Finite Population Correction Factor In Exercises 57 and 58, us...

Question

Finite Population Correction Factor In Exercises 57 and 58, use the information below.

In this section, you studied the construction of a confidence interval to estimate a population mean. In each case, the underlying assumption was that the sample size n was small in comparison to the population size N. When n ≥ 0.05N however, the formula that determines the standard error of the mean needs to be adjusted, as shown below.

[IMAGE]

Recall from the Section 5.4 exercises that the expression sqrt[(N-n)/(n-1)] is called a finite population correction factor. The margin of error is

[IMAGE]

Use the finite population correction factor to construct each confidence interval for the population mean.

a. c = 0.99, xbar = 8.6, σ = 4.9, N = 200, n = 25.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a company wants to estimate the average number of hours employees spend on training per month. Out of 500 employees, a random sample of 60 is taken. The sample mean is 6.23 hours, and the population standard deviation is 1.15 hours. Construct a 99% confidence interval, or the population mean using the finite population correction factor. Now, we're asked to calculate the 99% confidence interval for the population mean. Let's begin by looking at what information we're given in this problem. The first piece of information that we're given in the problem is going to be our population size. So we'll label that with a capital N, and that is going to be equal to 500. The next piece of information is going to be our sample size, which we'll label with a lowercase n, and that is going to be equal to 60. The next piece of information is our sample mean, which we'll label as X bar, and that's going to be equal to 6.23. And the final piece of information that we're given is our population standard deviation, which is going to be sigma, and that's going to be equal to 1.15. Now, in order to Calculate our 99% confidence interval. We'll first need to calculate the standard error, and we're gonna need to use the finite population correction factor with that. So we call that our standard error, SE. Is going to be equal to sigma divided by the square root of lowercase n, and then that fraction is multiplied by The square root of the quantity. Of N minus n in quantity, divided by the quantity of N minus 1. So, we're going to substitute in all of our values into this expression. That means our standard error is going to be equal to. 1.15 divided by the square root of 60. Multiplied by the square root of the quantity of 500 minus 60 in quantity, divided by the quantity of 500 minus 1 in quantity. And evaluating this expression, this is going to be approximately equal to. 0.1394. So here we just rounded to 4 decimal places. So we're gonna use the standard error to calculate our margin of error. So recall that our margin of error, E is going to be equal to Z, the alpha divided by 2. Multiplied by Our standard error SE. Now we just calculated our standard error, but we need to determine our Z score. And the Z value is something that we look up in our tables. Now, we're looking for the 99. percent confidence level. And that actually corresponds to a Z value of Z alpha divided by 2, that is equal to. 2.576. So we'll substitute that into our margin of error formula. And so that means that E is going to be equal to. 2.576. Multiplied by 0.13. 94. And when we evaluate this expression, This is going to be approximately equal to. 0.36. So now we can calculate the upper and lower bounds for our confidence interval. Now, our lower bound. is going to be X for minus E. Which is equal to 6.23 minus. 0.36. Which when you evaluate that person, that's going to be equal to. 5.87. And for the upper bound, we're gonna have X bar plus E. That's going to be equal to 6.23, plus 0.36. Which is equal to 6.59. So, we can write our 99% confidence interval using interval notation, and so we're gonna go from 5.87 hours. 2 6.59 hours. And so this here is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Finite Population Correction Factor (FPC)

Confidence Interval

Standard Error of the Mean (SEM)

Watch next