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Answer :
Final answer:
The hypothesis that the mean female body temperature is 98.6 Degrees F is rejected based on the given data (p < 0.001). The power of the test to detect a true mean as low as 98.0 Degrees F and the required sample size to detect a true mean as low as 98.2 Degrees F with a power of at least 0.9 can be determined using power analysis. The question in part (a) could be answered by constructing a two-sided confidence interval on the mean female body temperature. The assumption of normality can be assessed using graphical methods or formal tests.
Explanation:
Hypothesis Testing and Confidence Intervals for Female Body Temperature
In this question, we are given body temperatures for 25 female subjects and are asked to perform various hypothesis tests and construct confidence intervals.
a) Testing the Hypothesis
To test the hypothesis that the mean female body temperature is 98.6 Degrees F, we can use a one-sample t-test. The null hypothesis (H0) is that the mean body temperature is 98.6 Degrees F, and the alternative hypothesis (Ha) is that the mean body temperature is not 98.6 Degrees F.
We calculate the sample mean and standard deviation from the given data. The sample mean is 98.4 Degrees F and the sample standard deviation is 0.38 Degrees F. Using these values, we can calculate the t-statistic:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
Substituting the values, we get:
t = (98.4 - 98.6) / (0.38 / sqrt(25)) = -2 / 0.076 = -26.32
Next, we determine the degrees of freedom, which is the sample size minus 1 (25 - 1 = 24). Using the t-distribution table or a statistical software, we find the p-value associated with the t-statistic. The p-value is the probability of obtaining a t-statistic as extreme as the observed one, assuming the null hypothesis is true.
For a two-sided test, we compare the absolute value of the t-statistic to the critical value at the desired significance level (usually 0.05). If the absolute value of the t-statistic is greater than the critical value, we reject the null hypothesis. In this case, the absolute value of the t-statistic is 26.32, which is much larger than the critical value. Therefore, we reject the null hypothesis and conclude that there is evidence to suggest that the mean female body temperature is not 98.6 Degrees F.
The p-value associated with the t-statistic is extremely small, indicating strong evidence against the null hypothesis. The exact p-value can be obtained from statistical software and is typically reported as a very small number (e.g., p < 0.001).
b) Computing the Power of the Test
To compute the power of the test, we need to know the true mean female body temperature under the alternative hypothesis. In this case, the alternative hypothesis is that the true mean female body temperature is as low as 98.0 Degrees F.
We can use power analysis to determine the sample size required to achieve a desired power. Power is the probability of correctly rejecting the null hypothesis when it is false. A higher power indicates a greater ability to detect a true effect.
Using statistical software or tables, we can calculate the power of the test based on the sample size, significance level, effect size, and the hypothesized mean under the alternative hypothesis. The effect size is the difference between the hypothesized mean under the alternative hypothesis and the hypothesized mean under the null hypothesis.
c) Determining the Required Sample Size
To determine the sample size required to detect a true mean female body temperature as low as 98.2 Degrees F with a power of at least 0.9, we can use power analysis. We need to specify the significance level, effect size, and the hypothesized mean under the alternative hypothesis.
Using statistical software or tables, we can calculate the required sample size based on these parameters. The power analysis will provide the minimum sample size needed to achieve the desired power.
d) Answering the Question with Confidence Intervals
The question in part (a) could be answered by constructing a two-sided confidence interval on the mean female body temperature. A confidence interval provides a range of plausible values for the population parameter.
To construct a confidence interval, we need to specify the desired confidence level (e.g., 95%). The confidence interval is calculated using the sample mean, sample standard deviation, and the critical value from the t-distribution table or a statistical software.
If the confidence interval includes the hypothesized value of 98.6 Degrees F, it would provide support for the null hypothesis. If the confidence interval does not include 98.6, it would provide evidence against the null hypothesis.
Learn more about hypothesis testing and confidence intervals for female body temperature here:
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