10. Hypothesis Testing for Two Samples

A human resources department is comparing two employee training programs to see if they lead to different pass rates on a required certification exam. They randomly select two groups of employees. In Program A, 16 out of 20 employees passed the exam. In Program B, 30 out of 40 employees passed. Are the basic conditions met to conduct a 2-proportion hypothesis test?

A school administrator wants to compare the proportion of students who passed a standardized math exam in two different schools by taking samples from 2 classes. Assume the samples are random and independent. Calculate the $z$-score for testing whether there is a significant difference in the population proportions of student passing rates, but do not find a $P$-value or draw a conclusion for the hypothesis test.

Class A: 72 out of 120 students passed.

Class B: 65 out of 100 students passed.

A researcher using a survey constructs a 90% confidence interval for a difference in two proportions. According to the data, they calculate $p̂_1-p̂_2=0.09$ with a margin of error of 0.07. Should they reject or fail to reject the claim that there is no difference in these two proportions?

The data below is taken from two random, independent samples. Calculate the margin of error for a 99% confidence interval for the difference in population proportions.

$x_1=87$, $x_2=68$

$n_1=120$, $n_2=115$