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Numerade Educator

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Problem 20 Medium Difficulty

In a recent year, Delaware had the highest per capita annual income with $\$ 51,803$. If $\sigma=\$ 4850$, what is the probability that a random sample of 34 state residents had a mean income greater than $\$ 50,000 ?$ Less than $\$ 48,000 ?$

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alright in this problem, we're talking about the fact that Delaware has the highest per capital annual income, and we know that the average of that annual income is 51,803. We know that the standard deviation is $4850 and we're talking about the entire population of Delaware in this problem. We're also talking about the fact that we're going to take a random sample of 34 residents. So from that population, we're going to pull a sample and the sample size is 34. And then we're going to talk about the average of those 34 residents. So the average of the sample means and we've got to talk about the standard deviation of the sample means. And there's two questions that we're going to answer in this problem. The first one is what is the probability that the mean of those 34 residents that were chosen is greater than 50,000, and the second one is going to be less than 48,000. So let's work on the fact that we're doing the mean being greater than 50,000, so we're going to need to construct are bell shaped curve. Um, again, it's the bell shaped curve is because of our sample size being large enough, it's greater than 30 so the sample distribution will be normally distributed. The average of our sample means, based on the central limit theorem, will be equivalent to the average of the population, or, in this case, 51,803. And the standard deviation will be equivalent to the standard deviation of the population divided by the Santa Square root of the sample size or, in other words, 4850 divided by the square root of 34. So in order to solve our problem, we're going to put the average in the centre Rod Bell, and we want the fact that the average has to be greater than 50,000, which would fall about here on our bell, and we want to be greater. So we're going to need the standard Z score. So we're going to calculate the Z score for 50,000 and the formula for the sea scores e equals X bar minus mu sub x bar all over Cygnus of expert. So in this instance, the sea score would be just move that over a little bit. Disease score would be 50,000 minus 51 803 all over 4850. Divide by the square root of 34 which calculates out to be approximately negative 2.17 So negative 2.17 is up there. So when we're asking for the probability that the average is greater than 50,000, it's no different than saying What's the probability that the Z score is greater than negative 2.17? And that is the same thing as saying one, minus the probability that the Z score is less than negative 2.17 And then if you go to your standard normal table and look up negative 2.17 you will find that the area to the left is 0.1 50 When you subtract that from one, we get 10.9 eight five zero. So in summary of the first part of this problem, the probability that the 34 Delaware residents have an average income or a mean income greater than 50,000 is 0.98 50 Now let's go to part two of this, we're going to have to use the same information. I am going to screwed it up a little bit here. We're going to need the Z score associated with 48,000. So the second question is, what's the probability that the average is less than 48,000? We're going to calculate the Z score using that same formula we're gonna do 48,000 minus 51,803 over 400,850 divided by the square root of 34. And if you use your calculator, you will calculate that out to be approximately negative 4.57 So when we say that we're looking for the probability that the average is less than 48,000, since 48,000 has a Z score of negative 4.57 We could also write this problem as what's the probability that the Z score is less than negative 4.57 Now, if you look at your standard normal table, you're going to see that there is no negative 4.57 It's off the chart, and as you look as we get closer and closer to that value. The numbers are getting closer and closer to zero. So we're going to say that the probability that Z is less than negative 4.57 is close 20 So it's a highly unlikely event. So to summarize Part two of this question, the probability that X bar were the average of the 34 residents that were selected in a sample having a salary or income sorry of less than $48,000 is close 20