The required 90% confidence interval for adult males is
[tex]\text {CI} = (64.2, \: 70.6)\\\\[/tex]
The required 90% confidence interval for adult females is
[tex]\text {CI} = (72, \: 79.2)\\\\[/tex]
The confidence interval of male and female pulse rates do not overlap since the mean pulse rate of female is way greater than the mean pulse rate of males.
Step-by-step explanation:
We are given the pulse rates of adult females and adult males and we have to construct the 90% confidence interval of the mean pulse rate for males and females.
Let us first compute the mean and standard deviation of the given pulse rates data.
Using Excel,
=AVERAGE(number1, number2,....)
The mean pulse rate of adult males is found to be
[tex]\bar{x}_{male} = 67.4[/tex]
The mean pulse rate of adult females is found to be
[tex]\bar{x}_{female} = 75.6[/tex]
Using Excel,
=STDEV(number1, number2,....)
The standard deviation for adult male pulse rate is found to be
[tex]s_{male} = 11.9[/tex]
The standard deviation for adult female pulse rate is found to be
[tex]s_{female} = 13.5[/tex]
The confidence interval is given by
[tex]$ \text {CI} = \bar{x} \pm t_{\alpha/2}(\frac{s}{\sqrt{n} } ) $\\\\[/tex]
Where [tex]\bar{x}[/tex] is the sample mean, n is the sample size, s is the sample standard deviation and [tex]t_{\alpha/2}[/tex] is the t-score corresponding to a 90% confidence level.
The t-score corresponding to a 90% confidence level is
Significance level = α = 1 - 0.90 = 0.10/2 = 0.05
Degree of freedom = n - 1 = 40 - 1 = 39
From the t-table at α = 0.05 and DoF = 39
t-score = 1.685
The required 90% confidence interval for adult males is
[tex]\text {CI} = 67.4 \pm 1.685\cdot \frac{11.9}{\sqrt{40} } \\\\\text {CI} = 67.4 \pm 1.685\cdot 1.882\\\\\text {CI} = 67.4 \pm 3.17\\\\\text {CI} = 67.4 - 3.17, \: 67.4 + 3.17\\\\\text {CI} = (64.2, \: 70.6)\\\\[/tex]
Therefore, we are 90% confident that the actual mean pulse rate of adult male is within the range of 64.2 to 70.6 bpm
The required 90% confidence interval for adult females is
[tex]\text {CI} = 75.6 \pm 1.685\cdot \frac{13.5}{\sqrt{40} } \\\\\text {CI} = 75.6 \pm 1.685\cdot 2.1345\\\\\text {CI} = 75.6 \pm 3.60\\\\\text {CI} = 75.6 - 3.60, \: 75.6 + 3.60\\\\\text {CI} = (72, \: 79.2)\\\\[/tex]
Therefore, we are 90% confident that the actual mean pulse rate of adult female is within the range of 72 to 79.2 bpm
Comparison:
The confidence interval of male and female pulse rates do not overlap since the mean pulse rate of female is way greater than the mean pulse rate of males.
