Answer :
To solve this hypothesis testing problem, we'll follow several steps to find the test statistic and the p-value.
### Step 1: Gather the Data
We have the sample data points: 83.1, 97.4, 84.6, 74.3, and 98.
### Step 2: Understand the Hypotheses
We are testing the following hypotheses:
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 89.9\)[/tex]
- Alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu \neq 89.9\)[/tex]
This is a two-tailed test because the alternative hypothesis is testing for any difference (not specifying greater than or less than).
### Step 3: Calculate Sample Mean and Standard Deviation
Calculate the mean of the sample. The sample mean is:
[tex]\[
\bar{x} = \frac{83.1 + 97.4 + 84.6 + 74.3 + 98}{5} = 87.48
\][/tex]
Next, calculate the sample standard deviation ([tex]\(s\)[/tex]). Use the formula for the sample standard deviation with Bessel's correction ([tex]\(ddof=1\)[/tex]):
[tex]\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\][/tex]
For the given data, the standard deviation is approximately 10.24.
### Step 4: Calculate the Test Statistic
Use the formula for the t-statistic when the population standard deviation is unknown:
[tex]\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\][/tex]
Where [tex]\( n \)[/tex] is the number of data points, which is 5 in this case.
Substitute in the values:
[tex]\[
t = \frac{87.48 - 89.9}{10.24 / \sqrt{5}}
\][/tex]
The calculated test statistic is approximately [tex]\(-0.534\)[/tex].
### Step 5: Determine the Degrees of Freedom
The degrees of freedom ([tex]\(df\)[/tex]) for this test is [tex]\(n - 1 = 5 - 1 = 4\)[/tex].
### Step 6: Calculate the p-value
For a two-tailed test, calculate the p-value by finding the probability of observing a value at least as extreme as the test statistic in either tail of the distribution.
Given the absolute value of the t-statistic [tex]\(|t| = 0.534\)[/tex] and [tex]\(df = 4\)[/tex], the p-value is approximately [tex]\(0.6214\)[/tex].
### Conclusion
- Test Statistic: [tex]\(-0.534\)[/tex]
- p-value: [tex]\(0.6214\)[/tex]
Since the calculated p-value ([tex]\(0.6214\)[/tex]) is greater than the significance level ([tex]\(\alpha = 0.002\)[/tex]), we do not reject the null hypothesis. There is not enough evidence to conclude that the mean is different from 89.9.
### Step 1: Gather the Data
We have the sample data points: 83.1, 97.4, 84.6, 74.3, and 98.
### Step 2: Understand the Hypotheses
We are testing the following hypotheses:
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 89.9\)[/tex]
- Alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu \neq 89.9\)[/tex]
This is a two-tailed test because the alternative hypothesis is testing for any difference (not specifying greater than or less than).
### Step 3: Calculate Sample Mean and Standard Deviation
Calculate the mean of the sample. The sample mean is:
[tex]\[
\bar{x} = \frac{83.1 + 97.4 + 84.6 + 74.3 + 98}{5} = 87.48
\][/tex]
Next, calculate the sample standard deviation ([tex]\(s\)[/tex]). Use the formula for the sample standard deviation with Bessel's correction ([tex]\(ddof=1\)[/tex]):
[tex]\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\][/tex]
For the given data, the standard deviation is approximately 10.24.
### Step 4: Calculate the Test Statistic
Use the formula for the t-statistic when the population standard deviation is unknown:
[tex]\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\][/tex]
Where [tex]\( n \)[/tex] is the number of data points, which is 5 in this case.
Substitute in the values:
[tex]\[
t = \frac{87.48 - 89.9}{10.24 / \sqrt{5}}
\][/tex]
The calculated test statistic is approximately [tex]\(-0.534\)[/tex].
### Step 5: Determine the Degrees of Freedom
The degrees of freedom ([tex]\(df\)[/tex]) for this test is [tex]\(n - 1 = 5 - 1 = 4\)[/tex].
### Step 6: Calculate the p-value
For a two-tailed test, calculate the p-value by finding the probability of observing a value at least as extreme as the test statistic in either tail of the distribution.
Given the absolute value of the t-statistic [tex]\(|t| = 0.534\)[/tex] and [tex]\(df = 4\)[/tex], the p-value is approximately [tex]\(0.6214\)[/tex].
### Conclusion
- Test Statistic: [tex]\(-0.534\)[/tex]
- p-value: [tex]\(0.6214\)[/tex]
Since the calculated p-value ([tex]\(0.6214\)[/tex]) is greater than the significance level ([tex]\(\alpha = 0.002\)[/tex]), we do not reject the null hypothesis. There is not enough evidence to conclude that the mean is different from 89.9.
