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)

Step-by-step explanation:

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

$\overline{x}\pm z^*\dfrac{\sigma}{\sqrt{n}}$

, 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 :-

$20\pm (1.96)\dfrac{49}{\sqrt{450}}$

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

$\approx 20\pm 4.53$

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

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)