Question

The State Department of Education is developing a new test to measure the reading
level of sixth graders. To pilot this test, they select a random sample of 50 sixth
graders from within the state and have them take the test. The mean score for the
students in this sample is 70.5 points. Suppose that it is already known that the
standard deviation for all sixth graders in the state is 8.4 points.

A.Construct and interpret a 98% confidence interval to estimate the mean score on
the new reading test for all sixth graders in the state.

B.Does this interval provide convincing evidence to support a school district’s claim
that the mean score on this new reading test is less than 75 points?

C.Does this interval provide convincing evidence to support a claim that the mean
score on this new reading test is more than 70 points?

Answer 1

a)

sample mean, xbar = 70.5
sample standard deviation, σ = 8.4
sample size, n = 50

Given CI level is 98%, hence α = 1 - 0.98 = 0.02
α/2 = 0.02/2 = 0.01, Zc = Z(α/2) = 2.33

ME = zc * σ/sqrt(n)
ME = 2.33 * 8.4/sqrt(50)
ME = 2.77

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (70.5 - 2.33 * 8.4/sqrt(50) , 70.5 + 2.33 * 8.4/sqrt(50))
CI = (67.7321 , 73.2679)

2)

yes, because confidence interval does not contain 75

c)

N0, because confidence interval contain 70