Answer :
[tex]\(\pm 2.262\)[/tex]Final Answer:
The hypothesis test suggests that there is sufficient evidence to reject the claim that the population mean body temperature is 98.6°F, at a significance level of 0.05.
Explanation:
To test the hypothesis, we first calculate the sample mean and standard deviation. Given the sample:
[tex]x_1[/tex] = 98.5, [tex]x_2[/tex] = 98.7, [tex]x_3[/tex] = 99.0, [tex]x_4[/tex] = 96.9, [tex]x_5[/tex] = 98.9, [tex]x_6[/tex] = 98.8, [tex]x_7[/tex] = 97.9,
[tex]x_8[/tex] = 99.4, [tex]x_9[/tex] = 98.8, [tex]x_{10}[/tex] = 97.1
Calculate the sample mean:
[tex]\[\bar{x} = \frac{\sum_{i=1}^{10} x_i}{10} = \frac{98.5 + 98.7 + 99.0 + 96.9 + 98.9 + 98.8 + 97.9 + 99.4 + 98.8 + 97.1}{10} = 98.35\][/tex]
Next, calculate the sample standard deviation:
[tex]\[s = \sqrt{\frac{\sum_{i=1}^{10} (x_i - \bar{x})^2}{10 - 1}} = \sqrt{\frac{(98.5-98.35)^2 + (98.7-98.35)^2 + ... + (97.1-98.35)^2}{9}} \approx 0.8917\][/tex]
Then, compute the t-statistic using the formula:
[tex]\[t = \frac{\bar{x} - \mu}{s/\sqrt{n}} = \frac{98.35 - 98.6}{0.8917/\sqrt{10}} \approx -2.661\][/tex]
Now, determine the critical value from the t-distribution at a significance level of 0.05 with degrees of freedom (df = 10 - 1 = 9). For a two-tailed test, the critical value is approximately [tex]\(\pm 2.262\)[/tex].
Since the absolute value of the calculated t-statistic (2.661) exceeds the critical value (2.262), we reject the null hypothesis that the population mean body temperature is 98.6°F.
Therefore, we conclude that the true population mean body temperature is likely different from 98.6°F.
