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The resting heart rates for a sample of individuals are normally distributed with a mean of 70 and a standard deviation of 15.

5. About what percent of heart rates are greater than 85?
6. About what percent of heart rates are less than 40?
7. About what percent of heart rates are less than 85?
8. About what percent of heart rates are greater than 55?

Answer :

We are given that the resting heart rates follow a normal distribution with mean
[tex]$$\mu=70$$[/tex]
and standard deviation
[tex]$$\sigma=15.$$[/tex]

Below is a step-by-step approach for each part of the problem.

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1. To find the percent of heart rates greater than 85, we first calculate the corresponding [tex]$z$[/tex]-score.

The [tex]$z$[/tex]-score for a value [tex]$x$[/tex] is given by
[tex]$$
z = \frac{x-\mu}{\sigma}.
$$[/tex]

For [tex]$x=85$[/tex]:
[tex]$$
z = \frac{85-70}{15} = 1.0.
$$[/tex]

The cumulative probability up to [tex]$x=85$[/tex] is the probability that a heart rate is less than 85, which is approximately
[tex]$$\Phi(1.0) \approx 84.13\%.$$[/tex]
Thus, the probability of a heart rate being greater than 85 is
[tex]$$
1 - \Phi(1.0) \approx 1 - 0.8413 = 0.1587,
$$[/tex]
or about [tex]$15.87\%.$[/tex]

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2. For the percent of heart rates less than 40, we compute the [tex]$z$[/tex]-score for [tex]$x=40$[/tex]:

[tex]$$
z = \frac{40-70}{15} = -2.0.
$$[/tex]

The cumulative probability for [tex]$z=-2.0$[/tex] is approximately
[tex]$$\Phi(-2.0) \approx 2.275\%.$$[/tex]
Thus, about [tex]$2.28\%$[/tex] of heart rates are less than 40.

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3. To find the percent of heart rates less than 85, we use the [tex]$z$[/tex]-score computed for 85 (which is [tex]$1.0$[/tex]). The cumulative probability is:
[tex]$$
\Phi(1.0) \approx 84.13\%.
$$[/tex]
So, approximately [tex]$84.13\%$[/tex] of heart rates are less than 85.

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4. Finally, to determine the percent of heart rates greater than 55, we calculate the [tex]$z$[/tex]-score for [tex]$x=55$[/tex]:

[tex]$$
z = \frac{55-70}{15} = -1.0.
$$[/tex]

The cumulative probability for [tex]$z=-1.0$[/tex] is approximately
[tex]$$\Phi(-1.0) \approx 15.87\%.$$[/tex]
Thus, the probability that a heart rate is greater than 55 is
[tex]$$
1 - \Phi(-1.0) \approx 1 - 0.1587 = 0.8413,
$$[/tex]
or about [tex]$84.13\%.$[/tex]

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In summary, the results are:

[tex]$$
\begin{aligned}
\text{Percent of heart rates greater than 85: } & \approx 15.87\% \\
\text{Percent of heart rates less than 40: } & \approx 2.28\% \\
\text{Percent of heart rates less than 85: } & \approx 84.13\% \\
\text{Percent of heart rates greater than 55: } & \approx 84.13\%.
\end{aligned}
$$[/tex]

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