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Problem 15 Medium Difficulty

The average overseas trip cost is $\$ 2708$ per visitor. If we assume a normal distribution with a standard deviation of $\$ 405,$ what is the probability that the cost for a randomly selected trip is more than $\$ 3000 ?$ If we select a random sample of 30 overseas trips and find the mean of the sample, what is the probability that the mean is greater than $\$ 3000 ?$

Answer

0.2358; less than 0.0001

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Video Transcript

So then problem number 15. There are really two questions, and both of those questions revolve around the same given information. So the given information is that the average cost of an overseas trip is 2708 with a standard deviation of 405 and the other given information is that we're going to assume normal distribution. So because of the nature of this problem, we know that the average always goes in the center of our bell, and we're ready to ask our questions. So the question here is what is the probability that the cost said the probability that the cost of a randomly selected trip So it's one trip, one piece of data, So we're going to say X is more than which is greater than $3000. So 3000 on the bell would end up being to the right of 2708 and we're talking about being greater than so. We're looking for the area of the curve to the right of 2000 or of 3000. In order to solve this, we're going to need to switch the 3000 into its standard score or it Z score and you have a formula for Z score as X minus new, divided by sigma. So in this case, it would be 3000 minus 2708 all over 405 which is equivalent 2.72 So on our bow we can put a 0.72 at the same line as the 3000. So instead of us, thinking of this problem is being greater than 3000, we can also talk about it as being the probability where Z is greater than 0.72 When you look in the back of the book at the standard normal table, you're going to find that the table always goes to the left, the area to the left. So if we look up 0.7 to the area to the left of this line is 0.76 four to. So if we want the area to the right, we're gonna have to do one. Take away the area to the left with a probability that sees to the left of 10.72 because the entire curve represents an area of one. So, in essence, are problems gonna read one minus 10.7642 which would result in 0.23 five eight. So in summary, the probability that a randomly selected trip costs more than $3000 is 0.2358 Now, we're ready to tackle the second part of this problem. So I'm just gonna scoot up a little bit and we're going to discuss what was given are given was still that the average cost of a trip overseas was 2708 and our standard deviation was still 405. They want you to realize that this is information about the population, all trips overseas. And in the second part of this problem, where you're gonna be talking about a sample from the population and the sample from the population is going to be of size 30 it says if we select a random sample of 30 overseas trips, we want to ultimately find the probability that the mean of those trip costs is greater than 3000. Ultimately, that's what we're trying to solve Now. In order to solve this type of problem, we will need the central limit theorem, So we will need to find the average of the sample means and we will need to find and let me fix that one second here. The average of the sample means would be new X bar, and we will need to find the standard deviation of the sample means and the central limit. Theorem says that the average of the sample means is equivalent toothy average of the population, which in this case is 2708. And the standard deviation, or of the sample means or the standard error of the mean is equivalent to the standard deviation of the population divided by the square root of the sample size. So in this case, it would be 405 divided by the square root of 30 again, we're going to use our bell shaped curve, so we're gonna draw at our bell shaped curve and at the center of our bell shaped curve, we're gonna put our average of the sample means 72,708 and we're talking about when the average is greater than 3000. So we're going to want to put 3000 on our curva swell, and we're talking about being greater, So just like the first part of this problem, we needed to switch the 3000 into its standard equivalent. So we're going to also find a Z score. But this time the Z score is going to look slightly different because instead of a single data, we're talking about an average of 30 pieces of data, so it's gonna be X minus. The average of the sample means divided by the standard deviation of sample means. And we know that the X bar is 3000 minus. The average of the sample means 2708 and the standard deviation of the sample means is 400 over the square root of 30. And if you pick your calculator up and calculate that you end up with approximately a 3.95 so up here on our bell, we can put 3.95 And when we talk about to the right of 3000, it's no different than saying that we're to the right of 3.95 so we can rewrite our problem to read. The probability the Z is greater than 3.95 and greater than 3.95 is the same as one, minus the probability that Z is less than 3.95 And then when we go to the standard normal chart in the back of your textbook, the 3.95 is nowhere to be found. Your chart goes up to almost 3.5. Um, and if you look in that lower right hand corner, you've got a an area or a probability of 0.9998 So 3.95 is larger than what's there, so we know that the probability is going to be something greater than 0.9998 so we can make an assumption that it's it's really, really, really close toe one. So what we're gonna do is we're just going to use something that's really close toe one. We're gonna say one minus 10.99999 and we're going to get an answer of 0.1 or somewhere close. So in summary, our best bet is to just say that the probability that the average of those 30 trips is greater than $3000 is less than 0.1