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 ��̂��1-��̂��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.

A researcher using a survey constructs a 90% confidence interval ...

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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