Question

State the hypotheses, state the test statistic, state the P-value, make an decision, and state a conclusion.

A sample of five third graders took a reading test. They then went on their winter break, and took the test again when they returned. Following are the test scores for each of the students before and after the winter break. Can you conclude that the mean reading score was lower after the winter break? Use ? = 0.05. Assume the populations are normally distributed.

	1	2	3	4	5
Before Winter Break	67	68	78	75	84
After Winter Break	66	65	79	74	82

Answer 1

Here, we have to use paired t test.

The null and alternative hypotheses for this test are given as below:

Null hypothesis: H₀: the mean reading score was same after and before the winter break

Alternative hypothesis: H_a: the mean reading score was lower after the winter break.

H₀: µ_d = 0 versus H_a: µ_d < 0

This is a left tailed test.

Test statistic for paired t test is given as below:

t = (Dbar - µ_d)/[S_d/sqrt(n)]

From given data, we have

Dbar = 1.2

S_d = 1.4832

n = 5

df = n – 1 = 4

α = 0.05

t = (Dbar - µ_d)/[S_d/sqrt(n)]

t = (1.2 - 0)/[1.4832/sqrt(5)]

t = 1.8091

The p-value by using t-table is given as below:

P-value = 0.0724

P-value > α = 0.05

So, we do not reject the null hypothesis

There is not sufficient evidence to conclude that the mean reading score was lower after the winter break.