Question

Of the 130 people in Exercise 36 (Section 8.3\(), 65\) were female and 65 were male. \(^{11}\) The means and standard deviations of their temperatures (in degrees Fahrenheit) are shown here. Find a \(95 \%\) confidence interval for the difference in the average body temperatures for males versus females. Based on this interval, can you conclude that there is a difference in the average temperatures for males versus females? Explain.

Short answer

Answer: [Insert the confidence interval calculated in Step 3: Find the 95% confidence interval]. Since the confidence interval [does/does not] contain zero, it can be [concluded/not concluded] that there is a significant difference in the average body temperatures between males and females.

Step-by-step solution

Determine the information given

The exercise tells us that there are 130 people, of which 50% are female (65 individuals) and 50% are male (65 individuals). The means and standard deviations of their temperatures (in degrees Fahrenheit) are given.

Calculate the standard error

The formula for the standard error (SE) of the difference in means is: SE = sqrt( ( (standard deviation of females)^2 / n_females ) + ( (standard deviation of males)^2 / n_males ) ) Here, n_females and n_males are the sample sizes for females and males, respectively. Plug in the given values for the standard deviations and sample sizes to calculate the standard error.

Find the 95% confidence interval

The formula for the confidence interval for the difference between two means is: Confidence interval = (mean difference - critical value * SE, mean difference + critical value * SE) For a 95% confidence interval, the critical value is 1.96. Plug in the given values for the means, the calculated standard error, and the critical value to find the confidence interval.

Determine if there is a significant difference in the average temperatures

Based on the confidence interval, evaluate if there is a significant difference in the average temperatures for males versus females. If the confidence interval contains zero, it means that there might be no significant difference. If the entire range of the confidence interval falls above or below zero, it implies a significant difference between the two means. Based on this criterion, conclude whether there is a difference in the average temperatures between males and females.

Explanation

Based on the confidence interval results, explain if it can be concluded that there is a difference in the average temperatures for males versus females.

Key concepts

Standard Error Calculation

The standard error (SE) is a crucial statistic when comparing two sample means, such as the average body temperatures between males and females. It quantifies the variability of the estimated difference between the sample means. The lower the SE, the more precise is our estimate.

To calculate the SE for the difference in means, we use the formula:
\[\begin{equation}SE = \sqrt{\left(\frac{(SD_{females})^2}{n_{females}}\right) + \left(\frac{(SD_{males})^2}{n_{males}}\right)}\end{equation}\]
where SD_{females} and SD_{males} are the standard deviations for female and male temperatures, respectively, and n_{females} and n_{males} refer to the sample sizes.

To interpret the SE, imagine we are measuring the precision of a sharpshooter's aim. A smaller SE would mean the sharpshooter’s shots are closely grouped, indicating high precision. Similarly, a smaller SE in our context means that our estimate of the difference in average temperatures is precise and reliable.

Sample Size Significance

The sample size plays a pivotal role when making inferences about populations from samples. In the context of our exercise, both the male and female groups have equal sample sizes of 65. The significance of having adequately sized samples cannot be understated; it affects the standard error and, consequently, the width of our confidence interval.

Larger sample sizes typically result in a smaller standard error. Why is this important? Think of a sample as a snapshot of a population. A larger snapshot can give us a clearer picture. In statistical terms, more data points can provide a better estimate of the population parameter and make our confidence interval narrower, hence making our conclusions more robust.

Mean Difference Analysis

When we speak of 'mean difference analysis', we're looking at whether the observed difference between two sample means, in this case, the temperatures of males and females, is statistically significant or not. The analysis is rooted in the confidence interval we construct.

The confidence interval provides a range within which we expect the true difference in population means to lie, given a level of confidence (95% in our exercise). If this interval does not include zero, it suggests a significant difference between the groups. However, if zero is within this range—as it acts as the 'no difference' marker—it implies that any observed difference in sample means could simply be due to random chance.

Therefore, by analyzing the mean difference through the lens of the confidence interval, we can draw conclusions about the significance of the observed temperature difference between male and female individuals in our exercise.