Finding Probabilities In Exercises 15–18, the population mean ...

Question

Finding Probabilities In Exercises 15–18, the population mean and standard deviation are given. Find the indicated probability and determine whether the given sample mean would be considered unusual.

For a random sample of n=64, find the probability of a sample mean being less than 24.3 when Mu=24 and sigma=1.25.

For a random sample of n=36, find the probability of a sample mean being less than 12,750 or greater than 12,753 when mu=12750 and 1.7.

Accepted Answer

Hello everyone. Let's take a look at this question together. A shipment of oranges has a population mean weight of 150 g and a standard deviation of 6 g for a random sample of N equals 64 oranges, what is the probability that the sample mean weight is less than 150 g or greater? Than 151.5 g, would a sample mean above 151.5 g be considered unusual? Is it answer choice A 0.5228 and yes? Answer choice B 0.5228 and no. Answer choice C 0.0522 and yes, or answer choice D 0.0522 and no. So in order to solve this question, we have to determine the probability that the sample mean weight is less than 150 g or greater than 151.5 g, or a random sample of N equals 64 oranges, with a population mean weight of 150 g, and a standard deviation of 6 g. As well as if this sample mean above 151.5 g is considered unusual, and looking at the information provided in the question, we know that we are given a population mean of 150 g, a population standard deviation of 6 g, and a sample size n of. 64, and we know that we want to find the probability that the sample mean is less than 150 or is greater than 151.5 and determine if a sample mean greater than 151.5 is considered unusual. And the first step is to calculate the standard error or SE. Using the formula for the standard error, which is equal to the population standard deviation divided by the square root of the sample size, which substituting the values into the formula, our standard error SE is equal to 0.75. And now we compute the Z score for a sample mean equal to 151.5 using the formula for the Z score, which gives us Z equals 151.5, subtracted by 150 divided by 0.75, which, when simplified, our Z score for a sample mean equaling 151.5 is approximately 2. And now we find the probability of the sample mean being less than 150, plus the probability that the sample mean is greater than 101.5, which we I should note, since the distribution is symmetric around the mean equaling 150 and 151.5 is 1.5 above the mean, then we know that the probability that the sample mean is less than 150 is equal to 0.5 because 150 is the mean, and the probability that the sample mean is greater than 151.5 is equal to. The probability of Z being greater than 2, which is approximately 0.0228. And now we can add all of the probabilities. So the probability that the sample mean is less than 150, or the sample mean is greater than 151.5 is equal to 0.5 plus 0.0228, which is equal to 0.5228. Therefore, the probability that the sample mean weight is less than 150 g or greater than 151.5 g is 0.5228, and we know that a result is considered unusual if the probability is less than 5%, and the probability that the sample mean is greater than 151.5 is equal to 0.0228, which is less than 0.05, we can determine that, yes, a sample mean greater than 151.5 g is considered unusual. And looking at our answer choices, we can identify that answer choice A is the correct answer, as the probability that the sample mean weight is less than 150 g or greater than 151.5 g is 0.5228. And a sample mean above 151.5 g is considered unusual since it is less than 5%. So answer choice A is the correct answer, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Central Limit Theorem

Standard Error

Z-scores

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