Respuesta :
Answer:
Step-by-step 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
Answer:
95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community is [$1358.55 , $1641.45].
Step-by-step 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. = [tex]\frac{\bar X - \mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
where, [tex]\bar X[/tex] = mean monthly rent for a sample of 12 apartments = $1500
       [tex]\sigma[/tex] = standard deviation = $250
       n = sample of apartments = 12
       [tex]\mu[/tex] = 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, [tex]\mu[/tex] is ;
P(-1.96 < N(0,1) < 1.96) = 0.95 Â {As the critical value of z at 2.5% level of
                           significance are -1.96 & 1.96}
P(-1.96 < [tex]\frac{\bar X - \mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < 1.96) = 0.95
P( [tex]-1.96 \times {\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]{\bar X - \mu}[/tex] < [tex]-1.96 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ) = 0.95
P( [tex]\bar X-1.96 \times {\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\mu[/tex] < [tex]\bar X+1.96 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ) = 0.95
95% confidence interval for [tex]\mu[/tex] = [ [tex]\bar X-1.96 \times {\frac{\sigma}{\sqrt{n} } }[/tex] , [tex]\bar X+1.96 \times {\frac{\sigma}{\sqrt{n} } }[/tex]Â ]
                      = [ [tex]1500-1.96 \times {\frac{250}{\sqrt{12} } }[/tex] , [tex]1500+1.96 \times {\frac{250}{\sqrt{12} } }[/tex] ]
                      = [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].
