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=45, find the probability of a sample mean being greater than 551 when mu=550 and sigma=3.7.

For a random sample of n=45, find the probability of a sample mean being greater than 551 when mu=550 and sigma=3.7.

Accepted Answer

Hello everyone. Let's take a look at this question together. A researcher selects a random sample of N equals 50 from a population with a mean of 120 and a standard deviation of 4.2. What is the probability that the sample mean is less than 119? Would this sample mean be considered unusual? Is it answer choice A 0.461 and no? Answer choice B, 0.4061, and yes, answer choice C 0.0461, and no, or answer choice D 0.0461 and yes. So in order to solve this question, we have to determine the probability that the sample mean is less than 119 if a researcher selects a random sample of N equals 50 from a population with a mean of 120 and a standard deviation of 4.2 and determine if this probability is considered unusual. And so Looking at the information provided in the question, we know that we are given a population mean equal to 120, a population standard deviation equal to 4.2, and a sample size N equal to 50. And we want to find the probability that the sample mean is less than 119. And the first step in solving this problem is to Calculate the standard error using the formula for the standard error, which is equal to the population standard deviation divided by the square root of the sample size, so substituting the values, our standard error is 4.2 divided by the square root of 50. And then the next step is to convert to a Z score. So we have Z is. Equal to 119, subtracted by 120, divided by 4.2, divided by the square root of 50, which, when simplified, our Z is equal to approximately 1.684. And so now we need to find the probability that Z is less than -1.684, using interpolation between values. In the Z table. And then for interpolation from the Z table, a standard Z table typically gives the following values, which includes the probability that Z is less than -1.68, which equals 0.0465, and the probability that Z is less than -1.69, which is 0.0455. So our Z equals -1.684 lies 40% of the way from -1.68 to 1.69. So then we interpolate, which gives us the probability that Z is less than -1.684, is approximately 0.0465, subtracted by 0.40, multiplied by in parentheses 0.0. 465 subtracted by 0.0455. And when simplified, this gives us a value of 0.0461, so the probability that the sample mean is less than 119 is 0.0461, and we know that a result is typically considered unusual if the probability is less than 5%, and since our probability of 0.0. 461 is less than 5%. We know that the sample mean is in fact, considered unusual. And looking at our answer choices, we can identify that answer choice D is the correct answer, as the probability that the sample mean is less than 119 is 0.0461, and the sample mean is considered unusual since it is less than 5%. So answer choice D 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-Score

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