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.

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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

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

c. c = 0.95, xbar = 40.3, σ = 0.5, N = 300, n = 68.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a researcher wants to estimate the average height of plants in a greenhouse. She takes a random sample of 54 plants from a total population of 210 plants. The sample mean height is 15.75 centimeters, and the population standard deviation is 0.80 centimeters. Construct a 95% confidence interval with the population mean using the finite population correction factor. So we're asked to construct a 95% confidence interval for the population mean. So let's begin by looking at what information that we're given in the problem. And the first piece of information that we're given in the problem is going to be our sample size, which will denote with a lowercase n, and that's going to be equal to 54. Next piece of information that we're given in the problem is our population size, which we will denote as a capital N and that's going to be equal to 210. The next piece of information that we're given is going to be our sample mean, so that's going to be X bar, and that's going to be equal to 15.75. And the final piece of information that we're given is our population standard deviation, which we'll denote as sigma, and that's going to be equal to 0.80. Now, in order to calculate our confidence interval, we'll first need to determine the standard error using the finite population correction factor. And so, recall that our standard error, which is SE, is going to be equal to. Sigma divided by the square root of lowercase n, then that fraction is going to be multiplied by the square root of the quantity. Of N minus n in quantity, divided by the quantity of N minus 1. And so we can substitute in all of our values. Into this formula. So that means our standard error is going to be equal to. 0.80 divided by the square root of 54. And that's going to get multiplied by the square root of the quantity of 210 minus 54, in quantity divided by the quantity of 210 minus 1. And when we evaluate this expression, this is going to be approximately equal to. 0.094. And here we just round it to three decimal places. So, in order to construct our confidence interval, we'll need to calculate the margin of error. So we call that our margin of error E is going to be equal to. Z multiplied by our standard error, SE. Now, ZSA in this case, is going to be our critical value, which is something that we need to look up in our table. So for our 95% confidence interval, we'll look up. A value that corresponds to that 95% confidence level, and we see that Z star. Is going to be equal to. 1.96. So substituting in our values for our margin of error, we see that E is going to be equal to. 1.96. Multiplied by 0.094. And evaluating this expression, we see that our margin of error is going to be approximately equal to. 0.18. So now we use our margin varier to calculate our confidence interval. So for our lower bound of our interval, we're going to have X bar minus E. Which is going to be equal to. 15.75 minus 0.18. Which is going to be equal to. 15.57. And for our upper bound. We're going to have X bar plus E. This is going to be equal to 15.75. Plus 0.18, which is going to be equal to. 15.93. So we can write our 95% confidence interval using interval notation, and so we'll have the interval from 15.57 centimeters. 2 15.93. centimeters. So this here is going to be the answer to 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

Confidence Interval

Standard Error of the Mean

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