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Problem 18 Easy Difficulty

The average yearly Medicare Hospital Insurance benefit per person was $\$ 4064$ in a recent year. If the benefits are normally distributed with a standard deviation of $\$ 460,$ find the probability that the mean benefit for a random sample of 20 patients is
a. Less than $\$ 3800$
b. More than $\$ 4100$

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March 18, 2021

3. The times per week a student uses a computer lab are normally distributed, with a mean of 6.2 hours and a standard deviation of 0.9 hour. A student is randomly selected. Find the probability that the student uses a computer lab

Video Transcript

right in this problem. You were given information about medical hospital insurance benefits, and it says that the average yearly amount is 4006 or $4064 and that the standard deviation is $460 and that information is solely about the population. We're then going to select a sample off 20 patients. So we have to talk about the information about the sample and we're selecting 20 patients, so the sample size is 20. And if that's the case, we're going to talk about the fact that we need an average of the sample means and we're going to need the standard deviation of the sample means. And keep in mind that the average of the sample means is the same as the average of the population, which in this case is 4064 And the standard deviation of the sample means is equivalent to the standard deviation of the population divided by the square root of N. So in this case is going to be 460 divided by the square root of 20 and we have two questions. Part A is asking us to find the probability that the mean benefit for the random sample is less than 3800. So for part A, we're doing the probability that the mean is less than 3800. And then for part B, we're going to do that. The mean is more than 4100. So the first thing we're going to want to definitely do is talk about the bell shaped curve. We have 4064 right in the center. And for part, I were looking for 3800. We're less than 3800. All right, so we're gonna want to switch the 3800 Intuit Z score, so Z equals 3800 minus 4064 divided by the standard error of the mean, which is for 60 over the square root of 20. So the Z score associated with 3800 is negative 2.57 So on the bell, we can put negative 2.57 So being less than 3800 is the same as the probability that Z is less than negative 2.57 And if you look in your standard normal table, you're gonna find that the probability of less than the Z being less than negative 2.57 is 0.0 51 So that was for part a. Now we want to do part B and for part B, you're asked to find the probability that the average is greater than 4100. So 4100 would be like somewhere here. So you're looking for this so again, we're gonna need a Z score associated with 4100. So we're going to do 4100 minus the average of the sample means for 064 all over the standard deviation of the sample means which is 460 divided by the square root of 20. So that Z score is going to be 0.35 So 0.35 is equivalent to a 41 100. So being greater than 4100 is the same thing as the probability that Z is greater than 0.35 which can be written as one minus the probability that Z is less than 0.35 And when you use the standard normal table the area to the left of 0.35 would be 0.6 368 And when you do one minus that, you get a probability of being point 36 three to. So just in summary, the probability that the 12 selected patients haven't average um, hospital insurance benefit less than 3800 is 0.51 and the probability that the 20 patients selected have an average insurance benefit worth something greater than 4100 would be 0.3632