Answered

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 [tex]$\mu=19.4$[/tex]. A random sample of 35 "socially conscious" stocks gave a P/E ratio sample mean of [tex]$\bar{x}=17.7$[/tex], with sample standard deviation [tex]$s=5.2$[/tex]. 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 [tex]$\alpha=0.05$[/tex].

(a) What is the level of significance?

State the null and alternate hypotheses.
- [tex]$H_0: \mu=19.4$[/tex]
- [tex]$H_1: \mu \neq 19.4$[/tex]

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.
- The Student's [tex]$t$[/tex], since the sample size is large and [tex]$\sigma$[/tex] is unknown.

What is the value of the sample test statistic? (Round your answer to three decimal places.)
[tex]$\square$[/tex]



Answer :

GinnyAnswer
To answer the given questions, let's go through each part step-by-step:

### (a) What is the level of significance?
The level of significance, denoted by α, is 0.05. This is given in the problem statement.

### State the null and alternate hypotheses.
The hypotheses test needs to determine whether the mean P/E ratio of socially conscious stocks is different from the mean P/E ratio of the S&P Stock Index. The null hypothesis ([tex]\(H_0\)[/tex]) states that there is no difference, while the alternate hypothesis ([tex]\(H_1\)[/tex]) states that there is a difference. Thus:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\( \mu = 19.4 \)[/tex]
- Alternate Hypothesis ([tex]\(H_1\)[/tex]): [tex]\( \mu \neq 19.4 \)[/tex] (a two-tailed test since we are checking for any difference)

### (b) What sampling distribution will you use?
Since the population standard deviation ([tex]\(\sigma\)[/tex]) is unknown and the sample size is 35 (which is large enough, greater than 30), we use the Student's t-distribution for this test.

- The Student's [tex]\(t\)[/tex], since the sample size is large and [tex]\(\sigma\)[/tex] is unknown.

The correct choice is "The Student's [tex]\(t\)[/tex], since the sample size is large and [tex]\(\sigma\)[/tex] is unknown."

### What is the value of the sample test statistic?
The sample test statistic is calculated using the formula for the t-statistic:
[tex]\[ t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \][/tex]

Plugging in the values:
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 17.7
- Population mean ([tex]\(\mu\)[/tex]) = 19.4
- Sample standard deviation ([tex]\(s\)[/tex]) = 5.2
- Sample size ([tex]\(n\)[/tex]) = 35

After the calculations, the t-statistic is found to be approximately:
[tex]\[ t = -1.934 \][/tex]

### Additional Steps:
1. Degrees of Freedom: The degrees of freedom ([tex]\(df\)[/tex]) is [tex]\(n - 1\)[/tex], which equals 34.
2. P-value: For a two-tailed test, the p-value corresponding to the t-statistic is approximately 0.061.
3. Conclusion: Compare the p-value with the significance level [tex]\(\alpha\)[/tex].
- Since the p-value (0.061) is greater than the significance level (0.05), we fail to reject the null hypothesis.

### Summary:
- Level of significance: [tex]\( \alpha = 0.05 \)[/tex]
- Null Hypothesis: [tex]\( H_0: \mu = 19.4 \)[/tex]
- Alternate Hypothesis: [tex]\( H_1: \mu \neq 19.4 \)[/tex]
- Sampling distribution: The Student's [tex]\( t \)[/tex]-distribution
- The sample test statistic: [tex]\( -1.934 \)[/tex] (rounded to three decimal places)

Given these results, we do not have enough evidence to 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 the 0.05 level of significance.

Other Questions