In Exercises 55–60, find the indicated probabilities and inter...

Question

In Exercises 55–60, find the indicated probabilities and interpret the results.

The mean annual salary for Level 1 actuaries in the United States is about $72,000. A random sample of 45 Level 1 actuaries is selected. What is the probability that the mean annual salary of the sample is (a) less than $75,000? Assume sigma = $11,000.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a company reports that the mean monthly rent for a one bedroom apartment in a city is $1350 with the standard deviation of $125. If a random sample of 48 apartments is selected, what is the probability that the sample mean rent is less than $1390? And we're gonna have four possible choices as our answers. For choice A, we have 0.8366. For choice B, we have 0.9727. For choice C, we have 0.9867. And for choice D, we have 0.9940. Now, we're asked to find the probability that the sample mean rent is less than $1390. Now, in order to find that probability, we first want to convert the $1390 into a Z score. And to do that, we're going to need the mean of the sample means, as well as the standard deviation of sample means. So, recall that according to the central limit theorem, if random samples of size N where N is greater than equal to 30 are drawn from any population with a mean U and the standard deviation sigma. Then the sampling distribution of sample means approximates a normal distribution. So, in our case, we have a sample of 48 apartments. That means that our in value N is going to be equal to. 48 Which is actually greater than 30. So that means that the sampling distribution of sample means approximates a normal distribution. Therefore, our mean of sample means, which we'll label as m sub X bar, is just going to be equal to the population mean U, which we're given in the problem as $1350. Now, for the standard deviation of sample means, which we'll label as stigma sub X bar, That is just going to be equal to our population standard deviation, stigma, divided by the square root of N. And we're given the population standard deviation of $125 so that means this is going to be equal to $125. Divided by square root of 48. Which is going to be approximately equal to 18.042. So, we'll now use these two values to calculate the corresponding Z score for the $1390. So, we call your formula for calculating Z score, that's gonna be Z is equal to the quantity of X bar minus UX bar in quantity, divided by sigma sub X bar. Where X bar here is going to be the sample mean that we're looking for, which is going to be the 1390. So substituting in our values, this is gonna be equal to 1390 minus 1350, and that whole quantity is going to be divided by 18.042. And when we evaluate this expression, this is going to be approximately equal to. 2.217. So here we just rounded to three decimal places. So now we want to find the cumulative probability corresponding to the Z score, and for this, we're gonna use our standard normal distribution tables. And so when we look our Z value up in our table, we see that it falls between two of the entries. It falls between Z equal to 2.21. Which has a corresponding cumulative probability of 0.9864, and the other entry is Z is equal to 2.22, which has a corresponding cumulative probability of 0.9868. So to find the corresponding cumulative probability or R Z value, we're gonna need to use interpolation. So, we'll be looking for the cumulative probability P of Z less than 2.217. And this is going to be equal to, and here we call your interpolation formula. So, for the first term, we want to have the change in our cumulative probabilities, divided by our change in our Z scores from our table. So the numerator will have the quantity of 0.9868 minus 0.9864 in quantity, and this could be divided by The quantity, which is gonna be our change in our Z scores, so it's gonna be 2.22 minus 2.21. In quantity. And then we'll have to multiply that fraction by the quantity of our Z score that we're looking for, which is going to be 2.217. Minus the smaller of our two Z values, so that's gonna be 2.21 in quantity, and then we'll have to add to this the cumulative probability of the smaller of the two Z values from our table, which is going to be 0.9864. And so when we evaluate this expression, We get that it is approximately equal to. 0.9867. So here we've just rounded to 4 decimal places. Now, what we're interested in is the probability. Of P of X bar, less than. 1390, but this is exactly equal to the probability P of Z less than 2.217, which we just found is going to be equal to 0.9867. So this here is the answer that we're looking for for this problem. And when we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice Z. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Central Limit Theorem

Standard Error of the Mean

Z-Score

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