Answer: (1583.63, 1672.37)
Step-by-step explanation:
Given : Sample size : n= 83
Sample mean : [tex]\overline{x}=1628[/tex]
Sample standard deviation : [tex]\sigma=243[/tex]
The population standard deviation[tex](\sigma)[/tex] is unknown .
The confidence interval for population mean :
[tex]\overline{x}\pm t_{\alpha/2}\dfrac{s}{\sqrt{n}}[/tex]
For 90% confidence , significance level =[tex]\alpha=1-0.90=0.10[/tex]
Using t-distribution table , Critical t-value = [tex]t_{(\alpha/2, n-1)}=t_{0.05,82}=1.6636[/tex]
, where n-1 is the degree of freedom.
Now , 90% confidence interval for the mean square footage of all the homes in that city will be :-
[tex]1628\pm (1.6636)\dfrac{243}{\sqrt{83}}\\\\ 1628\pm (1.6636)(26.672715)\\\\\\\\\approx1628\pm 44.37\\\\=(1628- 44.37,\ 1628+ 44.37)\\\\=(1583.63,\ 1672.37)[/tex]
Hence, the 90% confidence interval for the mean square footage of all the homes in that city = (1583.63, 1672.37)
