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Assume that females have pulse rates that are normally distributed with a mean of mu equals 74.0 beats per minute and a standard deviation ofsigma equals 12.5 beats per minute. If 25 adult females are randomly​ selected, find the probability that they have pulse rates with a mean between 68 beats per minute and 80 beats per minute.

Question

Assume that females have pulse rates that are normally distributed with a mean of mu equals 74.0 beats per minute and a standard deviation of

sigma equals 12.5 beats per minute.
 
If 25 adult females are randomly​ selected, find the probability that they have pulse rates with a mean between 68 beats per minute and 80 beats per minute.
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Step 1

Given the pulse rates of females are normally distributed with mean 74 beats per minute and standard deviation 12.5 beats per minute. The total number of females in sample = n = 25.

According to central limit theorem the sample mean is normally distributed with mean µ and standard deviation   σ/√n.

The Z score is the distance from mean in terms of standard deviation. Its formula is shown below and for Z score for  X = 68 and X = 80 beats per minute is calculated as shown below.

For X= 68, Z = -2.4 and for X=80, Z = 2.4.

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Step 2

The probability that pulse rates with a mean between 68 and 80 beats per minute can be written as P( 68 ≤ X ≤ 80) = P( -2.4 ≤ Z ≤ 2....

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