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Numerade Educator



Problem 72 Easy Difficulty

Use a computer to compare a random sample to the population from which the sample was drawn. Consider the normal population with mean 75 and standard deviation 14. Answer questions a through $\mathrm{f}$ of Exercise 6.71 using $N(75,14).$



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Video Transcript

{'transcript': "So for this problem, we're gonna be continuing off a problem. 74 the same numbers, and we're going to find the mean and standard deviation a different way. Now the first thing we're going to need is to go back the problem 74 and copy down this table that we found gonna be using this a lot. This was the table that we made for our hissed a gram. And the way we found that was by going into table to an appendix B. Finding the equals 11 and the P equals 110.5 and copying down the intersection, which is this right over here. So I will write it down again. No probability of tax. We had a 0.569 a point free to 98.878 point 014 0.1 And then we had zero plus seven times. Then I'm also going to copy down the corresponding X values. So 01234 and five through 11. So now we can use our formula, Isa. So the first wine is going to be the formula for the mean, And you should have this one written down mean is equal to the sum of all X probability of X. So what this means is we're gonna take each X value, multiply it by its corresponding probability and at all of them up. So, I mean, is equal chill so we can ignore the X equals zero cause zero times the number zero, and we can ignore five to a well, been because we're again multiplying by zero. So we have point free to nine plus two times 20.87 plus free times 0.14 plus four times 0.0 wine. And this gives us 0.5 for nine. Now, to find standard deviation, we're gonna have a little bit more work to Dio. And that's because standard deviation is equal to the square root of variance. Well, how do we find variance? Variants is equal to me. Some of all X squared probability of acts minus the sum of all X probability of X, and this whole fame is square toe. I cannot make this look a little bit meter here there. So we need to find these two pieces and then subtract them. So let's go back into our problem. What's gonna be interesting is for this first piece we can actually just take what we did for the mean. All we have to do is square the X values, and for this second piece, you notice it's very similar to the mean. The only difference is the whole thing is squared, so variance is equal to point free to nine times. So two squared is four times 40.87 plus free squared is nine times 0.14 plus so 16 times 160.0 wine and then this whole feign. When we make this more clear, they got a little bit too close to each other. Race. There we go. And then this whole feign we're going to Let's put this in a bracket minus R means square disappoint five or nine squared, and this gives us 0.476 So find standard deviation. We just take the square root of 0.476 and that is equal 2.6 899 So this way was a little bit harder than the last line. Now compare the results with problems. 74 A. So what were our results in 74 A. We had 0.55 point 72 to 8. Let's go back what we have now. 0.5 for nine and 90.6899 You notice we got very similar answers because thes equations are the same feign. We're doing the same work. The only reason we got slightly off answers is because of all the rounding we had the Dio. So these equations give us the same result and we are done."}