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⎯⎯⎯x¯ = 272. We want to estimate the mean score μμ in the population of all young men. Consider the NAEP sample as an SRS from a Normal population with standard deviation σσ = 60.

(a) If we take many samples, the sample mean x⎯⎯⎯x¯ varies from sample to sample according to a Normal distribution with mean equal to the unknown mean score μμ in the population. What is the standard deviation of this sampling distribution?
(b) According to the 99.7 part of the 68-95-99.7 rule, 99.7% of all values of x⎯⎯⎯x¯ fall within _______ on either side of the unknown mean μμ. What is the missing number?
(c) What is the 99.7% confidence interval for the population mean score μμ based on this one sample? Note: Use the 68-95-99.7 rule to find the interval.

Answer 1

given that,

life sample of men n = 840,

x-bar = 272,

standard deviation σ = 60

(b) The missing number is 3

(c)

life sample of men n = 840

x-bar = 272

standard deviation s = 60

% = 99.7

Standard Error, SE = σ/√n = 60/√840 = 2.070196678

z-score = 2.967737925

Width of the certainty interim = z * SE = 2.96773792534179 * 2.07019667802706 = 6.143801194

Lower Limit of the certainty interim = x-bar - width = 272 - 6.1438011942975 = 265.8561988

Maximum Limit of the certainty interim = x-bar + width = 272 + 6.1438011942975 = 278.1438012

The certainty interim is [265.86, 278.14]

n = 840, x-bar = 272, σ = 60

(b) The missing number is 3

(c)

n = 840

x-bar = 272

s = 60

% = 99.7

Standard Error, SE = σ/√n = 60/√840 = 2.070196678

z-score = 2.967737925

Width of the certainty interim = z * SE = 2.96773792534179 * 2.07019667802706 = 6.143801194

Lower Limit of the certainty interim = x-bar - width = 272 - 6.1438011942975 = 265.8561988

Maximum Limit of the certainty interim = x-bar + width = 272 + 6.1438011942975 = 278.1438012

The certainty interim is [265.86, 278.14]