Finite Population Correction Factor If a simple random sample ...

Question

Finite Population Correction Factor If a simple random sample of size n is selected without replacement from a finite population of size (n>0.05N), and the sample size is more than 5% of the population size , better results can be obtained by using the finite population correction factor, which involves multiplying the margin of error E by [Image]. Refer to the weights of the M&M candies in Data Set 38 “Candies” in Appendix B.

b. Use only the red M&Ms and treat that sample as a simple random sample selected from the population of the 345 M&Ms listed in the data set. Find the 95% confidence interval estimate of the mean weight of all 345 M&Ms. Compare the result to the actual mean of the population of all 345 M&Ms.

Accepted Answer

All right, hello, everyone. So this question says, a quality control engineer has data on the weights of 120 jelly beans from a production batch. The true mean weight of all 120 candies is known to be 0.982 g. To reestimate the mean without using all data. She isolates the 40 green jelly beans. And computes a sample mean of X bar equals 0.985 g, and a sample standard deviation of S equals 0.02 0 g. Treat these 40 as a simple random sample from the population of 120. Construct a 95% confidence interval for the true mean weight. Applying the finite population correction factor. Finally, compare your interval to the known mean of 0.92 g. All right, so first, let's take a moment to recall what information has been given to us. First, we have the population size or N equal to 120. And we also have lowercase n, which is the sample size of 40. X bar, which is the mean of the sample, is equal to 0.985, and S, which is the standard deviation of the sample, is equal to 0.020. Recall that a 95% confidence interval corresponds to the central 95% of a normal distribution or a distribution. Leaving 5% in the two tails. The Z distribution is symmetric about 0. So the critical value for a 95% confidence interval. Corresponds to the value. At the upper 2.5%. Therefore, using our table as a reference, the critical value that is Z. For a 95% confidence interval is equal to approximately 1.96. And it's this information that's necessary for the following calculations. Or rather all of the above data. All right, so first, we have to find the standard error, but without the finite population correction. So we'll say, SE, not. This is equal to. S divided by the square root of lowercase n. Plugging in the information that we already have, that's 0.020. Divided by the square root of 40. And this equals approximately 0.00316 g. Now, let's apply the finite population correction or FPC. This is going to equal the square root of. Capital N subtracted by lowercase n. Divided by Uppercase N or N subtracted by 1. Once again, plugging in the information we already have. That's the square root of 120 subtracted by 40. And divided by 120 subtracted by 1. This gives you approximately 0.18. Or 0.189 rather. 819, I should say 0.819. With both of these data points on hand, we can now calculate the adjusted margin of error, taking into account the finite population correction. So he Is equal to Z of 0.025. Multiplied by the standard error not. And multiplied by The finite population correction. So that's 1.96. Multiplied by 0.00316. And multiplied 0.819. Evaluating this expression approximately equals 0.005 07 g. Now for the 95% confidence interval. Now for the confidence interval, that's going to be equal to the sample mean, added to and subtracted by E. So that's equal to 0.985, added to and subtracted by. 0.00507. After evaluating both numbers. You would see that the lower bound of the confidence interval is 0.980 g. And the upper bound of the confidence interval is 0.990 g. So here, looking at both values of the confidence interval, we can see that the true mean of 0.982 lies well within the interval. This confirms that the interval given. Is indeed consistent with the actual population mean, after the finite population correction is applied. This leads us now to our final answer. Which reads that mu, the true population mean, is between 0.980 and 0.990 g. The known mean lies well within the interval. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Finite Population Correction Factor

Confidence Interval

Simple Random Sampling

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