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Numerade Educator

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Problem 27 Medium Difficulty

A study of 800 homeowners in a certain area showed that the average value of the homes was $\$ 82,000$, and the standard deviation was $\$ 5000 .$ If 50 homes are for sale, find the probability that the mean of the values of these homes is greater than $\$ 83,500 .$

Answer

0.0143

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Filomena M.

November 5, 2020

. In a recent study of 15 class eight students, the mean number of hours per week that they played video games was 16.6 and the standard deviation of 2.8. Assume that the variable is normally distributed. A. Find the point estimate of the population mean.

Video Transcript

in this problem. We have a population of 800 homes and the average value of the home is $82,000 with a standard deviation of $5000 from that population. We're going to select a sample and the sample size is 50. Because we're saying if 50 homes are for sale, we want to find the probability that the mean is greater than 83,500. So because we selected a sample size that was large enough, no matter what shape the distribution of the population is, the sample distribution will be bell shaped or normal. Because we're dealing with a sample, we need to find the average of the sample means and we need to find the standard deviation of the sample means otherwise known as this um, standard error of the mean to find the average of the sample means the central limit theorem tells us it will be equivalent to the average of the population, and in this case it will be 82,000. The standard deviation of sample means, according to the central limit theorem, will be the standard deviation of the population divided by the square root off the sample size. So in this case, it will be 5000 over the square root of 50. So on our bell curve, we're going to start by putting the average the 82,000 in the center and we will need to calculate Z scores. So in this particular problem, we're talking about having a value of 83,500 and actually greater than that. And in order to find our Z score, we're going to have to apply the formula for using samples as X bar minus, um, use of Expo are divided by Cygnus X bar. So we need thes e score for 83,500. So we're going to say Z equals 83,500 minus 82,000, divided by 5000 divided by the square root of 50. And we do that. Our Z score turns out to be 2.12 so we can put a 2.12 up here. So when we're talking about the probability that the average of the 50 homes, um is greater than 83 5, we can also say that's the same as the probability that the Z score is greater than 2.12 and that's the same thing is saying one minus. The probability that the Z score is less than 2.12 were one minus 0.9830 and we got that 0.9830 by looking in the standard normal table in the back of your book, and the overall probability would be 0.170 So just to recap the probability of the 50 homes that are for sale in this a certain area having a valley an average value greater than 83,500 would be 0.170