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 z -score for testing whether there is a significant difference in the population proportions of student passing rates, but do not find a P 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.

Welcome back, everyone. Let's take a look at this practice problem. We're gonna continue talking about two proportion hypothesis tests. Alright? Let's check it out here. So a school administrator wants to compare a proportion of students who pass a standardized math exam in two different schools. So So you're trying to compare a difference in proportion and you're gonna take two samples from two different classes. So we're gonna be assuming that the samples are random and independent. So we're basically just gonna go ahead and take that for granted. Alright? We're gonna calculate the z score. We're actually not gonna do the entire hypothesis test. We're only just gonna calculate the z score for testing whether there is a significant difference in the population proportions. So notice how it says, don't find a p value or draw a conclusion. So it's basically just trying to get us to, you know, not actually do the entire hypothesis test and really just focus on one step, which is calculating the z score. Alright? However, you should always just start off these problems with, again, just go going through your assumptions and checking off your conditions. You can do this very, very quickly here. So are the samples random and independent? Yes. Because we're actually just told that they are explicitly. So that's easy. Are there greater than five successes and failures for each? We're told that in class a, 72 out of one twenty, so that's definitely correct, and 65 out of a hundred. So there's basically more than five successes and failures in both of them. Alright? So our basic conditions are met. Let's go ahead and take a look at the, z score. Alright? So, ultimately, what we wanna do is figure out the z score, which, remember, is just just long equation for two proportions. But, basically, even though it's got a lot of different letters, it's ultimately just a difference in sample proportions, p one hat minus p two hat minus then you have your, difference in population proportions, which is p one minus p two without the hats, and then it's gonna be divided by basically just some huge, square roots with some p's and q's inside of it. Right? So the square root goes, you have a p bar and q bar, which are not the same as the p hats and p's over there, over n one, plus p bar times q bar over n two. Alright? So this is again sort of like a weighted sort of standard deviation based on the pooled, proportions. Alright? So, again, we're gonna have to go ahead and look at all of these and and all these variables ultimately to figure out what is the z score. Alright? So there's a lot of things we have to figure out over here. Let's just go ahead and take a look at figuring out what are the p one hats and p two hats. Those come from your samples. Alright? So remember, what happens here is we're gonna go ahead. It's always a good idea just to build yourself a little table, pick which one is group one and group two. We're just gonna go ahead and pick class a as group one. So I'll just call x one, which is equal to 72, and n one is just the total of that sample, which is a 20. Alright? So we've got x two over here, which is equal to 65, and then n two is equal to a hundred. You could have flipped them, but I'm just gonna go ahead and do this. As long as you're consistent, it'll all all work out the same way. Alright? So I've got my x two and n two. Okay? So now what we're gonna do over here is, in order to figure out p one hats and p two hats, you're basically just gonna do 71 72 divided by a 20. Right? So it's 72 over one twenty, and this ends up being a p one hat of what I see here is this ends up being 0.6. Alright? So then this is p two hat over here, which ends up being 65 over a hundred, pretty straightforward, 0.65. Alright? So in other words, what we have to do here is we actually have to subtract these two. So in other words, I'm just gonna start writing these things out. Right? So I've got 0.6 minus 0.65. Those are my difference in sample proportions. Now I've got minus, then I've got p one minus p two without the hats. What is this? What is this? Do we actually have this information here in our problem? Remember, with a two sample hypothesis test, in fact, with all hypothesis test, the things that go inside of the second parenthesis here in the numerator is always the difference in the population proportion. Again, you don't know what that is, and that's the whole reason you're doing a hypothesis test. However, this number here actually comes from your initial hypothesis, your null hypothesis. Remember that your HO is that p one is equal to p two. That's always gonna be your default assumption here. Essentially, what that means here is that p one minus p two, the difference in the population proportion is actually zero. That's always what you're assuming here. Alright? So this thing always just ends up being zero inside of this parenthesis, because of your null hypothesis. Alright? So there's just a big fat zero over here, and then we're just gonna go ahead and and divide by the thing in the print in the, in the square roots. Alright? So now what we're gonna do here is we're gonna figure out what p hat p bar and q bar are. We figured out what p one hat and p two hat are, and those are our individual sample proportions. But remember, with a two sample hypothesis test, we always use in the denominator, we use the pooled sample proportions, which is basically just sort of like a weighted sample proportion. Alright? So let's go over here and figure this out here. So this is p bar, and remember, there's an equation for this. It's x one plus x two. It's the total number of successes divided by the total number of trials. So it's in other words, basically, your total number of successes, which is gonna be 72 plus 65 divided by the total number of trials in total, which is just gonna be a 20 plus a hundred. Alright? So if you go ahead and work this out, what you're gonna see here is that your p bar is actually just equal to 0.62. Alright? So remember, q bar is basically just the complement of that. And if p bar is 0.62, then q bar is just one minus that. You can do probably do this in your head, but this is just equal to 0.38. Alright? So we need these two numbers over here, remember, because we need to multiply them in the denominator of our z score equation. So this is basically just 0.62 times 0.38 divided by n one. Remember, n one is the total number in group one, which is a 20, plus and then you're gonna multiply those two numbers again. So this is 0.7 sorry. 0.62 times 0.38, and then divided by n two, which is the total number in group two, which is a hundred. Alright? So if you can multiply all of this stuff out here, it took a lot of numbers, but you were ultimately gonna get here as a z score of this is gonna be negative 0.76, and that's your final answer. Alright? So your z score ends up being negative 0.76, and you could continue on with the hypothesis test if you want to. We're gonna go ahead and stop here because that's all we were asked to do in this problem. Alright? So again, lots of different steps, lots of numbers to keep track of, but, ultimately, very straightforward calculation, just plug and chug. Alright? Thanks for watching. Let me know if you have any questions.

10. Hypothesis Testing for Two Samples

Two Proportions

Statistics

10. Hypothesis Testing for Two Samples

Two Proportions

Struggling with Statistics?

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Multiple Choice

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.

-0.76

1.07

-1.07

0.76

Verified step by step guidance

Step 1: Define the null hypothesis (H0) and alternative hypothesis (H1). H0: The population proportions of students passing the exam in the two schools are equal (p1 = p2). H1: The population proportions of students passing the exam in the two schools are not equal (p1 ≠ p2).

Step 2: Calculate the sample proportions for each class. For Class A, the sample proportion (p̂1) is calculated as the number of students who passed divided by the total number of students: p̂1 = 72 / 120. For Class B, the sample proportion (p̂2) is calculated as p̂2 = 65 / 100.

Step 3: Compute the pooled proportion (p̂) under the null hypothesis. The pooled proportion is calculated as the total number of successes divided by the total number of observations: p̂ = (72 + 65) / (120 + 100).

Step 4: Calculate the standard error (SE) for the difference in sample proportions. The formula for SE is: SE = sqrt(p̂ * (1 - p̂) * (1/n1 + 1/n2)), where n1 and n2 are the sample sizes for Class A and Class B, respectively.

Step 5: Compute the z-score using the formula: z = (p̂1 - p̂2) / SE. Substitute the values of p̂1, p̂2, and SE into the formula to calculate the z-score.

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Multiple Choice

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 researcher using a survey constructs a 90% confidence interval for a difference in two proportions. According to the data, they calculate ${\hat{p}}_{1} - {\hat{p}}_{2} = 0.09$ 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.

$x sub 1 equals 87$ , $x sub 2 equals 68$

$n sub 1 equals 120$ , $n sub 2 equals 115$

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Textbook Question

Denomination Effect A trial was conducted with 75 women in China given a 100-yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from “The Denomination Effect,” by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a 0.05 significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills.

a. Test the claim using a hypothesis test.

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Textbook Question

c. If the significance level is changed to 0.01, does the conclusion change?

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