Question

Of the 130 people in Exercise 8.37,65 were female and 65 were male. The means and standard deviation of their temperatures are shown below. $$\begin{array}{lrr} & \text { Men } & \text { Women } \\\\\hline \text { Sample Mean } & 98.11 & 98.39 \\\\\text { Standard Deviation } & 0.70 & 0.74\end{array}$$ 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: Yes, there is a statistically significant difference in the average body temperatures between males and females. The 95% confidence interval does not include 0 and it is negative, indicating that the average body temperature of males is lower than that of females.

Step-by-step solution

Find the Point Estimate (Difference in Means)

The point estimate for the difference in average temperatures between males and females is simply the difference in sample means, which we can denote as \(\bar{X}_m - \bar{X}_f\). Based on the given data, we can calculate this value as follows: $$ \bar{X}_m - \bar{X}_f = 98.11 - 98.39 = -0.28 \hspace{10mm} (1) $$

Calculate the Standard Error

Next, we need to find the standard error for the difference in means, which we can calculate using the formula: $$ SE = \sqrt{\frac{(s_m^2)}{n_m} + \frac{(s_f^2)}{n_f}} $$ Where \(s_m\) and \(s_f\) are the standard deviations for males and females, and \(n_m\) and \(n_f\) are the sample sizes. Using the given data, we can calculate the standard error: $$ SE = \sqrt{\frac{(0.70^2)}{65} + \frac{(0.74^2)}{65}} = 0.1258 \hspace{10mm} (2) $$

Find the Confidence Interval

Now, we can find the \(95\%\) confidence interval using the point estimate, standard error, and the critical value (\(z\)) for a \(95\%\) confidence level. The critical value for \(95\%\) confidence is \(z = 1.96\). We can calculate the margin of error (ME) and then the confidence interval: $$ ME = z \times SE = 1.96 \times 0.1258 = 0.2466 $$ Now, let's find the lower limit (LL) and upper limit (UL) of the confidence interval: $$ LL = (\bar{X}_m - \bar{X}_f) - ME = -0.28 - 0.2466 = -0.5266 $$ $$ UL = (\bar{X}_m - \bar{X}_f) + ME = -0.28 + 0.2466 = -0.0334 $$ Thus, the \(95 \%\) confidence interval for the difference in average body temperatures for males versus females is \((-0.5266, -0.0334)\).

Conclude based on the Confidence Interval

Since the \(95\%\) confidence interval does not include \(0\), we can confidently say that there is a statistically significant difference in the average body temperatures for males and females. The negative sign of the confidence interval values indicates that the average body temperature of males is lower than that of females.

Key concepts

Point Estimate

When we talk about a point estimate, we're referring to a single value as the best guess or estimate of a population parameter. In this exercise, the point estimate focuses on the difference between the average body temperatures of males and females. Specifically, we're looking at the difference between their sample means. The calculation is simple: you subtract the sample mean of men's temperatures from that of women's. In this case: 98.11 (men) - 98.39 (women) = -0.28.
This result means that, on average, men's temperatures are 0.28 degrees Fahrenheit lower than women's. It's a snapshot of the general temperature difference you'll apply inside your confidence interval calculations.
Key points about point estimates:

• They are derived from sample data.
• They provide a single value estimate for the population parameter.
• They often serve as the baseline for more complex statistical analyses, like confidence intervals.

Standard Error

Standard error measures how far the sample mean of the data is expected to be from the true population mean. In essence, it helps to give an idea of the accuracy of the point estimate. For differences in means, like in this exercise, the standard error combines the variability of both groups' estimates.
Mathematically, it's calculated with the formula:
\[ SE = \sqrt{\frac{(s_m^2)}{n_m} + \frac{(s_f^2)}{n_f}} \]
where \( s_m \) and \( s_f \) are the standard deviations of men's and women's temperatures, while \( n_m \) and \( n_f \) are their respective sample sizes. In this case, the standard error was found to be 0.1258.
This value helps us assess how much variation we might expect if we took many samples. A smaller standard error implies more precise estimates of the mean difference. Remember:

• Standard error decreases with larger sample sizes.
• It reflects the inherent variability in the sample data.
• It is crucial for constructing confidence intervals.

Statistical Significance

Statistical significance helps us understand whether an observed effect in the data is unlikely due to random chance. When examining the confidence interval, if it does not include 0, it suggests a statistically significant difference between groups.
In this scenario, the confidence interval calculated was \((-0.5266, -0.0334)\). Since zero is not within this range, we can confidently say there is a statistically significant difference in the average body temperatures of men and women.
This result signifies that the observed difference (with men averaging lower temperatures than women) is unlikely to occur just by random sampling variability.
Some important points on statistical significance:

• A confidence interval not including 0 indicates significance in this context.
• Statistical significance does not imply practical significance; even small differences can be statistically significant with large samples.
• The context of the data and its implications should be considered alongside statistical significance.