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Problem 20 Easy Difficulty

What’s wrong? A driving school wants to find out which of its two instructors is more effective at preparing students to pass the state’s driver’s license exam. An incoming class of 100 students is randomly assigned to two groups, each of size 50. One group is taught by Instructor A; the other is taught by Instructor B. At the end of the course, 30 of Instructor A’s students and 22 of Instructor B’s students pass the state exam. Do these results give convincing evidence that Instructor A is more effective?
Min Jae carried out the significance test shown below to answer this question. Unfortunately, he made some mistakes along the way. Identify as many mistakes as you can, and tell how to correct each one.
State: I want to perform a test of
$$H_{0} : p_{1}-p_{2}=0$$
$$H_{a} : p_{1}-p_{2}>0$$
where $p_{1}=$ the proportion of Instructor A's students that passed the state exam and $p_{2}=$ the proportion of Instructor B's students that passed the state exam. Since no significance level was stated, I'll use $\sigma=0.05$
Plan: If conditions are met, I'll do a two-sample $z$ test for comparing two proportions.
$\bullet$ Random The data came from two random samples of 50 students.
$\bullet$ Normal The counts of successes and failures in the two groups - $30,20,22$ , and $28-$ are all at least $10 .$
$\bullet$ Independent There are at least 1000 students who take this driving school's class.
Do: From the data, $\hat{p}_{1}=\frac{20}{50}=0.40$ and
$\hat{p}_{2}=\frac{30}{50}=0.60 .$ So the pooled proportion of successes is
$$\hat{p}_{C}=\frac{22+30}{50+50}=0.52$$
$\bullet$ Test statistic
$$z=\frac{(0.40-0.60)-0}{\sqrt{\frac{0.52(0.48)}{100}+\frac{0.52(0.48)}{100}}}=-2.83$$
Conclude: The P-value, $0.9977,$ is greater than $\alpha=$ $0.05,$ so we fail to reject the null hypothesis. There is not convincing evidence that Instructor A's pass rate is higher than Instructor B's.

Answer

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Video Transcript

problem. 20. The null hypothesis is that B one minus B two is equal to zero, and the alternative hypothesis is is that the one minus B two is bigger than zero, where be one proportion is X one over N ones for 30/50 she's open six and B two, which is equal to extra over in two, which is 22/50 which is Oh, point or four. The pooled proportion, which is X one plus x 2 30 plus 22 over in one plus in two she is 100. She's equal toe 1000.52 toe. Find the value off until the statistics, which is equal to the P one minus B two over a square root off 4.5 to 1 minus 4.5 52 square root off one over n. One plus one over and to which is equal to 1.6 eso. The probability is equal to the probability that that or the P value is equal, that the probability that that is bigger than 1.6, which is equal to the probability that set is smaller than negative 1.6, which is equal to open toe five, 48 and the P value is bigger than open toe five. So we say it to reject the null hypothesis is so there is no sufficient evidence to support support the claim.