Answer :
Final answer:
The student is asked to perform a statistical analysis on a set of body temperature data, split by gender. The analysis includes creating histograms, constructing confidence intervals, and conducting statistical tests of significance to evaluate differences from an assumed average body temperature of 98.6F.
Explanation:
This is a statistics problem involving data analysis. We are given a data set and asked to perform various statistical operations on it. The data set includes body temperatures, gender, and heart rates for a group of people, but we are only concerned with the body temperatures in this analysis. We are also given the gender of each individual (1=male, 2=female).
It's important to note that we cannot provide the actual calculations or graphical representations here, but we can explain the process. First, you would input the data into a statistical software tool, such as Excel or R, and separate the data by gender.
Next, for each gender, we prepare a histogram and a normal probability plot (NPP) of the temperatures. These graphical methods allow us to evaluate the normality of the data. Then, we construct 95% and 99% confidence intervals for the mean body temperature of each gender.
For example, if the 95% confidence interval for men does not include 98.6◦F, we would conclude that the true average body temperature for men is not 98.6◦F. The same process would be followed for a 99% confidence interval and for the data corresponding to women.
Finally, you would conduct a statistical test at a significance level of α=0.99 to determine whether the average temperature for each gender is different from 98.6◦F. If the p-value from the test is less than 0.01 (the complement of the given alpha level), we would reject the hypothesis that the average temperature is 98.6◦F.
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"The correct options are as follows:
For part (b), using the men's data, the 95% confidence interval for the mean body temperature of men does not include 98.6°F, and the 99% confidence interval also does not include 98.6°F. This suggests that the true mean body temperature for men is likely different from 98.6°F.
For part (c), using the women's data, the 95% confidence interval for the mean body temperature of women does not include 98.6°F, and the 99% confidence interval also does not include 98.6°F. This suggests that the true mean body temperature for women is likely different from 98.6°F.
For part (d), the statistical test with α = 0.01 for men's average temperature shows that there is a significant difference between the sample mean and 98.6°F, indicating that the average temperature for men is different from the widely believed value.
For part (e), the statistical test with α = 0.01 for women's average temperature shows that there is a significant difference between the sample mean and 98.6°F, indicating that the average temperature for women is different from the widely believed value.
To arrive at the conclusions, we would perform the following steps:
1. **Histogram and Normal Probability Plot (NPP)**:
- For men and women separately, we would plot histograms of their body temperatures to visually assess the distribution.
- We would also create NPPs to check if the data points follow a straight line, which would indicate a normal distribution.
2. **Confidence Intervals**:
- Calculate the sample mean and standard deviation for men and women separately.
- Use the t-distribution to calculate the confidence intervals, as the sample size is finite and the population standard deviation is unknown.
- For a 95% confidence interval, we would use the t-value from the t-distribution table corresponding to 64 degrees of freedom (n-1) and a 5% significance level.
- For a 99% confidence interval, we would use the t-value from the t-distribution table corresponding to 64 degrees of freedom and a 1% significance level.
- The confidence interval is given by:
[tex]\[ \text{CI} = \bar{x} \pm t_{\alpha/2, n-1} \times \frac{s}{\sqrt{n}} \][/tex]
where [tex]\(\bar{x}\)[/tex] is the sample mean, [tex]\(s\)[/tex] is the sample standard deviation, \(n\) is the sample size, and [tex]\(t_{\alpha/2, n-1}\)[/tex] is the t-value for the given confidence level and degrees of freedom.
3. **Hypothesis Testing**:
- Set up the null hypothesis ([tex]\(H_0\)[/tex]) that the mean body temperature is 98.6°F.
- Set up the alternative hypothesis ([tex]\(H_1[/tex]) that the mean body temperature is not 98.6°F (two-tailed test).
- Calculate the test statistic using the formula:
[tex]\[ t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \][/tex]
where [tex]\(\mu_0\)[/tex] is the hypothesized mean (98.6°F).
- Compare the calculated test statistic to the critical t-value from the t-distribution table for α = 0.01 and 64 degrees of freedom.
- If the absolute value of the test statistic is greater than the critical value, reject the null hypothesis, indicating that the average temperature is significantly different from 98.6°F.
By performing these calculations, we can determine whether the confidence intervals include the value of 98.6°F and whether the hypothesis tests lead us to reject the notion that 98.6°F is the true mean body temperature for men and women. The provided data snippet suggests that both the confidence intervals and the hypothesis tests will show that the true mean body temperature for both genders is likely different from 98.6°F."
