LGBT Identification In a survey of 15,349 U.S. adults, 860 ide...

Question

LGBT Identification In a survey of 15,349 U.S. adults, 860 identify as lesbian, gay, bisexual, or transgender. Construct a 95% confidence interval for the population proportion of U.S. adults who identify as lesbian, gay, bisexual, or transgender. (Adapted from Gallup)

Accepted Answer

Hello. In this problem, we are told that in a survey of 12,500 college students, 710 of them reported that they participate in campus clubs. We want to construct a 95% confidence interval for the population proportion of college students who participate in campus clubs. So, just to list out some given information, first, we are told that we have a total sample size of 12,500 college students. So we will set N equal to 12,500. Now we are told that 710 of them reported that they participate in campus clubs, so we will label X as 710. Now, in order to construct a 95% confidence interval, the first thing we want to do is we want to define our point estimate. The point estimate is the estimate of college students who participate in campus clubs. That is going to be defined as 710 divided by 12,500, and that is going to give us an approximate point estimate of 0.0568. Now, the next thing we want to do is we want to go ahead and determine the standard error for the confidence interval. 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 0.0568, multiplied by 1 minus 0.0568, and this will be divided by the sample size, which is 12,500. With the help of a calculator, we will get an approximate standard error. An approximate standard error of 0.0021. Now, the next thing we want to do is you want to determine if we can apply a normal distribution to this data. We have a sufficiently large sample size, but notice that our point estimate is close to 0. So what we need to do is we need to check to see if a normal distribution is possible. Now, in order to check, we want to determine if the point estimate and its complement can be applied to a normal distribution. In order to determine this, we can take our sample size and multiply it to our point estimate, and we will do the same thing to the complement. And what we are checking is to see if both of them are greater than 5. If they are both greater than 5, then we can apply a normal distribution. So starting with the first part, the sample size, which is 12,500, Multiplied to our point estimate, which is 0.0568, gives us a product of 710 rounded. This value is greater than 5, so that's the first condition met. Now, for the complement, that is going to be the sample size, 12,500, multiplied by 1 minus 0.0568, and that is going to give us an estimate of 11,790. This value is greater than 5, and since both of the values were greater than 5, we can apply a normal distribution. So now that we know that we can apply a normal distribution, we want to go ahead and find a Z critical value for the 95% confidence interval. Now, by using a Z table, we will get a Z critical value for 95% confidence interval that is going to be 1.96. Now that we have the standard error and the critical value, we will go ahead and find the margin of error. Now, the margin of error is defined as the standard error multiplied by the critical value. That is going to be 0.0021, multiplied by 1.96, and that is going to give us an approximate margin of error of 0.0041. Now that we have our margin of error and the point estimate, we can start to construct the confidence interval. Now the confidence interval is defined as our point estimate plus or minus our margin of error. So the end points for the confidence interval is going to be 0.0568 plus or minus 0.0041. And by adding and subtracting these spikes together, we get a lower boundary for the interval of 0.0527, and an upper boundary of the interval of 0.0609. And this is going to be our 95% confidence interval. And with that being said, the solution to this problem is going to be B. 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.

LGBT Identification In a survey of 15,349 U.S. adults, 860 identify as lesbian, gay, bisexual, or transgender. Construct a 95% confidence interval for the population proportion of U.S. adults who identify as lesbian, gay, bisexual, or transgender. (Adapted from Gallup)

Key Concepts

Population Proportion

Confidence Interval

Standard Error

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