Question

The heights of 40 randomly chosen men are measured and found to follow a normal distribution. An average height of 175 cm is obtained. The standard deviation of men's heights is 20 cm. Calculate a 95% two-sided confidence interval for the heights, and interpret (in a sentence) what it means.

Answer 1

95% two-sided confidence interval for the true mean heights of men is [168.8 cm , 181.2 cm].

Explanation:

We are given that the heights of 40 randomly chosen men are measured and found to follow a normal distribution.

An average height of 175 cm is obtained. The standard deviation of men's heights is 20 cm.

Firstly, the pivotal quantity for 95% confidence interval for the true mean is given by;

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

where,
`\bar X` = sample average height = 175 cm

`\sigma` = population standard deviation = 20 cm

n = sample of men = 40

Here for constructing 95% confidence interval we have used One-sample z test statistics as we know about population standard deviation.

So, 95% confidence interval for the true mean,
`\mu` is ;

P(-1.96 < N(0,1) < 1.96) = 0.95 {As the critical value of z at 2.5%

level of significance are -1.96 & 1.96}

P(-1.96 <
`(\bar X-\mu)/((s)/(√(n) ) )` < 1.96) = 0.95

P(
`-1.96 * }{(\sigma)/(√(n) ) }` <
`{\bar X-\mu}` <
`1.96 * }{(\sigma)/(√(n) ) }` ) = 0.95

P(
`\bar X-1.96 * }{(\sigma)/(√(n) ) }` <
`\mu` <
`\bar X+1.96 * }{(\sigma)/(√(n) ) }` ) = 0.95

95% confidence interval for
`\mu` = [
`\bar X-1.96 * }{(\sigma)/(√(n) ) }` ,
`\bar X+1.96 * }{(\sigma)/(√(n) ) }` ]

= [
`175-1.96 * }{(20)/(√(40) ) }` ,
`175+1.96 * }{(20)/(√(40) ) }` ]

= [168.8 cm , 181.2 cm]

Therefore, 95% confidence interval for the true mean height of men is [168.8 cm , 181.2 cm].

The interpretation of the above interval is that we are 95% confident that the true mean height of men will be between 168.8 cm and 181.2 cm.