Chapters 20-21 Summative Assessment Part 2

Question

1. A textbook company is conducting research to evaluate the effectiveness of their new curriculum for AP Statistics. They select a random sample of 1008 students who used a traditional curriculum and find that 634 of these students passed the AP exam. In a random sample of 825 students who used the new curriculum, 561 students passed the AP exam. 

1. Do these data provide convincing evidence that the proportion of students who pass the AP exam is higher with the new curriculum than the traditional curriculum? 

1. The textbook company’s statisticians would like to increase the power of their test. Describe two ways they could do so. 

Answer 1

Hello Deiondre,

1) We are comparing the proportion who passed using the new curriculum vs the traditional curriculum.

First, let's state the two Hypotheses.

1. H₀: p_N=p_T
2. H_a: p_N>p_T

This is a one-sided two-proportion z-test.

Traditional: p_T=634/1008=0.629

New: p_N=561/825=0.680

Difference: p_N-p_T=0.051

Pooloed Proportion: Since we are testing the difference...

p_hat=(634+561)/(1008+825)=0.652

Standard error (pooled)

SE=√(p_hat(1-p_hat)((1/1008)+(1/825)))=0.02236

Test Statistic:

z=(p_hat_N-p_hat_T)/SE=2.28

P-value for a one-tailed test:

P(Z>2.28)≈0.0113

Since p≈0.011 < 0.05, there is covincing evidence that the proportion of students who pass the AP exam is higher with the new curriculum than with the traditional curriculum.

2) There are 4 ways to increase the power of their test.

1, Increase the sample size: larger samples reduce variability (smaller standard error), making it easier to detect true differences.

2, Increase the significance of α: Using a larger α (such as 0.10 instead of 0.05) increases the rejection region and makes the test more sensitive.

3, Reduce variability in sampling: better randomization

4, Increase the true difference between the groups: not controllable, but valid.

I really hope this helps you out with your SAT!!!

-Andrew Evans