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Suppose that for a certain individual, calorie intake at breakfast is a random variable withexpected value 500 and standard deviation $50,$ calorie intake at lundom with expected value 900 and standard deviation $100,$ and calorie intake at dinner is a random variable withexpected value 2000 and standard deviation $180 .$ Assuming that intakes at different meals are independent of each other, what is the probability that average calorie intake per day over thenext $\left(365$ -day) year is at most 3500$?\left[$ Hint: Let $X_{i}, Y_{i},$ and $Z_{i}$ denote the three calorie intakes on \right.\right. day $i$ . Then total intake is given by $\sum\left(X_{i}+Y_{i}+Z_{i}\right) . ]$

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Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 1

Jointly Distributed Random Variables

Probability Topics

The Normal Distribution

University of North Carolina at Chapel Hill

Cairn University

University of St. Thomas

Boston College

Lectures

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01:49

Suppose that the mean calo…

02:42

The amount of calories adu…

01:34

The number of calories in …

05:09

A group of individuals wit…

01:44

A random sample of 10 choc…

00:59

The New York Times collect…

01:39

Calories in a Standard Siz…

01:27

So we're trying to find this probability down here, which is the probability that the average amount of calories consumed on a given day over the course of a year is 3500 or less. And we're gonna find this, thi this normal distribution of this tea by just adding up these X I, Y I, and C. I. Normal distributions, which are the normal distributions for the amount of calories consumed at breakfast, lunch and dinner. And so we can say t is a normal distribution with a mean of our three means added together, which is 500 plus 900 plus 2000, which is 3400 and a variance of our variances added together, which is 44,900. And now we can use the central limit theorem to actually find this probability. And so what we want to do is we want to set up a z. The Z score where we have X minus you over standard deviation divided by the square of our end. And then we're just going to plug this into this probability and use the standard normal table from there. So if we do this, we have P of Z is listener equal to Our exes 3500, so we have 30 500 minus are you? Is 3400? And this is all divided by Our standard deviation is the square root of our variants. So we have the square root of 44,900, Divided by the Square Root of 3 65. And if we simplify this equation, we'll see that we have P of Z is less than or equal to 100 over 11.09, which then we simplify further, is around the probability that C is less than or equal to 9.02. And so I can already see that this is a very big number for a standard normal table distribution, since the mean of a standard normal table is zero and we have a value of Z is less than or equal to nine. So this is going to be pretty much one. It's going to be pretty much guaranteed that we're going to have a average amount of calories consumed less than or equal to 3500 calories per day over the course of a year.

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