Use technology to find the standard deviation of the set of 36...

Question

Use technology to find the standard deviation of the set of 36 sample means. How does it compare with the standard deviation of the ages found in Exercise 5? Does this agree with the result predicted by the Central Limit Theorem?

Accepted Answer

Hello everyone. Let's take a look at this question together. A biologist records the weights of 64 different sample means, each from a sample of 25 fish. The standard deviation of the original fish weights is 5.5 g. If the standard deviation of the sample means is calculated to be 1.1 g, does this match the central limit theorem prediction? What is the standard deviation predicted by the theorem? Is it answer choice A? Yes, it matches. The predicted standard deviation is 1.1 g. Answer choice B, no, it does not match. The predicted standard deviation is 0.88 g. Answer choice C, yes, it matches the predicted standard deviation is 5.5 g, or answer choice D, no, it does not match the predicted standard deviation. is 1.5 g. So in order to solve this question, we have to determine what is the standard deviation predicted by the central limit theorem to determine if that standard deviation predicted by the theorem matches our standard deviation of the sample means, which was calculated to be 1.1 g. And so we can recall that in order to determine whether the calculated standard deviation. Of the sample means, which was 1.1 g, matches the central limit theorem prediction, we compare it to the standard error of the mean predicted by the central limit theorem. And so we calculate our central limit theorem prediction, where we know that the central limit theorem states that the standard deviation of the sample means, also called the standard error of the mean, or SE, which Is that the standard error of the mean is equal to our standard deviation divided by the square root of our sample size, where we know that from the information provided in the question, the standard deviation is 5.5 g, and our sample size and is 25. And now we can calculate the predicted standard error using the values and the formula, so substituting the values. Our standard error of the mean is 5.5 divided by the square root of 25, which is equal to 5.5 divided by 5, giving us a value of 1.1 g for our standard error of the mean. And so our predicted standard deviation is 1.1 g, which looking at the information provided in the question, we know that the standard deviation of the sample means was calculated. To be 1.1 g. Therefore, this does in fact match the central limit theorem prediction. And looking at our answer choices, we can identify that answer choice A is the correct answer, as the standard deviation predicted by the theorem does in fact match the calculated standard deviation, which was 1.1 g. So answer choice A is correct, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Use technology to find the standard deviation of the set of 36 sample means. How does it compare with the standard deviation of the ages found in Exercise 5? Does this agree with the result predicted by the Central Limit Theorem?

Key Concepts

Standard Deviation

Central Limit Theorem (CLT)

Sample Means

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