Question

Most dermatologists recommend using sunscreens that have a Sun Protection Factor (SPF) of at least 30. One of the authors wanted to find out whether the SPF of sunscreens used by students at her school exceeds this value on average? To collect data, the author surveyed students in her introductory statistics course and found that in a sample of 52 students the mean SPF was 34.55 and the standard deviation of SPF was 20.12. The data were not strongly skewed.​ Use the Theory-Based Inference applet to find and report the 95% confidence interval for the parameter of interest. Round answers to 4 decimal places, e.g. 0.7523. ​We are 95% confident that the mean SPF level of sunscreen used by all Cal Poly students is between and .

Answer 1

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.

Explanation:

Given : Significance level :
`\alpha: 1-0.95=0.05`

Sample size : n= 52 , which is a large sample (n>30), so we use z-test.

By using z-value table,

Critical value:
`z_(\alpha/2)=1.96`

Sample mean :
`\overline{x}= 34.55`

Standard deviation :
`\sigma= 20.12`

The confidence interval for population means is given by :-

`\overline{x}\pm z_(\alpha/2)(\sigma)/(√(n))`

i.e.
`34.55\pm(1.96)( 20.12)/(√(52))`

`=34.55\pm5.46867829455\\\\\approx34.55\pm5.4687\\\\=(34.55-5.4687, 34.55+5.4687)=(29.0813,\ 40.0187)`

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.