Question

A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 185 milligrams with s = 17.6 milligrams. Construct a 95% confidence interval for the true mean cholesterol content of all such eggs.

Answer 1

95% confidence interval for the true mean cholesterol content of all such eggs is [173.82 , 196.18].

Explanation:

We are given that a laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 185 milligrams with s = 17.6 milligrams.

Assuming the population has a normal distribution.

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

P.Q. =
`(\bar X - \mu)/((s)/(√(n) ) )` ~
`t_n_-_1`

where,
`\bar X` = sample mean amount of cholesterol = 185 milligrams

s = sample standard deviation = 17.6 milligrams

n = sample of chicken eggs = 12

`\mu` = true mean

Here for constructing 95% confidence interval we have used t statistics because we don't know about population standard deviation.

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

P(-2.201 <
`t_1_1` < 2.201) = 0.95 {As the critical value of t at 11 degree of

freedom are -2.201 & 2.201 with P = 2.5%}

P(-2.201 <
`(\bar X - \mu)/((s)/(√(n) ) )` < 2.201) = 0.95

P(
`-2.201 * {(s)/(√(n) ) }` <
`{\bar X - \mu}` <
`2.201 * {(s)/(√(n) ) }` ) = 0.95

P(
`\bar X-2.201 * {(s)/(√(n) ) }` <
`\mu` <
`\bar X+2.201 * {(s)/(√(n) ) }` ) = 0.95

95% confidence interval for
`\mu` = [
`\bar X-2.201 * {(s)/(√(n) ) }` ,
`\bar X+2.201 * {(s)/(√(n) ) }` ]

= [
`185-2.201 * {(17.6)/(√(12) ) }` ,
`185+2.201 * {(17.6)/(√(12) ) }` ]

= [173.82 , 196.18]

Therefore, 95% confidence interval for the true mean cholesterol content of all such eggs is [173.82 , 196.18].