Question

Assume that females have pulse rates that are normally distributed with a mean of beats per minute and a standard deviation of beats per minute.
a. If 1 adult female is randomly​ selected, find the probability that her pulse rate is between beats per minute and beats per minute.
b. If 4 adult females are selected, find the probability that they have pulse rates with a mean between 68 beats per minute and 76 beats per minute.
c. Why can the normal distribution be used in heartbeat even the sample side does not exceed 30?

Answer 1

Answer:

a) This is the p-value of Z when X = A subtracted by the p-value of Z when X = B.

b) P-value of Z when X = 76 subtracted by the p-value of X = 68.

c) Because the underlying distribution(pulse rates of females) is normal.

Step-by-step explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this question:

Mean $\mu$, standard deviation $\sigma$.

standard deviation of beats per minute.

a. If 1 adult female is randomly​ selected, find the probability that her pulse rate is between A beats per minute and B beats per minute.

This is the p-value of Z when X = A subtracted by the p-value of Z when X = B.

b. If 4 adult females are selected, find the probability that they have pulse rates with a mean between 68 beats per minute and 76 beats per minute.

Sample of 4 means that we have $n = 4, s = \frac{\sigma}{\sqrt{4}} = 0.5\sigma$

The formula for the z-score is:

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{X - \mu}{0.5\sigma}$

$Z = 2\frac{X - \mu}{\sigma}$

This probability is the p-value of Z when X = 76 subtracted by the p-value of X = 68.

c. Why can the normal distribution be used in heartbeat even the sample side does not exceed 30?

Because the underlying distribution(pulse rates of females) is normal.