Answer :
To solve this problem, we need to perform a hypothesis test for the mean. Here's a step-by-step solution:
1. State the Hypotheses:
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 63.8\)[/tex]
- Alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu \neq 63.8\)[/tex]
2. Identify the Significance Level:
- The significance level is [tex]\(\alpha = 0.10\)[/tex].
3. Collect the Sample Data:
- Sample data: 58.1, 83.2, 81.8, 97.7, 68.7, 87.1
4. Calculate the Sample Mean ([tex]\(\bar{x}\)[/tex]):
[tex]\[
\bar{x} = \frac{58.1 + 83.2 + 81.8 + 97.7 + 68.7 + 87.1}{6} = 79.43
\][/tex]
5. Calculate the Sample Standard Deviation ([tex]\(s\)[/tex]):
- First, find the deviations from the mean, square them, and get the mean of these squares.
- Then take the square root (using [tex]\(n-1\)[/tex] for an unbiased estimate):
[tex]\[
s \approx 14.363
\][/tex]
6. Calculate the Sample Size ([tex]\(n\)[/tex]):
- The number of observations in the sample is [tex]\(n = 6\)[/tex].
7. Calculate the Test Statistic (t-value):
[tex]\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} = \frac{79.43 - 63.8}{14.363 / \sqrt{6}} = 2.732
\][/tex]
8. Degrees of Freedom:
[tex]\[
\text{Degrees of Freedom} = n - 1 = 6 - 1 = 5
\][/tex]
9. Calculate the p-value:
- Since it is a two-tailed test, we multiply the tail probability by 2.
- Using the t-distribution tables or software:
[tex]\[
p\text{-value} = 0.0412
\][/tex]
10. Make a Decision:
- Compare the p-value to [tex]\(\alpha\)[/tex]:
- [tex]\( p\text{-value} = 0.0412\)[/tex] is less than [tex]\(\alpha = 0.10\)[/tex].
11. Conclusion:
- Since the p-value is less than or equal to [tex]\(\alpha\)[/tex], we reject the null hypothesis. This suggests that there is sufficient evidence to conclude that the population mean is different from 63.8. The p-value indicates the likelihood of observing such a sample mean or more extreme if the null hypothesis were true.
Thus, the test statistic is 2.732, the p-value is 0.0412, and the p-value is less than or equal to [tex]\(\alpha\)[/tex].
1. State the Hypotheses:
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 63.8\)[/tex]
- Alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu \neq 63.8\)[/tex]
2. Identify the Significance Level:
- The significance level is [tex]\(\alpha = 0.10\)[/tex].
3. Collect the Sample Data:
- Sample data: 58.1, 83.2, 81.8, 97.7, 68.7, 87.1
4. Calculate the Sample Mean ([tex]\(\bar{x}\)[/tex]):
[tex]\[
\bar{x} = \frac{58.1 + 83.2 + 81.8 + 97.7 + 68.7 + 87.1}{6} = 79.43
\][/tex]
5. Calculate the Sample Standard Deviation ([tex]\(s\)[/tex]):
- First, find the deviations from the mean, square them, and get the mean of these squares.
- Then take the square root (using [tex]\(n-1\)[/tex] for an unbiased estimate):
[tex]\[
s \approx 14.363
\][/tex]
6. Calculate the Sample Size ([tex]\(n\)[/tex]):
- The number of observations in the sample is [tex]\(n = 6\)[/tex].
7. Calculate the Test Statistic (t-value):
[tex]\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} = \frac{79.43 - 63.8}{14.363 / \sqrt{6}} = 2.732
\][/tex]
8. Degrees of Freedom:
[tex]\[
\text{Degrees of Freedom} = n - 1 = 6 - 1 = 5
\][/tex]
9. Calculate the p-value:
- Since it is a two-tailed test, we multiply the tail probability by 2.
- Using the t-distribution tables or software:
[tex]\[
p\text{-value} = 0.0412
\][/tex]
10. Make a Decision:
- Compare the p-value to [tex]\(\alpha\)[/tex]:
- [tex]\( p\text{-value} = 0.0412\)[/tex] is less than [tex]\(\alpha = 0.10\)[/tex].
11. Conclusion:
- Since the p-value is less than or equal to [tex]\(\alpha\)[/tex], we reject the null hypothesis. This suggests that there is sufficient evidence to conclude that the population mean is different from 63.8. The p-value indicates the likelihood of observing such a sample mean or more extreme if the null hypothesis were true.
Thus, the test statistic is 2.732, the p-value is 0.0412, and the p-value is less than or equal to [tex]\(\alpha\)[/tex].