Question

Students were given a pre-test. Then after they listened to a certain type of music they were given a similar post test. If we define the difference as the post test score minus the pre test score and we find a 95% confidence interval for the mean of the differences to be [-1.34, 4.56]. Which of the following is true? It seems like music has affected the scores and the post test scores are significantly higher. It seems like music has affected the scores and the post test scores are significantly lower. There seems to be insufficient evidence to show that music affected the scores since zero is in the confidence interval. The average scores in the post test were on average about 4.56 points higher than the pre-test.

Answer 1

The 95% confidence interval includes zero, which indicates "There seems to be insufficient evidence to show that music affected the scores since zero is in the confidence interval". We cannot assert a positive or negative effect as the interval does not exclusively lie above or below zero.

Step-by-step explanation:

When evaluating the effect of music on test scores using a confidence interval, we look at where zero falls within that interval. Given the 95% confidence interval for the mean of the differences is [-1.34, 4.56], zero is included within this range. This suggests that we do not have sufficient statistical evidence to claim that the music has had a significant impact on test scores, as the confidence interval includes the possibility of no change (zero difference).

If the music had consistently improved the test scores, the interval would be entirely above zero. Conversely, if the test scores had consistently decreased, the interval would be entirely below zero. Because the interval straddles zero, some scores might have improved, some might have degraded, and some might have stayed the same following exposure to the music.

Thus, we can't conclusively say that the music has either increased or decreased the scores based on this 95 percent confidence interval alone.

Answer 2

The only one that is true is:

"There seems to be insufficient evidence to show that music affected the scores since zero is in the confidence interval."

Step-by-step explanation:

It seems like music has affected the scores and the post test scores are significantly higher. It seems like music has affected the scores and the post test scores are significantly lower.

False. Although the sample results show a higher value in the post test scores, the confidence interval is what gives estimations about the population. As the value 0 and negative values are included in the interval, the difference of scores is not evidently more than 0.

There seems to be insufficient evidence to show that music affected the scores since zero is in the confidence interval.

True. If there was enough evidence to show that music affected the scores, both bounds of the confidence interval should be negative (the music lowers the scores) or positive (the music improves the scores).

The average scores in the post test were on average about 4.56 points higher than the pre-test.

False. The average scores in the post test can not be 4.56 points higher than the pre-test because this is the value of the upper bound of the confidence interval.

The difference between the two averages is the average of the bounds of the confidence interval: (4.56+(-1.34)/2 = 1.61.