Question

Assume that females have pulse rates that are normally distributed with a mean of u = 75.0 beats per minute and a standard deviation of sigma = 12.5 beats per minute. If 1 adult female is randomly​ selected, find the probability that her pulse rate is between 69 beats per minute and 81 beats per minute.

Answer 1

`P(69<X<81)=P((69-\mu)/(\sigma)<(X-\mu)/(\sigma)<(81-\mu)/(\sigma))=P((69-75)/(12.5)<Z<(81-75)/(12.5))=P(-0.48<z<0.48)`

And we can find this probability with this difference:

`P(-0.48<z<0.48)=P(z<0.48)-P(z<-0.48)=0.684-0.316= 0.368`

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Solution to the problem

Let X the random variable that represent the pulse rates of a population, and for this case we know the distribution for X is given by:

`X \sim N(75,12.5)`

Where
`\mu=75` and
`\sigma=12.5`

We are interested on this probability

`P(69<X<81)`

And the best way to solve this problem is using the normal standard distribution and the z score given by:

`z=(x-\mu)/(\sigma)`

If we apply this formula to our probability we got this:

`P(69<X<81)=P((69-\mu)/(\sigma)<(X-\mu)/(\sigma)<(81-\mu)/(\sigma))=P((69-75)/(12.5)<Z<(81-75)/(12.5))=P(-0.48<z<0.48)`

And we can find this probability with this difference:

`P(-0.48<z<0.48)=P(z<0.48)-P(z<-0.48)=0.684-0.316= 0.368`

Answer 2

36.88% probability that her pulse rate is between 69 beats per minute and 81 beats per minute.

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (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 pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 75, \sigma = 12.5`

Find the probability that her pulse rate is between 69 beats per minute and 81 beats per minute.

This is the pvalue of Z when X = 81 subtracted by the pvalue of Z when X = 69.

X = 81

`Z = (X - \mu)/(\sigma)`

`Z = (81 - 75)/(12.5)`

`Z = 0.48`

`Z = 0.48` has a pvalue of 0.6844

X = 69

`Z = (X - \mu)/(\sigma)`

`Z = (69 - 75)/(12.5)`

`Z = -0.48`

`Z = -0.48` has a pvalue of 0.3156

0.6844 - 0.3156 = 0.3688

36.88% probability that her pulse rate is between 69 beats per minute and 81 beats per minute.