Chapter 7, Problem 7.20TY

a.

To determine

To obtain: The percent of healthy adults who have resting pulse rates that are below 50 beats per minute.

Answer to Problem 7.20TY

The percent of healthy adults who have resting pulse rates that are below 50 beats per minute is 1.25%.

Explanation of Solution

Given info:

The resting pulse rates for healthy adults follow a normal distribution with mean 69 beats per minute and standard deviation 8.5 beats per minute.

Calculation:

Define the random variable x as resting pulse rates for healthy adults.

The formula for the standardized score is

$z=\frac{x-\mu }{\sigma }$

The healthy adults who have resting pulse rates that are below 50 beats per minute is denoted as $x<50$.

Subtract the mean and then divide by the standard deviation to transform the value of x into standard normal z. The standardized score is

$\begin{array}{c}\frac{x-69}{8.5}<\frac{50-69}{8.5}\\ <-\frac{19}{8.5}\\ z<-2.24\end{array}$

The percent of healthy adults who have resting pulse rates that are below 50 beats per minute is obtained by finding the area to left of –2.24.

Use Table A: Standard normal cumulative proportions to find the area to the left of –2.24.

Procedure:

• Locate –2.2 in the left column of the A-2 Table.
• Obtain the value in the corresponding row below 0.04.

That is, $P\left(z\le -2.24\right)=0.0125$

Thus, the percent of healthy adults who have resting pulse rates that are below 50 beats per minute is 1.25%.

b.

To determine

To obtain: The percent of healthy adults who have resting pulse rates that exceed 85 beats per minute.

Answer to Problem 7.20TY

The percent of healthy adults who have resting pulse rates that exceed 85 beats per minute is 3.01%.

Explanation of Solution

Calculation:

The formula for the standardized score is

$z=\frac{x-\mu }{\sigma }$

The healthy adults who have resting pulse rates that exceed 85 beats per minute is denoted as $x>85$.

Subtract the mean and then divide by the standard deviation to transform the value of x into standard normal z.

$\begin{array}{c}\frac{x-69}{8.5}>\frac{85-69}{8.5}\\ >\frac{16}{8.5}\\ z>1.88\end{array}$

Where, standardized score $z=\frac{x-69}{8.5}$

The percent of healthy adults who have resting pulse rates that exceed 85 beats per minute is obtained by finding the area to right of 1.88.

But, the Table A: Standard normal cumulative proportions apply only for cumulative areas from the left.

Use Table A: Standard normal cumulative proportions to find the area to the left of 1.88.

Procedure:

• Locate 1.8 in the left column of the A-2 Table.
• Obtain the value in the corresponding row below 0.08.

That is, $P\left(z<1.88\right)=0.9699$

The area to the right of 1.88 is

$\begin{array}{c}P\left(z>1.88\right)=1-P\left(z<1.88\right)\\ =1-0.9699\\ =0.0301\end{array}$

Thus, the percent of healthy adults who have resting pulse rates that exceed 85 beats per minute is 3.01%.

c.

To determine

To obtain: The specification limits for the central 80% of resting pulse rates.

Answer to Problem 7.20TY

The specification limits for the central 80% of resting pulse rates lie between 58.1 beats per minute and 79.9 beats per minute.

Explanation of Solution

Calculation:

The central 80% represents the area of 0.80 in between the values of x.

The remaining area from central 80% is 20% $(=100\%-80\%)$.

The area 20% represents the area of 10% to the left of lower limit and 10% to right of upper limit.

For lower limit:

Use Table A: Standard normal cumulative proportions to find z score.

Procedure:

• Locate the nearest value of 0.10 in the body of the Table A-2.
• Move left until the first column and note the value as –1.2.
• Move upward until the top row is reached and note the value as 0.08.

Thus, the z score is –1.28.

The formula to find the lower limit is

$x=\mu +\left(z\times \sigma \right)$

Substitute 69 for μ, 8.5 for σ, and –1.28 for z

$\begin{array}{c}x=69+\left(-1.28\times 8.5\right)\\ =69-10.88\\ =58.12\\ =58.1\end{array}$

Thus, the lower limit is 58.1.

For upper limit:

Here, the Table A: Standard normal cumulative proportion applies only for cumulative areas from the left. Thus,

$\begin{array}{c}\left(\begin{array}{l}\text{Cumulative proportion }\\ \text{to the left of }z\text{-score}\end{array}\right)=1-\left(\begin{array}{l}\text{Cumulative proportion }\\ \text{to the right of }z\text{-score}\end{array}\right)\\ =1-0.10\\ =0.90\end{array}$

Procedure:

• Locate the nearest value of 0.90 in the body of the Table A-2.
• Move left until the first column and note the value as 1.2.
• Move upward until the top row is reached and note the value as 0.08.

Thus, the z score is 1.28.

The formula to find the upper limit is

$x=\mu +\left(z\times \sigma \right)$

Substitute 69 for μ, 8.5 for σ,   and 1.28 for z

$\begin{array}{c}x=69+\left(1.28\times 8.5\right)\\ =69+10.88\\ =79.88\\ \mathrm{=79.9}\end{array}$

Thus, the upper limit is 79.9.

Hence, the specification limits for the central 80% of resting pulse rates are (58.1, 79.9).