Answer: We are 95% confident that the mean SPF level of sunscreen used by all Cal Poly students is between 29.0813 and 40.0187.
Step-by-step explanation:
Given : Significance level : [tex]\alpha: 1-0.95=0.05[/tex]
Sample size : n= 52 , which is a large sample (n>30), so we use z-test.
By using z-value table,
Critical value: [tex]z_{\alpha/2}=1.96[/tex]
Sample mean : [tex]\overline{x}= 34.55[/tex]
Standard deviation : [tex]\sigma= 20.12[/tex]
The confidence interval for population means is given by :-
[tex]\overline{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}[/tex]
i.e. [tex]34.55\pm(1.96)\dfrac{ 20.12}{\sqrt{52}}[/tex]
[tex]=34.55\pm5.46867829455\\\\\approx34.55\pm5.4687\\\\=(34.55-5.4687, 34.55+5.4687)=(29.0813,\ 40.0187)[/tex]
Now, the 95% confidence interval for the ppopulation mean = . (29.0813, 40.0187)
Hence, We are 95% confident that the mean SPF level of sunscreen used by all Cal Poly students is between 29.0813 and 40.0187.
