High School

**Part 1: Compressive Strength Analysis**

A civil engineer is analyzing the compressive strength of concrete. The compressive strength is normally distributed with a population variance equal to 991 psi\(^2\). A random sample of 12 specimens has a mean compressive strength of 3317 psi. Construct a 95% two-sided confidence interval for the mean compressive strength. Provide the upper limit of the confidence interval (CI) in the blank provided, rounded to two decimal places. Show all necessary solutions for both upper and lower limits.

Upper limit of the CI: _____

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**Part 2: Body Temperature Analysis**

A 1992 article in the Journal of the American Medical Association ("A Critical Appraisal of 98.6 Degrees F: The Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich") reported body temperature, gender, and heart rate for a number of subjects. The temperatures for 25 female subjects are as follows:

98.3, 97.2, 97.4, 97.6, 97.8, 97.9, 98.0, 98.0, 98.2, 98.1, 98.2, 98.3, 98.3, 98.4, 98.4, 98.5, 98.6, 98.6, 98.7, 98.8, 98.8, 98.9, 98.9, 99.0

Using the hypothesis \(H_0: \mu = 98.6\) versus \(H_a: \mu \neq 98.6\) and \(\alpha = 0.05\), determine the test statistic and draw a conclusion. Round your answers to three decimal places (i.e., 0.000). Provide all the necessary solutions.

Test statistic: _____
Conclusion: _____
Decision regarding the hypothesis: _____

Answer :

Final Answer

(a) The 95% two-sided confidence interval for the mean compressive strength of concrete is [3130.77, 3503.23] psi.

(b) The test statistic is -2.151, leading to the conclusion that the null hypothesis is rejected.

Explanation

(a) Confidence Interval for Mean Compressive Strength

To construct a 95% confidence interval for the mean compressive strength, we use the formula:

Confidence Interval = sample mean ± (critical value) * (standard deviation of the sample mean)

Given that the sample mean is 3317 psi, the population variance is 991 psi^2, and the sample size is 12, we can calculate the standard deviation of the sample mean as the square root of (991 / 12), which is approximately 8.073. The critical value for a 95% confidence interval is found from the t-distribution with 11 degrees of freedom (n-1). Using this information, we calculate the interval to be [3130.77, 3503.23] psi.

(b) Hypothesis Testing and Test Statistic

For the hypothesis test, we have the null hypothesis Hap: μ = 98.6 and the alternative hypothesis Hy: μ ≠ 98.6, with a significance level (α) of 0.05. We conduct a two-tailed t-test.

We calculate the test statistic using the formula:

Test Statistic = (sample mean - hypothesized mean) / (sample standard deviation / √sample size)

Substituting the given values, we find the test statistic to be -2.151.

Since the test statistic falls in the rejection region (outside the critical values of the t-distribution for a 95% confidence level), we reject the null hypothesis. This suggests that there is significant evidence to conclude that the mean body temperature is not equal to 98.6°F.

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