Question

in the unted states, the height of men are normally distributed with the mean 69 inches and standard deviation 2.8 inches. If 16 men are randomly selected. what is the probability that their mean height is less than 68 inches

Answer 1

Probability that their mean height is less than 68 inches is 0.0764.

Explanation:

We are given that in the united states, the height of men are normally distributed with the mean 69 inches and standard deviation 2.8 inches.

Also, 16 men are randomly selected.

Let
`\bar X` = sample mean height

The z-score probability distribution for sample mean is given by;

Z =
`(\bar X-\mu)/((\sigma)/(√(n) ) )` ~ N(0,1)

where,
`\mu` = population mean height = 69 inches

`\sigma` = population standard deviation = 2.8 inches

n = sample of men = 16

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) 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.

So, probability that the mean height of 16 randomly selected men is less than 68 inches is given by = P(
`\bar X` < 68 inches)

P(
`\bar X` < 68 inches) = P(
`(\bar X-\mu)/((\sigma)/(√(n) ) )` <
`(68-69)/((2.8)/(√(16) ) )` ) = P(Z < -1.43) = 1 - P(Z
`\leq` 1.43)

= 1 - 0.9236 = 0.0764

Now, in the z table the P(Z x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 1.43 in the z table which has an area of 0.92364.

Therefore, probability that their mean height is less than 68 inches is 0.0764.