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The population of female pulse rates is normally distributed with mean equal to 74.0 beats per minute and standard deviation equal to 12.5 beats per minute.

Let the population mean pulse rate be $渭=74.0\text{beats}\text{per}\text{minute}$.

Let the population standard deviation of beats per minute $蟽=12.5\text{beats}\text{per}\text{minute}$.

The z-score for a given sample observation has the following expression:

$z=\frac{x-渭}{蟽}$

The z-score for the sample mean has the following expression:

$z=\frac{\stackrel{炉}{x}-渭}{\frac{蟽}{\sqrt{n}}}$

a.

The sample value should lie between 72 beats per minute and 76 beats per minute.

The probability that the sample value will lie between 72 beats per minute and 76 beats per minute is computed below:

$$\begin{gathered} P\left(72<x<76\right)=P\left(\frac{72-渭}{蟽}<\frac{x-渭}{蟽}<\frac{76-渭}{蟽}\right) \\ =P\left(\frac{72-74}{12.5}<z<\frac{76-74}{12.5}\right) \\ =P\left(-0.16<z<0.16\right) \\ =P\left(z<0.16\right)-P\left(Z<-0.16\right) \end{gathered}$$

Using the standard normal table, the required probability can be computed as:

$$\begin{gathered} P\left(72<x<76\right)=P\left(z<0.16\right)-P\left(Z<-0.16\right) \\ =P\left(z<0.16\right)-\left(1-P\left(Z<0.16\right)\right) \\ =0.5636-\left(1-0.5636\right) \\ =0.1272 \end{gathered}$$

Therefore, the probability that the pulse rate of the selected female is between 72 beats per minute and 76 beats per minute is equal to 0.1272.

b.

Let the sample size be equal to n = 4

The sample mean should lie between 72 beats per minute and 76 beats per minute.

The probability that the sample mean will lie between72 beats per minute and 76 beats per minute is equal to:

$$\begin{gathered} P\left(72<\stackrel{炉}{x}<76\right)=P\left(\frac{72-渭}{\frac{蟽}{\sqrt{n}}}<\frac{\stackrel{炉}{x}-渭}{\frac{蟽}{\sqrt{n}}}<\frac{76-渭}{\frac{蟽}{\sqrt{n}}}\right) \\ =P\left(\frac{72-74}{\frac{12.5}{\sqrt{4}}}<z<\frac{76-74}{\frac{12.5}{\sqrt{4}}}\right) \\ =P\left(-0.32<z<0.32\right) \\ =P\left(z<0.32\right)-P\left(Z<-0.32\right) \end{gathered}$$

Using the standard normal table, the required probability can be computed as:

$$\begin{gathered} P\left(72<\stackrel{炉}{x}<76\right)=P\left(z<0.32\right)-P\left(Z<-0.32\right) \\ =0.6255-0.3745 \\ =0.2510 \end{gathered}$$

Therefore, the probability that for a sample of 4 females, their mean pulse rate is between 72 beats per minute and 76 beats per minute is equal to 0.2510.

c.

Although the sample size equal to 4 is less than 30 and is not large, it is given that the population of female pulse rates is normally distributed.

Hence, the sample mean female pulse rate can be assumed to follow the normal distribution.