You want to rent an unfurnished onebedroom apartment in Bost

You want to rent an unfurnished one-bedroom apartment in Boston next year. The mean monthly rent for a random sample of 10 apartments advertised in the local newspaper is $1024. Assume that the standard deviation is $275. Find the 90%, 95%, and 99% confidence intervals for the mean monthly rent for this category of apartments. Look at the 95% confidence interval and say whether the following statement is true or false. “This interval describes the price of 95% of the rents of all the unfurnished one-bedroom apartments in the Boston area.” Be sure to explain your answer.

Solution

AT 90%
CI = x ± Z a/2 * (sd/ Sqrt(n))
Where,
x = Mean
sd = Standard Deviation
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
CI = Confidence Interval
Mean(x)=1024
Standard deviation( sd )=275
Sample Size(n)=10
Confidence Interval = [ 1024 ± Z a/2 ( 275/ Sqrt ( 10) ) ]
= [ 1024 - 1.64 * (86.963) , 1024 + 1.64 * (86.963) ]
= [ 881.381,1166.619 ]

AT 95%
Confidence Interval = [ 1024 ± Z a/2 ( 275/ Sqrt ( 10) ) ]
= [ 1024 - 1.96 * (86.963) , 1024 + 1.96 * (86.963) ]
= [ 853.553,1194.447 ]

AT 99%
Confidence Interval = [ 1024 ± Z a/2 ( 275/ Sqrt ( 10) ) ]
= [ 1024 - 2.58 * (86.963) , 1024 + 2.58 * (86.963) ]
= [ 799.636,1248.364 ]

This interval describes the price of 95% of the rents of all the unfurnished one-bedroom apartments in the Boston area.”
Yes , since mean 1024 lies in the interval [ 853.553,1194.447 ]