Question

You want to rent an unfurnished one-bedroom apartment in Boston next year. The mean monthly rent for a random sample of 12 apartments advertised in the local newspaper is $1500. Assume that the standard deviation is $250. Find a 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community. (Round your answers to two decimal places.)

Answer 1

95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community is [$1358.55 , $1641.45].

Explanation:

We are given that the mean monthly rent for a random sample of 12 apartments advertised in the local newspaper is $1500. Assume that the standard deviation is $250.

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

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

where,
`\bar X` = mean monthly rent for a sample of 12 apartments = $1500

`\sigma` = standard deviation = $250

n = sample of apartments = 12

`\mu` = population mean monthly rent

Here for constructing 95% confidence interval we have used z statistics because we know about population standard deviation.

So, 95% confidence interval for the population​ 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)/((\sigma)/(√(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) ) }` ]

= [
`1500-1.96 * {(250)/(√(12) ) }` ,
`1500+1.96 * {(250)/(√(12) ) }` ]

= [1358.55 , 1641.45]

Hence, 95% confidence interval for the true mean monthly rent for unfurnished one-bedroom apartments available for rent in this community is [$1358.55 , $1641.45].

Answer 2

Explanation:

We want to determine a 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community.

Number of sample, n = 12

Mean, u = $1500

Standard deviation, s = $250

For a confidence level of 95%, the corresponding z value is 1.96.

We will apply the formula

Confidence interval

= mean ± z ×standard deviation/√n

It becomes

1500 ± 1.96 × 250/√12

= 1500 ± 1.96 × 72.17

= 1500 ± 141.45

The lower end of the confidence interval is 1500 - 141.45 = 1358.55

The upper end of the confidence interval is 1500 + 141.45 = 1641.45

Therefore, with 95% confidence interval, the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community is between $1358.55 and $1641.45