Question

Every 2 years, the National Assessment of Educational Progress (NAEP) assesses nationally representative samples of students in public and private schools in grades 4, 8, and 12.

In one year, the NAEP sample of 1600 students in grade 8 in public schools had a mean quantitative score = 280. Because the sample size is large, the sample s is close to the population standard deviation, σ, so take σ = 64.

a) Give a 95% confidence interval for the mean score μ in the population of all young adults. (7 pts.)

b) Suppose you had the same result from with sigma=64 and x-bar= 280, but now the result comes from a sample of 400 young adults. Give the 95% confidence interval for μ. (4 pts.)

c) Then suppose that a sample of 6400 young adults had produced the sample mean x-bar= 280 with sigma=64, and again give the 95% confidence interval for μ. (4 pts.)

d) In the 3 questions above, what are the margins of error for samples of size 400, 1600, and 6400? How does increasing the sample size affect the margin of error of a confidence interval? (Be sure to make mental distinctions between standard deviation, standard error, and margin of error -- they are related but different items.)

Answer #1

a) At 95% confidence level, the critical value is
z_{0.025} = 1.96

The 95% confidence interval is

= 276.864, 283.136

b) The 95% confidence interval is

= 273.728, 286.272

c)The 95% confidence interval is

= 278.432, 281.568

d) For n = 400, margin of error = 6.272

For n = 1600, margin of error = 3.136

For n = 6400, margin of error = 1.568

As the sample size increases, the margin of error decreases.

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