What is the probability that a random sample of 17 pregnancies has a mean gestation period within 9 days of the mean?( length of pregnancies of a certain animal are approximately normally distributed with mean of 295 days and standard deviation of 15 days)

Using the normal distribution and the central limit theorem, it is found that there is a 0.9864 = 98.64% probability that a random sample of 17 pregnancies has a mean gestation period within 9 days of the mean.

In a normal distribution with mean and standard deviation , the z-score of a measure X is given by:

In this problem:

Gestation period within 9 days of the mean is between 286 and 304 days, and the probability is the p-value of Z when X = 304 subtracted by the p-value of Z when X = 286.

X = 304:

By the Central Limit Theorem

has a p-value of 0.9932

X = 286:

has a p-value of 0.0068

0.9932 - 0.0068 = 0.9864.

0.9864 = 98.64% probability that a random sample of 17 pregnancies has a mean gestation period within 9 days of the mean.

A similar problem is given at link

0.9864

Step-by-step explanation:

The average sample mean is 295, and the standard deviation is:

σ = 15 / √17

σ = 3.64

The z-scores are:

z = (x − μ) / σ

z₁ = (-9) / 3.64

z₁ = -2.47

z₂ = (9) / 3.64

z₂ = 2.47

So the probability is:

P(-2.47 < Z < 2.47) = P(Z < 2.47) − P(Z < -2.47)

P(-2.47 < Z < 2.47) = 0.9932 − 0.0068

P(-2.47 < Z < 2.47) = 0.9864