Question

Educators introduce a new math curriculum and measure 45 students' scores before and at the end of the academic year using an exam widely believed to accurately measure students understanding.

The average score at the start the year was 65%, and after was 75%.

a) In words, what are the null and alternative hypothesis tests that the educators would most likely be interested in testing?


b) What does a hypothesis test tell us that we can't learn from simply noting that students improved 10 percentage points?

c) Suppose the p-value from this hypothesis test was 0.54. Explain how to interpret this value in this context

Answer 1

a) H0:
`\bar X_(before) \geq X_(after)`

H1:
`\bar X_(before)< \bar X_(after)`

Or equivalently:

H0:
`\bar X_(before) - X_(after) \geq 0`

H1:
`\bar X_(before)- \bar X_(after) <0`

b) The hypothesis test can tell to us if the difference is significantly at some confidence level selected. And that's more important than just the simple difference obtained between 75-65= 10%, because when we do inference we are analyzing the population of interest and not just the sample. And if our conclusion for the test is that the null hypothesis is rejected we can conclude that we have an improvement on the scores for the population at some significance level.

c) The p value is a measure of probability in order to compare with a significance level provided if we can reject or not the null hypothesis.

For this case the p value since is a left tailed test would be given by:

`P(t <t_c) =0.54`

And using any significance level
`\alpha =0.01,0.05,0.1` we fail to reject the null hypothesis because
`p_v >\alpha`. And on that case we can't conclude that the scores after are significantly better than before.

Explanation:

Part a

For this case we have this information given:

`\bar X_(before)= 65\%`

`\bar X_(after)= 75\%`

For this case the most appropiate system of hypothesis would be:

H0:
`\bar X_(before) \geq X_(after)`

H1:
`\bar X_(before)< \bar X_(after)`

Or equivalently:

H0:
`\bar X_(before) - X_(after) \geq 0`

H1:
`\bar X_(before)- \bar X_(after) <0`

Part b

The hypothesis test can tell to us if the difference is significantly at some confidence level selected. And that's more important than just the simple difference obtained between 75-65= 10%, because when we do inference we are analyzing the population of interest and not just the sample. And if our conclusion for the test is that the null hypothesis is rejected we can conclude that we have an improvement on the scores for the population at some significance level.

Part c

The p value is a measure of probability in order to compare with a significance level provided if we can reject or not the null hypothesis.

For this case the p value since is a left tailed test would be given by:

`P(t <t_c) =0.54`

And using any significance level
`\alpha =0.01,0.05,0.1` we fail to reject the null hypothesis because
`p_v >\alpha`. And on that case we can't conclude that the scores after are significantly better than before.