Question

refer to the accompanying data set and construct a 90% confidence interval estimate of the mean pulse rate of adult females, then do the same for adult males. compare the results.
click the icon to view the pulse rates for adult females and adult males.
construct a 90% confidence interval of the mean pulse rate for adult females
72.0 bpm < μ < 79.5 bpm
(round to one decimal place as needed.)
construct a 90% confidence interval of the mean pulse rate for adult males
bpm < μ < bpm
(round to one decimal place as needed.)

Explanation:

Step1: Calculate the mean and standard deviation for adult males

Using a calculator or statistical software on the male pulse - rate data:
Let \(n_m\) be the number of male data points. Counting the male data, \(n_m = 40\)
The mean \(\bar{x}_m=\frac{\sum_{i = 1}^{n_m}x_{i,m}}{n_m}\approx69.4\)
The standard deviation \(s_m=\sqrt{\frac{\sum_{i = 1}^{n_m}(x_{i,m}-\bar{x}_m)^2}{n_m - 1}}\approx11.0\)

Step2: Find the critical value

For a \(90\%\) confidence interval and \(n_m-1 = 39\) degrees of freedom (using a \(t\) - distribution, since the population standard deviation is unknown). The critical value \(t_{\alpha/2}\) with \(\alpha=1 - 0.90=0.10\) and \(\alpha/2 = 0.05\). Using a \(t\) - table or calculator, \(t_{0.05,39}\approx1.685\)

Step3: Calculate the margin of error

The margin of error \(E_m=t_{\alpha/2}\frac{s_m}{\sqrt{n_m}}=1.685\times\frac{11.0}{\sqrt{40}}\approx2.9\)

Step4: Construct the confidence interval

The confidence interval is \(\bar{x}_m - E_m<\mu<\bar{x}_m+E_m\)
\(69.4 - 2.9<\mu<69.4 + 2.9\)

Answer:

\(66.5\) bpm \(<\mu<72.3\) bpm