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Numerade Educator

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Problem 55 Medium Difficulty

Findings from a survey of American adults conducted by Yankelovich Partners for the International Bottled Water Association indicate that Americans on the average drink 6.18-ounce servings of water a day [http://www.pangaeawater.com/]. Assuming that the number of 8-ounce servings of water is approximately normally distributed with a standard deviation of 1.4 servings, what proportion of Americans drink
a. more than the recommended 8 servings?
b. less than half the recommended 8 servings?

Answer

a .0869; b .0668

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Video Transcript

okay for this problem. We have some normally distributed data, so we should always sketch are a normal distribution and label It are mean and standard deviation so we can see what we're looking at here. So I brought a diagram to first stuff on then right? Are probability statements in the I'm calculate the the probabilities. So we know the 6.1, uh, eight ounce servings of water is the mean here, and the standard deviation of these water servings is 1.4 ounces. All right, so plus or minus so 6.1. And there's one standard deviation above 12 above and a couple below wouldn't have a nicer diagram about here in a minute. But I like to sketched out for the direction. And the question here wants us to know the probability that you have more than eat no recommended eight servings. Okay, so 6.1 servings is the average of the mean, so into the probability that we drink X is greater than hey, cereals casts of 6.1 is here, and they will be up to 7.5. So just give a general direction of more than this will be. The area of the curve is what we're finding, whether it's from a nap letter from the table. So I'm just gonna put a little mark over here. That's each the diagram there and with my math calculator. And then for 2nd 1 we're going to look at the direction and then calculate the probability less than half of the recommended beats serving. So less than half less than half, uh, remember recommended eight servings. So that probability it's less than before servings. All right, so 6.1 is here, 1.4 below. So just a rough estimate of less than half is going to be the area to left over here, all right, and he's graphing calculator a table to get dizzy scores. I like to use tablets very visual, and I like the way it asks for the equation. But technically, what it's doing is just taking the easy scores associated with each of those numbers and friend of probabilities associated with them. As you can see, that's like of our sketch. 6.1 is our mean and 1.4 is a standard deviation. So the apple it will panel review it on the Komeito or clean away what we sketched over there already and way want to. Let's go from the top here we want to. Between is the default here relative to the left from value are actually go to the right of the value eight because we want toe part. They asked for numbers that are larger than eight if we have the meaning of 6.1 and the standard deviation. So there is a nice sketch. If you have some nice you may our label out these standard deviations on your paper for this way. Look like we end up with an answer of approximately 0.874 Drink more than eight. Serving today. Yes, today. So there's that and let's go over here and just change. Looking to the left for Boise is our green. The left, the value means we want to look below for So we're looking at a table. You get dizzy score. Um, and then you'll get the decimal. We're about to get here. I took in this button, so that's 0.66 each. Oh, well so a little less than 6.66 point 6% which makes sense pretty small senses, Couple center deviations away and this is, ah, important sent drink more than what's needed if the 6.1 is the mean suggested servings.