Question

We took a simple random sample of 2nd graders and 3rd graders and asked them to pass a simple spelling test. Our prediction was that the 3rd graders would generally perform better. Our null hypothesis then results in a question like: are the proportions that pass the same? (and then we can either reject that or not passed on the results. 23 2nd graders took the test and 12 of them passed. 35 3rd graders took the test and 20 of them passed. What is the 95% confidence interval for difference in the 2nd and 3rd grader passing proportions. Can we reject the claim that 3rd graders and 2nd graders perform the same on the test? Explain

Answer 1

Here, , n1 = 23 , n2 = 35
p1cap = 0.5217 , p2cap = 0.5714

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.5217 * (1-0.5217)/23 + 0.5714*(1-0.5714)/35)
SE = 0.1336

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.5217 - 0.5714 - 1.96*0.1336, 0.5217 - 0.5714 + 1.96*0.1336)
CI = (-0.3116 , 0.2122)

we cannot reject the claim because confidence interval contains 0.