Question

About 130,000 high school students took the AP Statistics exam in 2010. The free-response section of the exam consisted of five open-ended problems and an investigative task. Each free-response question is scored on a 0 to 4 scale (with 4 being the best). For one of the problems, a random sample of 30 student papers yielded the scores that are graphed in the dot plot of part (a) in the previous problem. The mean score for this sample is x̄ = 1.267 and the standard deviation is s = 1.230.

(a) Find and interpret the standard error of the mean.

(b) Construct and interpret a 99% confidence interval to estimate the mean score on this question. Use the four-step process.

Answer 1

Solution :

Given that,

sample mean = = 1.267

sample standard deviation = s = 1.230

sample size = n = 30

Degrees of freedom = df = n - 1 = 30 - 1 = 29

a) SE = s / n = 1.230 / 30 = 0.2246

interpretation = If we were to draw several samples of size 30 from students papers yielded the scores that are graphed in the dot plot of part (a) in the previous problem construct a sampling distribution of the sample means, we would end up with a sample mean of 1.267 and standard error of 0.2246

b) 1) point estimate = sample mean = = 1.267

2) At 99% confidence level

= 1 - 99%

=1 - 0.99 =0.01

/2 = 0.005

t/2,df = t_0.005,29 = 2.756

3) Margin of error = E = t_/2,df * SE

= 2.756 * 0.2246

Margin of error = E = 0.619

4) The 99% confidence interval estimate of the population mean is,

  ± E

= 1.267 ± 0.619

= ( 0.648, 1.886 )