Chapter 15, Problem 15.32E

(a)

To determine

To find: The approximate distribution of $\overline{x}$ according to the central limit theorem.

Answer to Problem 15.32E

The approximate distribution of $\overline{x}$ is normal with mean $\mu =8.8\text{}\text{beats}\text{per}\text{five}\text{seconds}$ and the standard deviation is $\frac{\sigma }{\sqrt{n}}=0.2041\text{beats}\text{per}\text{five}\text{seconds}$.

Explanation of Solution

Given info:

The data shows that the mean heart rate $\left(\mu \right)$ is 8.8 beats per five seconds and the standard deviation $\left(\sigma \right)$ is 1 beat per five seconds. The distribution of heart rates is not normal.

Calculation:

Central limit theorem:

It states that as the sample size n becomes large, the sampling distribution of $\overline{x}$ follows normal distribution with mean $\mu$ and standard deviation $\frac{\sigma }{\sqrt{n}}$.

Mean of the Sampling distribution of $\overline{x}$:

The mean of the sampling distribution of $\overline{x}$ is $\mu$. That is, 8.8.

Thus, the mean of the average heart rate $\overline{x}$ is 8.8 beats per five seconds.

Standard deviation of the Sampling distribution of $\overline{x}$:

The standard deviation can be calculated by using the formula $\frac{\sigma }{\sqrt{n}}$.

Substitute 1 for $\sigma$ and 24 for n

$\begin{array}{c}\frac{\sigma }{\sqrt{n}}=\frac{1}{\sqrt{24}}\\ =\frac{1}{4.8990}\\ =0.2041\end{array}$

Thus, the standard deviation of $\overline{x}$ is 0.2041 beats per five seconds.

Hence, by the central limit theorem,the approximate distribution of $\overline{x}$ is normal with mean $8.8\text{}\text{beats}\text{per}\text{five}\text{seconds}$ and standard deviation is $0.2041\text{beats}\text{per}\text{five}\text{seconds}$.

(b)

To determine

To find: The approximate probability that $\overline{x}$ less than 8.

Answer to Problem 15.32E

The approximate probability of $\overline{x}$ less than 8 is 0.

Explanation of Solution

Calculation:

STATE:

The mean heart rate $\left(\mu \right)$ is 8.8 beats per five seconds and the standard deviation $\left(\sigma \right)$ is 1 beat per five seconds. What is the approximate probability that $\overline{x}$ is less than 8?

PLAN:

The distribution of heart rates is not normal. Then, the central limit theorem would be used to find the distribution of mean heart rate.

SOLVE:

Let X be the number of heart beats per five seconds and $\overline{x}$ be the mean number of beats per five seconds.

From the part a, $\overline{x}$ is normal with mean $\mu =8.8\text{}\text{beats}\text{per}\text{five}\text{seconds}$ and the standard deviation is $\frac{\sigma }{\sqrt{n}}=0.2041\text{beats}\text{per}\text{five}\text{seconds}$.

Then,

$\begin{array}{c}P\left(\overline{x}<8\right)=P\left(\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{8-8.8}{\frac{1}{\sqrt{24}}}\right)\\ =P\left(Z<\frac{-0.8}{0.2041}\right)\\ =P\left(Z<-3.92\right)\\ \approx 0\left[\begin{array}{l}\text{From}\text{Table}\text{A:}\text{Standard}\text{Normal}\\ \text{Cumulative}\text{Proportions}\end{array}\right]\end{array}$

Thus, the approximate probability of $\overline{x}$ less than 8 is 0.

CONCLUSION:

The approximate probability of $\overline{x}$ less than 8 is 0.

(c)

To determine

To find: The approximate probability that the heart rate of a runner is less than 100 beats per minute.

Answer to Problem 15.32E

The approximate probability that the heart rate of a runner is less than 100 beats per minute is 0.0107.

Explanation of Solution

Calculation:

The number of five seconds in one minute is $\frac{60}{5}=12$.

The mean heart rate for five seconds is $\frac{100}{12}=8.33\text{beats}\text{per}\text{five}\text{seconds}$.

Then,

$\begin{array}{c}P\left(\overline{x}<8\right)=P\left(\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{8.33-8.8}{\frac{1}{\sqrt{24}}}\right)\\ =P\left(Z<\frac{-0.47}{0.2041}\right)\\ =P\left(Z<-2.30\right)\\ =0.0107\left[\begin{array}{l}\text{From}\text{Table}\text{A:}\text{Standard}\text{Normal}\\ \text{Cumulative}\text{Proportions}\end{array}\right]\end{array}$

Thus, the approximate probability that the heart rate of a runner is less than 100 beats per minute is 0.0107.