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

Accepted Answer

Hello everyone, let's take a look at this question together. A population has a mean of 50 and a standard deviation of 4. If a random sample of size 100 is taken, what is the probability that the sample mean is greater than 50.7? Would this sample mean be considered unusual? Is it answer choice A 0.0401, and it is unusual. Answer choice B, 0.0401, and it is not unusual. Answer choice C 0.4010, and it is not unusual, or answer choice D 0.4010, and it is considered unusual. So in order to solve this question, we have to determine the probability that the sample mean is greater than 50.7, where we have a random sample size of 100 taken from a population that has a mean of 50 and a standard deviation of 4, as well as determine. If the sample mean is considered unusual, and based on the information provided in the question, we note that we are given a population mean that is equal to 50, a population standard deviation that is equal to 4, and a sample size and that is equal to 100. And we know that we want to find the probability that the sample mean is greater than 50.7, as well as determine whether a sample mean greater than 50.7 would be considered unusual. And so the first step we know is to calculate the standard error or SE, which is given using the following formula for our standard error, which is equal to the population. deviation divided by the square root of our sample sizen, which we substitute the value of 4 for the population standard deviation and 100 for our sample size. So our standard error is equal to 0.4, and then we convert to a Z score using the formula for the Z score, which gives us Z is equal to 50.7, subtracted by 50. Divided by 0.4, which, when simplified, our Z score is 1.75. And then using the Z score, we then find the probability, and since we want the probability of our sample mean being greater than 50.7, we know that we are looking for the probability of a Z score greater than 1.75, which using a Z table or a calculator, the probability of Z being greater than 1.75 is equal to 1 minus the probability of Z being less than 1.75 and substituting the values, we have 1 minus 0.9599, which is equal to 0.0401. So the probability that the sample mean is greater than 50.7 is 0.0401, and we know that a result is Considered unusual if its probability is less than 5%, so a P that is less than 0.05, and since the probability that our sample mean is greater than 50.7 is 0.0401, which is less than 0.05, a sample mean greater than 50.7 is considered unusual. Therefore, looking at our Answer choices, we can identify that answer choice A is the correct answer, as the probability that the sample mean is greater than 50.7 is 0.0401, and this sample mean is in fact 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.

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Key Concepts

Central Limit Theorem

Standard Error

Z-Score

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