Translating Statements In Exercises 29–34, translate the state...

Question

Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.

In a survey of 2094 U.S. adults who have used an online dating app, 57% said their personal experience with online dating was positive. The survey’s margin of error is ±3.6%. (Source: Pew Research Center)

Accepted Answer

Welcome back, everyone. In this problem, a survey of 2,094 video game users found that 57% said their overall experience with multiplayer games has been positive. The survey reports a margin of error of plus or minus 3.6%. Translate this information into a confidence interval and approximate the confidence level of the interval. A says the interval goes from 0.507 to 0.626 with an approximate confidence level of 95%. B says from 0.563 to 0.639 and approximately 98%. C from 0.498 to 0.535, approximately 99%. And the D from 0.534 to 0.606, approximately 99.9%. Now, let's break this problem down into two parts. Let's first start by finding the confidence interval. What do we already know? Well, so far we know that the sample proportion. Is 57% or 0.57%. And the survey reported a margin of error, ME of plus or minus 3.6%. So that's 0.036. How can this help us to figure out our confidence interval? Well, recall that our confidence interval. OK. Is equal to our sample proportion plus or minus or margin of error. So since we have both of these values, then we can solve for our confidence interval. Because no, that means it's going to be equal to 0.57 plus or minus 0.036. So the lower bone is going to be 0.57 minus 0.036, and the upper bone is 0.57 plus 0.036. Now when we go ahead and solve here, we should get that to be equal to 0.534 for the lower bound and 0.606 for the upper bound. This is the confidence interval for the given information. Now let's move on to the second part of our problem. Let's try to approximate the confidence level. How can we figure the confidence level out? Well, if you note here, we were given the margin of error, OK, and recall. Now the margin of error is equal to the corresponding Z24 or confidence level multiplied by the square root of the sample proportion multiplied by 1 minus the sample proportion divided by our sample size N. OK, where we've already seen earlier that our sample size N is 2,094. Now here if we can figure out what the Z score would be using this formula since we already know the margin of error, the sample proportion, and the sample size, then we can find the confidence level that corresponds to the Z score. So let's rearrange our formula to solve for a Z. Because no Z is gonna be equal to the margin of error. Divided by our term under the square root. OK. And now we can go ahead and substitute. So we know our margin of error is 0.036. And And our sample proportion is 0.57. 1 minus 0.57 is 0.43. So under a square, we're gonna have 0.57 multiplied by 0.43 divided by 20094. Now when you go ahead and solve here, we should get this to be approximately 3.33. So that's the Z square. Now our Z score, if you notice here, since Z is approximately 3.33, OK. It corresponds. OK, to a confidence level. Of approximately. 99.9%. Now how do we know that? Well, if we just visualize it quickly here, remember, the Z score represents the number of standard deviations we are away from the mean. So if our Z score is 3.33, and we know that beyond 3 standard deviations from the mean, we have 99 or sorry, within. Three standard deviations of the mean. There is 99% of our data. Then that means within 3.33 standard deviations of our mean it must be higher than 99%. It must almost be 100, and this is a very high level of confidence and suggests that the results are quite precise. So, in summary, our confidence interval goes from 0.534 to 0.606, and it has a confidence level of approximately 99.9%. If we look back at our answer choices, we can tell D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Confidence Interval

Margin of Error

Level of Confidence

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