Question

The National Assessment of Educational Progress (NAEP) gave a test of basic arithmetic and the ability to apply it in everyday life to a sample of 840 men 21 to 25 years of age. Scores range from 0 to 500; for example, someone with a score of 325 can determine the price of a meal from a menu. The mean score for these 840 young men was x⎯⎯⎯ = 272. We want to estimate the mean score μ in the population of all young men. Consider the NAEP sample as an SRS from a Normal population with standard deviation σ = 60. (a) If we take many samples, the sample mean x⎯⎯⎯ varies from sample to sample according to a Normal distribution with mean equal to the unknown mean score μ in the population. What is the standard deviation of this sampling distribution? (b) According to the 68 part of the 68-95-99.7 rule, 68% of all values of x⎯⎯⎯ fall within _______ on either side of the unknown mean μ. What is the missing number?

Answer 1

a) 2.0702

b) 2.07

Explanation:

Given:

Sample, n = 840

standard deviation, σ = 60

a) The standard deviation of the sampling distribution, will be:

= 2.0702

Standard deviation of the sampling distribution is 2.0702

b) According to the 68 part of the 68-95-99.7 rule, 68% of all values of x fall within 1 standard deviation on either side of the unknown mean μ.

Therefore, missing number will be:

= 1 * 2.0702

= 2.0702 ≈ 2.07

Answer 2

a) 2.0702

b) 2.07

Explanation:

Given:

Sample, n = 840

standard deviation, σ = 60

a) The standard deviation of the sampling distribution, will be:

`= (\sigma)/(√(n))`

`= (60)/(√(840))`

= 2.0702

Standard deviation of the sampling distribution is 2.0702

b) According to the 68 part of the 68-95-99.7 rule, 68% of all values of x fall within 1 standard deviation on either side of the unknown mean μ.

Therefore, missing number will be:

`1 * [(\sigma)/(√(n))]`

= 1 * 2.0702

= 2.0702 ≈ 2.07