Answer :
To test the hypothesis that the population mean body temperature is not 98.6 degrees Fahrenheit, we start by formulating the null and alternative hypotheses:
- Null Hypothesis [tex](H_0)[/tex]: [tex]\mu = 98.6[/tex] degrees F
- Alternative Hypothesis [tex](H_a)[/tex]: [tex]\mu \neq 98.6[/tex] degrees F
This is a two-tailed test because we are looking for any significant difference from 98.6, either higher or lower.
Step-by-Step Solution
Calculate the sample mean [tex](\bar{x})[/tex]:
[tex]\bar{x} = \frac{98.9 + 98.1 + 99.0 + 96.1 + 98.7 + 98.1 + 97.3 + 99.3 + 98.4 + 97.5}{10} = 98.14[/tex]
Calculate the sample standard deviation [tex](s)[/tex]:
First, find the sum of the squared deviations from the mean:
[tex]\sum (x_i - \bar{x})^2 = (98.9 - 98.14)^2 + (98.1 - 98.14)^2 + \, \ldots \, + (97.5 - 98.14)^2 = 7.64[/tex]Then compute the sample standard deviation:
[tex]s = \sqrt{\frac{7.64}{10 - 1}} = \sqrt{0.8489} \approx 0.921[/tex]Calculate the test statistic (t):
Use the formula for the t-statistic:
[tex]t = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{98.14 - 98.6}{0.921 / \sqrt{10}} \approx \frac{-0.46}{0.291} \approx -1.581[/tex]Determine the critical value and p-value:
With a significance level [tex]\alpha = 0.05[/tex] and [tex]n-1 = 9[/tex] degrees of freedom, use a t-distribution table or calculator to find the critical values. For a two-tailed test, the critical values are approximately [tex]\pm 2.262[/tex].
To find the p-value, use a t-distribution calculator or table. The absolute value of the test statistic is [tex]1.581[/tex]. The p-value is found by determining the probability of getting a t-value as extreme or more extreme than [tex]-1.581[/tex]. The p-value is approximately [tex]0.148[/tex].
Decision:
Since the p-value [tex]0.148[/tex] is greater than [tex]0.05[/tex], we fail to reject the null hypothesis [tex](H_0)[/tex]. There is not enough statistical evidence to suggest that the population mean body temperature is different from 98.6 degrees F.
In conclusion, based on the sample data, we do not have strong evidence to say the mean body temperature of the population differs from 98.6 degrees Fahrenheit.
