Question

Assume that an English comprehension test is given to a random sample of 12 students before and after they complete an English course. The tests taken before the course had   x? =105.75 points and s = 4.07 points.  The tests were taken after the course had x? = 113.08 points and s = 3.63 points.

Complete a hypothesis test to determine if the course improved the test score using a significance level ? = 0.05.

Answer 1

Solution:-

n = 12

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u₁< u₂
Alternative hypothesis: u₁ > u₂

Note that these hypotheses constitute a one-tailed test.  

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = sqrt[(s₁²/n₁) + (s₂²/n₂)]
SE = 1.574
DF = 11

t = [ (x₁ - x₂) - d ] / SE

t = - 4.66

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is thesize of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.

The observed difference in sample means produced a t statistic of - 4.66.

Therefore, the P-value in this analysis is less than 0.0001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that course improved the test score.