Q. 74 The number of calories in a 1-ou... [FREE SOLUTION]

91影视

The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 837 pages

ISBN: 9781319113339

Chapter 6: Q. 74 (page 395)

The number of calories in a 1-ounce serving of a certain breakfast cereal is a random variable with mean 110 and standard deviation 10. The number of calories in a cup of whole milk is a random variable with mean 140 and standard deviation 12. For breakfast, you eat 1 ounce of the cereal with $\frac{1}{2}$ cup of whole milk. Let T be the random variable that represents the total number of calories in this breakfast.
The standard deviation of T is

Short Answer

Expert verified

The random variable indicating the total number of calories in the breakfast's total number of calories has a standard deviation is $鈮� 11.66$

Step by step solution

Given information

1- ounces of breakfast cereal :

The mean value is $渭_{X} = 110$

The variance value is $蟽_{X} = 10$

Determine the cup of whole milk

The mean value is $渭_{Y} = 140$

The variance value is $蟽_{Y} = 12$

Calculations

If X and Y are not mutually exclusive, then

The property of mean is $渭_{aX + bY} = {补渭}_{X} + {产渭}_{Y}$

The property of variance is $蟽_{aX + bY}^{2} = a^{2} 渭_{X}^{2} + b^{2} 渭_{Y}^{2}$

T denotes the total number of calories consumed at breakfast ( 1 ounce of the cereal and half cup of whole milk).

That is,

$T = X + \frac{Y}{2}$

As a result, The mean of T is

$渭_{T} = 渭_{X + Y / 2} = 渭_{X} + \frac{1}{2} 渭_{Y} = 110 + \frac{1}{2} (140) = 180$

The resultant variance of T is

$蟽_{T} = 蟽_{X + \frac{Y}{2}} = \sqrt{蟽_{X}^{2} + \frac{1}{2^{2}} 蟽_{Y}^{2}} = \sqrt{(10)^{2} + \frac{1}{4} (12)^{2}} 鈮� 11 .66$

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