Question

The average height of 20-year-old American women is normally distributed with a mean of 64 inches and standard deviation of 4 inches. If you take a random sample of 4 women from this population and compute their average height. How likely this average height would be shorter than 63 inches

Answer 1

Answer:

Probability that average height would be shorter than 63 inches = 0.30854 .

Step-by-step explanation:

We are given that the average height of 20-year-old American women is normally distributed with a mean of 64 inches and standard deviation of 4 inches.

Also, a random sample of 4 women from this population is taken and their average height is computed.

Let X bar = Average height

The z score probability distribution for average height is given by;

                Z = $\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }$ ~ N(0,1)

where, $\mu$ = population mean = 64 inches

           $\sigma$ = standard deviation = 4 inches

           n = sample of women = 4

So, Probability that average height would be shorter than 63 inches is given by = P(X bar < 63 inches)

P(X bar < 63) = P( $\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }$ < $\frac{63-64}{\frac{4}{\sqrt{4} } }$ ) = P(Z < -0.5) = 1 - P(Z <= 0.5)

                                                        = 1 - 0.69146 = 0.30854

Hence, it is 30.85%  likely that average height would be shorter than 63 inches.