Question

Socially conscious investors screen out stocks of alcohol and tobacco makers, firms with poor environmental records, and companies with poor labor practices. Some examples of "good," socially conscious companies are Johnson and Johnson, Dell Computers, Bank of America, and Home Depot. The question is, are such stocks overpriced? One measure of value is the P/E, or price-to-earnings ratio. High P/E ratios may indicate a stock is overpriced. For the S&P Stock Index of all major stocks, the mean P/E ratio is μ = 19.4. A random sample of 36 "socially conscious" stocks gave a P/E ratio sample mean of x = 17.8, with sample standard deviation s = 5.4. Does this indicate that the mean P/E ratio of all socially conscious stocks is different (either way) from the mean P/E ratio of the S&P Stock Index? Use α = 0.05.

Answer 1

The mean P/E ratio of socially conscious stocks is different from the mean P/E ratio of the S&P Stock Index.

Step-by-step explanation:

The question asks whether socially conscious stocks are overpriced compared to the S&P Stock Index. To determine this, we will conduct a hypothesis test using the P/E ratio as the measure of value. The null hypothesis states that the mean P/E ratio of socially conscious stocks is the same as the mean P/E ratio of the S&P Stock Index, while the alternative hypothesis states that they are different. We will use a significance level of 0.05.

To perform the hypothesis test, we will calculate the test statistic and compare it to the critical value. The test statistic is calculated as (sample mean - population mean)/(sample standard deviation/sqrt(sample size)). For the given data, the test statistic is (17.8 - 19.4)/(5.4/sqrt(36)) = -2.88.

Next, we will compare the test statistic to the critical value. Since we are conducting a two-tailed test, the critical value is ±1.96 for a significance level of 0.05. Since the test statistic (-2.88) falls outside the critical value range (-1.96 to 1.96), we reject the null hypothesis and conclude that the mean P/E ratio of socially conscious stocks is different from the mean P/E ratio of the S&P Stock Index at a significance level of 0.05.

Answer 2

`t=(17.8-19.4)/((5.4)/(√(36)))=-1.78`

`p_v =2*P(t_((35))<-1.78)=0.084`

If we compare the p value and the significance level given
`\alpha=0.05` we see that
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, so we can conclude that the mean P/E ratio of all socially conscious stocks it's not significantly different from the mean P/E ratio of the S&P Stock Index (19.4) at 5% of signficance.

Step-by-step explanation:

1) Data given and notation

`\bar X=17.8` represent the P/E ratio sample mean

`s=5.4` represent the sample standard deviation

`n=36` sample size

`\mu_o =19.4` represent the value that we want to test

`\alpha=0.05` represent the significance level for the hypothesis test.

z would represent the statistic (variable of interest)

`p_v` represent the p value for the test (variable of interest)

2) State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if the mean P/E ratio of all socially conscious stocks is different (either way) from the mean P/E ratio of the S&P Stock Index (19.4) :

Null hypothesis:
`\mu =19.4`

Alternative hypothesis:
`\mu \\eq 19.4`

Since we don't know the population deviation, is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:

`t=(\bar X-\mu_o)/((s)/(√(n)))` (1)

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".

3) Calculate the statistic

We can replace in formula (1) the info given like this:

`t=(17.8-19.4)/((5.4)/(√(36)))=-1.78`

4) P-value

First we need to calculate the degrees of freedom given by:

`df=n-1=36-1=35`

Since is a two-sided test the p value would be:

`p_v =2*P(t_((35))<-1.78)=0.084`

5) Conclusion

If we compare the p value and the significance level given
`\alpha=0.05` we see that
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, so we can conclude that the mean P/E ratio of all socially conscious stocks it's not significantly different from the mean P/E ratio of the S&P Stock Index (19.4) at 5% of signficance.