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Answer :
To test the hypothesis Hâ‚€: [tex]\mu = 98.6[/tex] versus Hâ‚: [tex]\mu \neq 98.6[/tex], we will use the steps for hypothesis testing. This involves calculating the test statistic and the p-value using the provided sample data.
Step 1: Define the Null and Alternative Hypotheses
- Null Hypothesis (Hâ‚€): [tex]\mu = 98.6[/tex]
- Alternative Hypothesis (Hâ‚): [tex]\mu \neq 98.6[/tex]
This is a two-tailed test since we are checking for the possibility of the mean being both less than and greater than 98.6.
Step 2: Calculate the Sample Mean and Standard Deviation
Calculate the sample mean ([tex]\bar{x}[/tex]):
[tex]\bar{x} = \frac{\sum x_i}{n} = \frac{97.8 + 97.2 + \ldots + 99.0}{25}[/tex]
Calculate the sample standard deviation (s):
[tex]s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}[/tex]
Step 3: Compute the Test Statistic
Use the formula for the t-test statistic:
[tex]t = \frac{\bar{x} - \mu}{s / \sqrt{n}}[/tex]
Where:
- [tex]\mu[/tex] is the mean under the null hypothesis (98.6)
- [tex]n[/tex] is the sample size (25)
Step 4: Obtain the P-value
After calculating the test statistic, compare it to a t-distribution with [tex]n - 1 = 24[/tex] degrees of freedom. The p-value is the probability of observing the test statistic as extreme as, or more extreme than, the observed value, under the null hypothesis.
Step 5: Make a Decision
Compare the p-value to the significance level [tex]\alpha = 0.05[/tex]:
- If p-value [tex]\leq \alpha[/tex], reject the null hypothesis.
- If p-value [tex]> \alpha[/tex], fail to reject the null hypothesis.
Assuming the calculations show a p-value less than 0.05, we would reject the null hypothesis and conclude that there is a significant difference between the sample mean of body temperatures and 98.6.
Please perform the calculations for exact values of the sample mean, standard deviation, test statistic, and p-value.
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