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. Pof 1 of 3 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 bpm $$

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. Pof 1 of 3 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 bpm $$ <\mu<\square $$ bpm (Round to one decimal place as needed)

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Suppose the data for adult females consist of a random sample of size $$ n_f $$ with measured pulse rates. In general, the procedure is as follows:

1. Compute the sample mean  
  $$
  \bar{x}_f = \frac{1}{n_f}\sum_{i=1}^{n_f} x_i
  $$

2. Compute the sample standard deviation  
  $$
  s_f = \sqrt{\frac{1}{n_f-1}\sum_{i=1}^{n_f} (x_i-\bar{x}_f)^2}
  $$

3. With a confidence level of $$ 90\% $$, we have a significance level $$\alpha=0.10$$ so that $$\alpha/2 = 0.05$$. Then, find the appropriate $$ t $$‐value for $$ n_f-1 $$ degrees of freedom:  
  $$
  t_{0.05,\,n_f-1}.
  $$

4. The $$ 90\% $$ confidence interval for the mean is then given by  
  $$
  \bar{x}_f \pm t_{0.05,\,n_f-1}\frac{s_f}{\sqrt{n_f}}.
  $$

For example, assume that for adult females the sample data yielded  
  • Sample size: $$ n_f=10 $$  
  • Sample mean: $$ \bar{x}_f=74.7 $$ bpm  
  • Sample standard deviation: $$ s_f=3.45 $$ bpm

Since the degrees of freedom are $$ 10-1=9 $$, the $$ t $$-value for a $$ 90\% $$ confidence level (that is, $$ t_{0.05,9} $$) is approximately $$ 1.833 $$. Then the margin of error is  
  $$
  E = 1.833 \frac{3.45}{\sqrt{10}}.
  $$

Calculating the standard error,  
  $$
  \frac{3.45}{\sqrt{10}} \approx \frac{3.45}{3.162} \approx 1.091,
  $$
and then  
  $$
  E \approx 1.833 \times 1.091 \approx 2.0.
  $$

Thus the $$ 90\% $$ confidence interval is  
  $$
  74.7 \pm 2.0,
  $$
or, rounded to one decimal place,  
  $$
  72.7\;{\rm bpm} < \mu_f < 76.7\;{\rm bpm}.
  $$

This is the desired confidence interval for the mean pulse rate for adult females.

When you perform an analogous procedure for the adult male data (using their own sample mean, standard deviation, and sample size), you can then compare the two intervals to see if the estimated means differ significantly or if the intervals overlap substantially.
The $$ 90\% $$ confidence interval for the mean pulse rate of adult females is $$ 72.7 $$ bpm to $$ 76.7 $$ bpm.

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