Question

Suppose Diana, an educational researcher at a local university, wants to test the impact of a new Spanish course that integrates cultural-immersion teaching techniques with standard teaching practices. She selects a simple random sample of 62 freshmen and divides them into 31 pairs, matched on IQ and high school GPA. She randomly selects one member of each pair to take the new course, while the other member in the pair takes the traditional course. Next, Diana records the course grade, tallied on a scale from 0 to 4, for all sample members at the end of the semester, and she computes the difference in grades between the members in each matched pair by subtracting the traditional course grade from the new course grade. She wants to determine if the new Spanish course improves or weakens student performance. She runs a matched-pairs ?- test to test the null hypothesis, ?0:?=0 , against the alternative hypothesis, ?1:?≠0 , where ? is the mean course grade difference for the student population. The sample statistics for Diana's test are summarized in the table. Variable description Sample mean Sample standard deviation Standard error estimate traditional course grade ?⎯⎯⎯trad= 3.45773 ?trad=2.30766 SEtrad=0.41447 new course grade ?⎯⎯⎯new=3.32772 ?new=2.21794 SEnew=0.39835 difference (new − traditional) ?⎯⎯⎯=−0.13001 ?=0.34221 SE=0.06146 Although Diana does not know the standard deviation of the underlying population of course grade differences, she assumes that the population is normally distributed because the sample data are symmetric, single-peaked, and contain no outliers.

Compute the ?- statistic for Diana's matched-pairs ?- test. Provide your answer with precision to three decimal places.

Compute the ?-value of the ?- statistic using software. You may find some software manuals helpful. Provide your answer precise to three decimal places.

Select the accurate statement regarding Diana's test decision if she tests at a significance level of ?=0.05 . A)Diana should reject the null hypothesis because the ?-value is greater than the value ?=0.05 .B) Diana should fail to reject the null hypothesis because the ?-value is greater than 0.025. C)Diana should fail to reject the null hypothesis because the ?-value is greater than the value ?=0.05 .D) Diana should fail to reject the null hypothesis because the mean grade difference is too small. E)Diana should reject the null hypothesis because the ?-value is less than the value ?=0.05 .

Answer 1

t-statistic

The t-statistic for matched-pair t -test is obtained using the formula,

Where,

?-value

The P-value for the t statistic is obtained using the t distribution table for t = -2.11536 and degree of freedom = 31-1=30,

In excel use function =T.DIST.2T(2.11536,30) to calculate the P-value.

Conclusion:

Correct Answer: E)Diana should reject the null hypothesis because the ?-value is less than the value ?=0.05 .

Explanation: Since,

Reject Ho. There is a statistically sufficient evidence at the level of significance to conclude that there is a significant difference in mean of difference.