91Ó°ÊÓ

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 body temperatures for 25 female subjects follow: 97.8,97.2 97.4,97.6,97.8,97.9,98.0,98.0,98.0,98.1,98.2,98.3,98.3 98.4,98.4,98.4,98.5,98.6,98.6,98.7,98.8,98.8,98.9,98.9 and 99.0 . a. Test the hypothesis \(H_{0}: \mu=98.6\) versus \(H_{1}: \mu \neq 98.6\) using \(\alpha=0.05 .\) Find the \(P\) -value. b. Check the assumption that female body temperature is normally distributed. c. Compute the power of the test if the true mean female body temperature is as low as 98.0 . d. What sample size would be required to detect a true mean female body temperature as low as 98.2 if you wanted the power of the test to be at least \(0.9 ?\) e. Explain how the question in part (a) could be answered by constructing a two-sided confidence interval on the mean female body temperature.

Step by step solution

01

Calculate Mean and Standard Deviation

First, calculate the sample mean \( \bar{x} \) and the sample standard deviation \( s \) for the given body temperature data. The sample mean is the sum of all temperatures divided by the number of samples (25). Then, use the formula for the standard deviation \( s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2} \) after computing each squared deviation from the mean.

02

Perform Hypothesis Test

For part (a), set up a two-tailed t-test for the hypothesis \( H_0: \mu = 98.6 \) versus \( H_1: \mu eq 98.6 \). Use the test statistic \( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \). Calculate the t-value and compare it to the critical t-value for \( \alpha = 0.05 \) with 24 degrees of freedom. Also, find the corresponding p-value from the t-distribution.

03

Check Normality Assumption

For part (b), construct a Q-Q plot or use a normality test such as the Shapiro-Wilk test on the temperature data to check for normality. The data are considered normally distributed if they closely follow the straight line in a Q-Q plot or if the p-value from the Shapiro-Wilk test is above the significance level (commonly \( \alpha = 0.05 \)).

04

Calculate the Power of the Test

For part (c), determine the power of the test if the true mean \( \mu \) is 98.0. Use the formula for power including the sample size, estimated standard deviation, true mean, significance level \( \alpha \), and critical t-value. Power is the probability of correctly rejecting \( H_0 \) when \( \mu = 98.0 \).

05

Determine Required Sample Size

For part (d), use the desired power level and the alternative mean \( \mu = 98.2 \) to determine the required sample size. Use the formula \( n = \frac{(Z_{\beta} + Z_{\alpha/2})^2 \times s^2}{(\mu - \mu_0)^2} \), where \( Z_{\beta} \) is the z-score for the desired power.

06

Construct Confidence Interval

For part (e), calculate the 95% confidence interval using the sample mean, standard deviation, and t-distribution critical value. The interval formula is \( (\bar{x} - t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}, \bar{x} + t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}) \). If 98.6 lies within this interval, \( H_0 \) is not rejected; otherwise, it is.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution

A normal distribution is a key concept in statistics. It's often referred to as a "bell curve" because of its shape. Understanding this distribution is crucial because many statistical tests assume that data follow this pattern. In a normal distribution, data points are symmetrically spread around the mean, forming a bell-shaped curve. Most values cluster around the central peak, and there are fewer extreme values in the tails.

Checking if a dataset like body temperature follows a normal distribution is important for hypothesis testing. This can be done using graphical methods, like a Q-Q plot, or statistical tests, such as the Shapiro-Wilk test. If data conform to a normal distribution, the results of hypothesis testing are more reliable. So, when analyzing body temperature data, ensure that it resembles the normal distribution before proceeding with the hypothesis test. This helps to justify the use of parametric tests, like the t-test, which rely on this assumption.

Sample Size Calculation

Sample size calculation is critical in designing studies and experiments. It determines the number of observations needed to detect an effect of a certain size with a specified power and significance level. Having an adequate sample size ensures that the study results are valid and not due to random chance.

In hypothesis testing, like in testing body temperatures, selecting the correct sample size helps in detecting differences between the hypothesized mean and the true population mean. The formula used for sample size calculation includes the desired power of the test, the standard deviation, the difference between the hypothesized mean and the true mean, and the significance level.

• The smaller the effect size you want to detect, the larger the sample size needed.
• A higher desired power also requires a larger sample size.
• If the standard deviation is high, more samples are needed to achieve the same accuracy.

Calculating the appropriate sample size beforehand helps in ensuring the reliability and validity of the test outcomes.

Confidence Interval

A confidence interval provides a range of values that is likely to contain the population parameter. For instance, constructing a confidence interval for the mean body temperature gives us a range within which the true average temperature likely falls.

A 95% confidence interval is commonly used and gives us a range around the sample mean, suggesting that if we were to take many samples, 95% of the calculated intervals would contain the true population mean. The confidence interval is given by:\[(\bar{x} - t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}, \bar{x} + t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}})\]where \( \bar{x} \) is the sample mean, \( t_{\alpha/2, n-1} \) is the t-value from the t-distribution, \(s\) is the standard deviation, and \(n\) is the sample size. If 98.6 °F falls within this interval, it suggests that the body temperature is consistent with the hypothesized mean. Otherwise, it suggests a significant difference, lending support to the alternative hypothesis.

p-value

The p-value is a statistic that helps us determine the significance of our hypothesis test results. It provides a measure of the probability that the observed data would occur under the null hypothesis. A small p-value indicates that such results would be very unlikely if the null hypothesis were true.

In the context of testing body temperatures, we compare the p-value with our significance level (usually \( \alpha = 0.05 \)). If the p-value is less than the significance level, we reject the null hypothesis. This means there is enough evidence to suggest that the true mean body temperature isn't 98.6 °F.

• A small p-value (e.g., <0.05) suggests strong evidence against the null hypothesis.
• A large p-value suggests that the data are consistent with the null hypothesis, meaning any observed differences could be due to random chance.

Thus, the p-value quantifies the strength of evidence against the null hypothesis, guiding us in making informed conclusions based on statistical analysis.