Question

In a random sample of 11 residents of the state of New York, the mean waste recycled per person per day was 2.9 pounds with a standard deviation of 0.98 pounds. Determine the 80% confidence interval for the mean waste recycled per person per day for the population of New York. Assume the population is approximately normal. Step 1 of 2 : Find the critical value that should be used in constructing the confidence interval. Round your answer to three decimal places.

Answer 1

80% confidence interval for the mean waste recycled per person per day for the population of New York is [2.495 , 3.305].

Explanation:

We are given that a random sample of 11 residents of the state of New York, the mean waste recycled per person per day was 2.9 pounds with a standard deviation of 0.98 pounds.

Firstly, the pivotal quantity for 80% confidence interval for the true mean waste recycled per person per day for the population of New York is given by;

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

where,
`\bar X` = sample mean waste recycled per person per day = 2.9 pounds

s = sample standard deviation = 0.98 pounds

n = sample of residents = 11

`\mu` = true mean waste recycled per person per day

Here for constructing 80% confidence interval we have used t statistics because we don't know about population standard deviation.

So, 80% confidence interval for the true mean,
`\mu` is ;

P(-1.372 <
`t_1_0` < 1.372) = 0.80 {As the critical value of t at 10 degree of

freedom are -1.372 & 1.372 with P = 10%}

P(-1.372 <
`(\bar X - \mu)/((s)/(√(n) ) )` < 1.372) = 0.80

P(
`-1.372 * {(s)/(√(n) ) }` <
`{\bar X - \mu}` <
`1.372 * {(s)/(√(n) ) }` ) = 0.80

P(
`-\bar X -1.372 * {(s)/(√(n) )` < -
`\mu` <
`-\bar X +1.372 * {(s)/(√(n) )` ) = 0.80

P(
`\bar X -1.372 * {(s)/(√(n) )` <
`\mu` <
`\bar X +1.372 * {(s)/(√(n) )` ) = 0.80

80% confidence interval for
`\mu` = [
`\bar X -1.372 * {(s)/(√(n) )` ,
`\bar X +1.372 * {(s)/(√(n) )` ]

= [
`2.9 -1.372 * {(0.98)/(√(11) )` ,
`2.9 -1.372 * {(0.98)/(√(11) )` ]

= [2.495 , 3.305]

Therefore, 80% confidence interval for the mean waste recycled per person per day for the population of New York is [2.495 , 3.305].