Question

An SRS of 450 450 high school seniors gained an average of ¯ x = 20 x¯=20 points in their second attempt at the SAT Mathematics exam. Assume that the change in score has a Normal distribution with standard deviation σ = 49 σ=49 . (a) Find a 95 % 95% confidence interval for the mean change in score μ μ in the population of all high school seniors. (Enter your answers rounded to two decimal places.)

Answer 1

Answer: (15.47, 24.53)

Explanation:

We know that the confidence interval for population mean is given by :_

`\overline{x}\pm z^*(\sigma)/(√(n))`

, where n= sample size.

`\sigma` = standard deviation.

`\overline{x}`= sample mean.

z*= Critical value.

Given : n= 450

`\overline{x}=20`

`\sigma=49`

Critical value for 95% confidence = z*=1.96 [From z-value table]

Then, the 95% confidence interval will be :-

`20\pm (1.96)(49)/(√(450))`

`\approx 20\pm (1.96)(2.31)`

`\approx 20\pm 4.53`

`=(20-4.53,\ 20+4.53)=(15.47,\ 24.53)`

Hence, the 95% confidence interval for the mean change in score μ μ in the population of all high school seniors. : (15.47, 24.53)