Question

An SRS of 350 high school seniors gained an average of ¯ x = 21 points in their second attempt at the SAT Mathematics exam. Assume that the change in score has a Normal distribution with standard deviation σ = 52 . (a) Find a 99 % confidence interval for the mean change in score μ in the population of all high school seniors. (Enter your answers rounded to two decimal places.) lower bound of confidence interval: upper bound of confidence interval:

Answer 1

The 99% confidence interval for the mean change in score in the population of high school seniors ranges from approximately 13.89 to 28.11 points.

Step-by-step explanation:

To find the 99% confidence interval for the mean change in score μ in the population of all high school seniors, we'll use the formula for the confidence interval of the mean for a population when the standard deviation is known:

Confidence interval = μ ± (z*·σ/√n)

Where μ is the mean, σ is the standard deviation, n is the sample size, and z* is the z-score associated with the desired confidence level. Here, the sample mean (μ) is 21, the standard deviation (σ) is 52, and the sample size (n) is 350. The z-score for a 99 percent confidence level is approximately 2.576.

Confidence interval = 21 ± (2.576*52/√350)

First, calculate the margin of error:

Margin of error = 2.576 * (52/√350) ≈ 7.11

Then, calculate the confidence interval:

Lower bound = 21 - 7.11 = 13.89 (rounded to two decimal places)

Upper bound = 21 + 7.11 = 28.11 (rounded to two decimal places)

Therefore, the 99 percent confidence interval for the mean change in score μ is approximately from 13.89 to 28.11 points.

Answer 2

Upper bound: 26.45

Lower bound: 15.55

Step-by-step explanation:

We are given the following in the question:

Sample mean,
`\bar{x}` = 21

Sample size, n = 350

Alpha, α = 0.05

Population standard deviation, σ = 52

95% Confidence interval:

`\mu \pm z_(critical)(\sigma)/(√(n))`

Putting the values, we get,

`z_(critical)\text{ at}~\alpha_(0.05) = 1.96`

`21 \pm 1.96((52)/(√(350)) ) = 21 \pm 5.45 = (15.55,26.45)`

Upper bound: 26.45

Lower bound: 15.55