Answer :
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.
### (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.
