Bisexual Idenfitication In a survey of 692 lesbian, gay, bisex...

Question

Bisexual Idenfitication In a survey of 692 lesbian, gay, bisexual, or transgender U.S adults, 378 said that they consider themselves bisexual. Construct a 90% confidence interval for the population proportion of lesbian, gay, bisexual, or transgender U.S. adults who consider themselves bisexual. (Adapted from Gallup)

Accepted Answer

Hello, in this problem, we are told that out of 780 employees surveyed at a large company, 324 indicated they work remotely at least once a week. We want to construct a 90% confidence interval for the proportion of employees who work remotely at least once a week. Now, to list out some given information, first, we are told that we have a total sample size of 780 employees. And we are told that out of this sample, 324 indicate that they work at least once a week remotely. So, in order to construct a confidence interval for the workers that work at least once a week remotely, we want to first define our point estimate. Now, our point estimate is defined as the population we're interested in divided by the total sample size. That is going to be 324 divided by 780, and that is going to give us a point estimate of 0.4154. Next, we are going to define or find our standard error for the confidence interval. Now, the standard error is defined as the square root of the point estimate, multiplied by its complement and divided by the sample size. That is going to be the square root of 0.4154, multiplied by 1 minus 0.4154, and this will be divided by the sample size, which is 780. That is going to give us an approximate standard error. Of 0.0178. So, now that we have this standard error and the point estimate, we want to determine if we can apply a normal distribution to this data. Now, since N is a sufficiently large sample, and P is nowhere near 0 and 1, we can go ahead and use a Z normal distribution. So the next thing we want to do is you want to find a Z critical value for a 90% confidence level. And by using a Z table, we get a Z critical value of 1.645. Now that we have the Z critical value and the standard error, we can use this to find the margin of error. Now, the margin of error is defined as the standard error multiplied by Z critical value. That is going to be 0.0178, multiplied by a critical value, which is 1.645, and that is going to give us an approximate margin of error of 0.0290. Now that we have the margin of error and the point estimate, we can go ahead and construct the confidence interval. Now the confidence interval is defined as the point estimate plus or minus the margin of error. So our boundaries for the confidence interval is going to be defined as 0.4154 plus or minus the margin of error, which is 0.0290. And by adding and subtracting these values together, we get a lower boundary of the confidence interval of 0.3864, and an upper boundary of 0.4444. And this is going to be our 90% confidence interval. And with that being said, the solution to this problem is going to be A. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Population Proportion

Confidence Interval

Sample Size and Its Impact

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