Question

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 mean of T is

Short answer

The resultant mean T is 180.

Step-by-step solution

Given information

To make one ounce of breakfast cereal,

The mean value is ${\mathrm{\mu }}_{\mathrm{X}}=110$

The variance value is ${\mathrm{\sigma }}_{\mathrm{X}}=10$

Find the cup of whole milk

The mean value is ${\mathbf{\mu }}_{\mathbf{Y}}\mathbf{=}\mathbf{140}$

The variance value is ${\mathrm{\sigma }}_{\mathrm{Y}}=12$

Calculations

If X and Y are not mutually exclusive, then

The property of mean is ${\mathrm{\mu }}_{\mathrm{aX}+\mathrm{bY}}={\mathrm{a\mu }}_{\mathrm{X}}+{\mathrm{b\mu }}_{\mathrm{Y}}$

The property of variance is ${\mathrm{\sigma }}_{\mathrm{aX}+\mathrm{bY}}^2={\mathrm{a}}^2{\mathrm{\mu }}_{\mathrm{X}}^2+{\mathrm{b}}^2{\mathrm{\mu }}_{\mathrm{Y}}^2$

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

That is,

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

As a result, The mean of T is

${\mathrm{\mu }}_{\mathrm{T}}={\mathrm{\mu }}_{\mathrm{X}+\mathrm{Y}/2}={\mathrm{\mu }}_{\mathrm{X}}+\frac{1}{2}{\mathrm{\mu }}_{\mathrm{Y}}=110+\frac{1}{2}\left(140\right)=180$

The resultant variance of T is

${\mathrm{\sigma }}_{\mathrm{T}}={\mathrm{\sigma }}_{\mathrm{X}+\frac{\mathrm{Y}}{2}}=\sqrt{{\mathrm{\sigma }}_{\mathrm{X}}^2+\frac{1}{2^2}{\mathrm{\sigma }}_{\mathrm{Y}}^2}=\sqrt{(10{)}^2+\frac{1}{4}(12{)}^2}\approx 11.66$