Answer: a) (0.0509, 0.0541)
b) (0.05224, 0.05276)
Step-by-step explanation:
Formula to find the confidence interval for population mean :-
[tex]\overline{x}\ \pm\ t_{n-1,\ \alpha/2}\dfrac{\sigma}{\sqrt{n}}[/tex]
We assumed that 30-year fixed mortgage rates are normally distributed .
Given : Sample size : [tex]n=28[/tex] , which is less than 30 so it is a small sample , so we apply t-test.
Sample mean :[tex]\overline{x}=0.0525[/tex]
Standard deviation : [tex]\sigma= 0.005[/tex]
a) Significance level : [tex]\alpha=1-0.90=0.1[/tex]
Critical value : [tex]t_{n-1,\alpha}=t_{27,0.05}=1.703[/tex]
Now, the 90% confidence intervals for the population mean 30-year fixed mortgage rate will be:-
[tex]0.0525\ \pm\ (1.703)\dfrac{ 0.005}{\sqrt{28}}\\\\\approx0.0525\pm0.0016\\\\=(0.0525-0.0016, 0.0525+0.0016)\\\\=(0.0509, 0.0541)[/tex]
b) Significance level : [tex]\alpha=1-0.99=0.01[/tex]
Critical value : [tex]t_{n-1,\alpha}=t_{27,0.005}=0.2771[/tex]
Now, the 99% confidence intervals for the population mean 30-year fixed mortgage rate will be:-
[tex]0.0525\ \pm\ (0.2771)\dfrac{ 0.005}{\sqrt{28}}\\\\\approx0.0525\pm0.00026\\\\=(0.0525-0.00026, 0.0525+0.00026)\\\\=(0.05224, 0.05276)[/tex]
