A researcher using a survey constructs a 90% confidence interval for a difference in two proportions. According to the data, they calculate p ^ 1 − p ^ 2 = 0.09 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?

Alright, folks. Welcome back. Let's continue talking about confidence intervals and how we use them in a two proportion hypothesis test. Let's check out our problem here. So we have a researcher who's using a survey survey, and they're gonna construct a 90% confidence interval for the difference between two proportions. Alright? Now Now according to the data, they calculate the difference in sample proportions. Remember, that's p one hat minus p two hat is equal to point o nine with a margin of error of 0.07. They don't give us the full confidence interval, just the point estimate and the margin of error. Alright? Now we're supposed to use this data to figure out, should they reject or fail to reject the claim that there is no difference in the two proportions. Alright? So one of the things you should notice about this problem is that we actually don't know a whole lot about what's going on here. We don't know what their proportions represent, what data is being collected, how how many people we're talked to, etcetera. This happens sometimes in problems where you're just gonna be given some numbers and asked very simply to come up with a conclusion. That's what's going on here. Right? So, presumably, the survey is done, some research that calculates these numbers. We're supposed to figure out, do you reject or fail to reject? Alright? So remember here that your confidence interval your confidence interval is always just going to be the point estimates. This is basically your best guess, p one hat minus p two hat. That's gonna be the you're gonna take that and you subtract the margin of error, and then you're gonna take p one hat minus p two hat, and then you're gonna sub and then you're gonna add the margin of error. Right? So basically, your confidence interval is always just gonna be in between these two numbers. Now remember, when we build these confidence intervals, we're always looking for the number zero, whether that's included or not in our interval. And, basically, what happens here is if your interval does not include zero, so if your interval does not include zero, that means that you have enough or you have sufficient evidence to basically say that your confidence interval you're 90% confidence confident that because zero is outside of our interval, that actually there is a significant difference between these two proportions. So if your interval does not include zero, you reject, and then the opposite is true. If your confidence interval that you create here does include the number zero, therefore, you have it's plausible that there actually is no difference in the, two proportions, and therefore, you're gonna fail to reject. Alright? It's a weird sort of double negative, but because it doesn't include because it includes zero, you're trying to you have enough evidence to suggest that there's no difference in the proportions, and then you fail to reject, which means that, basically, it's possible that it's actually true. Alright? That's basically what you're looking for here. Alright? Now we're actually told we're not gonna calculate the p one hat minus p two hat in the margin of error because it's actually already given to us. Basically, what we have here is our confidence interval, is that we have 0.09 minus the margin of error, which is 0.07, and then 0.09 on the upper bound, which is, plus 0.07 for the margin of error. Alright? So basically, if you work out these two numbers, what you're gonna see here is that this is 0.02, the confidence interval, and this is gonna be on the upper bound 0.16. Alright? So now we take a look at this interval here, and does it include zero or does it not? And clearly, here we can see here that it does not include zero. So it does not include zero. So oops. Include zero. Alright. So therefore, what happens is because this does not include zero, you're gonna reject. So you're gonna reject this claim here. We're basically 90% confident that the real difference in these two proportions, again, whatever they are, the real difference actually is significant and not zero. We have evidence to suggest that there is a real difference in the proportions over here. Okay? So the answer to this question is that we're gonna reject the null hypothesis. Alright? That's it for this one, folks. Let me know if you have any questions, and let's move on.

10. Hypothesis Testing for Two Samples

Two Proportions

Statistics

10. Hypothesis Testing for Two Samples

Two Proportions

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Multiple Choice

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?

Reject

Fail to reject

There is not enough information to answer the question.

Verified step by step guidance

Step 1: Understand the problem. The researcher is testing whether there is a difference in two proportions. The null hypothesis (H₀) is that there is no difference between the two proportions, i.e., p₁ - p₂ = 0. The alternative hypothesis (Hₐ) is that there is a difference, i.e., p₁ - p₂ ≠ 0.

Step 2: Recall the interpretation of a confidence interval. A 90% confidence interval provides a range of values within which the true difference in proportions (p₁ - p₂) is likely to fall, with 90% confidence. If the interval does not include 0, we reject the null hypothesis.

Step 3: Use the given information to construct the confidence interval. The point estimate for the difference in proportions is p̂₁ - p̂₂ = 0.09, and the margin of error is 0.07. The confidence interval is calculated as: [ (p̂₁ - p̂₂) - margin of error, (p̂₁ - p̂₂) + margin of error ].

Step 4: Substitute the values into the formula. The confidence interval becomes: [ 0.09 - 0.07, 0.09 + 0.07 ]. Simplify the bounds of the interval to determine whether 0 is included.

Step 5: Analyze the interval. If 0 is not within the confidence interval, reject the null hypothesis (H₀). If 0 is within the interval, fail to reject the null hypothesis. If the interval cannot be determined from the given information, conclude that there is not enough information to answer the question.

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Textbook Question

Can Dogs Detect Malaria? A study was conducted to determine whether dogs could detect malaria from socks worn by malaria patients and socks worn by patients without malaria. Among 175 socks worn by malaria patients, the dogs made correct identifications 123 times. Among 145 socks worn by patients without malaria, the dogs made correct identifications 131 times (based on data presented at an annual meeting of the American Society of Tropical Medicine, by principal investigator Steve Lindsay). Use a 0.05 significance level to test the claim of no difference between the two rates of correct responses.

b. Test the claim by constructing an appropriate confidence interval.

Textbook Question

Equivalence of Hypothesis Test and Confidence Interval Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 10 having a common attribute. The second sample consists of 2000 people with 1404 of them having the same common attribute. Compare the results from a hypothesis test of p1=p2 (with a 0.05 significance level) and a 95% confidence interval estimate of p1-p2

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?

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.

Multiple Choice

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.

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

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

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.

Textbook Question

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

Textbook Question

Clinical Trials of OxyContin OxyContin (oxycodone) is a drug used to treat pain, but it is well known for its addictiveness and danger. In a clinical trial, among subjects treated with OxyContin, 52 developed nausea and 175 did not develop nausea. Among other subjects given placebos, 5 developed nausea and 40 did not develop nausea (based on data from Purdue Pharma L.P.). Use a 0.05 significance level to test for a difference between the rates of nausea for those treated with OxyContin and those given a placebo.

c. Does nausea appear to be an adverse reaction resulting from OxyContin?

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