistic=F-ratio=MST/MSE=8.5452

Critical value=3.8853

Since F-ratio>Critical value or p-value=0.0049<0.05 hence we reject null hypothesis at 5% level of significance and conclude that there is sufficient evidence that at least one mean P/E ratio is different from other means P/E ratio.