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 (b) more than $68,000? Assume sigma = $11,000.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says the company reports that the monthly mean rent for a one bedroom apartment in a city is $1350 with a standard deviation of $125. If a random sample of 48 apartments is selected, what is the probability that the sample mean rent is more than $1320? And here we're given four possible choices as our answers. For choice A, we have 0.9103. For choice B, we have 0.9221. For choice C, we have 0.9468, and for choice D, we have 0.9518. Now we're asked to find the probability that the sample mean rent is more than $1320. So, the first thing we want to do is to convert our $1320 into a Z score. And in order to do that, I will need the mean of sample means, as well as the standard deviation of sample means. Now recall that according to the central limit theorem, if random samples of size N, where N is greater than or equal to 30 are drawn from any population with a mean mu and a standard deviation sigma, then the sampling distribution of sample means approximates a normal distribution. So, if we look at our case here, we're sampling 48 apartments. That means that N is going to be equal to 48 in our case, which is actually greater than 30. Which means that the sampling distribution of sample means approximates a normal distribution. So, our mean of sample means, which we'll label as mu sub X bar, it's just gonna be equal to our population mean, which is mu. Which we're given the problem is going to be equal to the $1350. Now, for the standard deviation of sample means, which we'll label as stigma sub X bar, that is going to be equal to our population standard deviation sigma divided by the square root of N. Which we're given our population standard deviation, that is $125 but this is going to be equal to $125. Divided by the square root of 48, which when we evaluate this expression, this is going to be approximately equal to 18.042. So now we can use these two values to calculate the Z score for the $1320. So, if you call your formula for calculating a Z score, that's gonna be Z is equal to. X minus use of X bar. Divided by, and that whole quantity divided by sigma sub X bar. Where export here is going to be our $1320 since that is the mean that we're actually looking for. And so when we substitute in the values, we get that Z is going to be equal to the quantity of $1320 minus or mus of X bar, we have $1350 in quantity, and that's going to be divided by 18.042 for Cigna sub X bar. And when we evaluate this expression, this is going to be approximately equal to. -1.663. So here we just rounded to three decimal places. So now we want to find the cumulative probability corresponding to our Z score here. And for this, we'll just use our standard normal distribution tables. And when we look at our table, we see that our Z score falls between two of the entries. It falls between Z. equal to -1.67. Which has a cumulative probability corresponding to. 0.0475. And the other entry is going to be Z is equal to -1.66, which has a cumulative probability of 0.0485. Now, since our Z value actually falls in between two of the entries in our table, we're actually going to use interpolation to find our cumulative probability. So, our cumulative probability that we're looking for, which will be P of Z less than minus 1.663, is going to be equal to, and here we call your formula for interpolation. So when we do interpolation, we'll first, the first term we're going to have is going to be our change in our cumulative probabilities, divided by our change in these values from our table. So, in the numerator, we'll have the quantity of 0.0485 minus 0.0475, and that quantity is going to be divided by the quantity of minus 1.66 minus 1.67 in quantity. And then this is going to get multiplied by the quantity, and here we'll have to look at the difference between the Z value that we're looking for, which is -1.663. Minus, and here we'll choose the smaller of the two Z values that we have from our table, which is going to be -1.67. In quantity, then we'll have to add to this the cumulative probability from Z equal to -1.67. Which is going to be 0.0475. And so when we evaluate this expression, this is going to be approximately equal to. 0.0482. Now, the problem is asking us to find the probability that the sample mean rent is greater than $1320. So, we're looking for P of X bar. Greater than 1320. But that is equivalent to finding the cumulative probability with Z greater than -1.663. So this is going to be equal to P of Z greater than -1.663. Which is equal to 1 minus P of Z less than minus 1.663. So here we just take the compliment. And we just found that cumulative probability. So, that means that this is going to be equal to 1 minus and or P of Z less than -1.663, will have the value of 0.0482. And so when we evaluate this expression, we get 0.9518. So this here is the answer that we're looking for for this problem. And if we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice D. 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

Z-Score

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