91Ó°ÊÓ

The National Center for Health Statistics' (NCHS) objective is to produce statistical data that will influence activities and policies aimed at improving Americans' health. NCHS, as the nation's primary health statistics agency, sets the standard for reliable, timely, and relevant data.

$25.2\%$of males never eat breakfast.

$23.6\%$of females never eat breakfast.

200 men and 200 women are chosen at random.

Let

M denote the number of men that never eat breakfast

W denote the number women that never eat breakfast

Note that

M is a binomial random variable

With

$p=0.252$and $n=200$

$W$is a binomial random variable

With

$p=0.236andn=200$

Then compute

$E\left[M\right]=200(0.252)=50.4$

And

$E\left[W\right]=200(0.236)=47.2$

Also,

$Var\left[M\right]=200(0.252)(1-0.252)≈37.7$

And

$Var\left[W\right]=200(0.236)(1-0.236)≈36.1$

Thus,

We need to approximate

M by a normal random variable

With

$\mathrm{μ}=50.4\text{and}{\mathrm{σ}}^2≈37.7$

W by a normal random variable

With

$\mathrm{μ}=47.2\text{and}{\mathrm{σ}}^2≈36.1$

Now,

We need to compute $P\{110≤M+W\}$.

Approximate $M+W$by a nomal distribution

With

$\mathrm{μ}=50.4+47.2=97.6$

And

${\mathrm{σ}}^2≈37.7+36.1=73.8$

Thus,

$P=\{110≤M+W\}=0.0749$

Let

M denote the number of men that never eat breakfast

W denote the number women that never eat breakfast

Note that

M is a binomial random variable

With

$p=0.252\text{and}n=200$

$W$is a binomial random variable

With

$p=0.236\text{and}n=200$

Then compute

$E\left[M\right]=200(0.252)=50.4$

And

$E\left[W\right]=200(0.236)=47.2$

Also,

$Var\left[M\right]=200(0.252)(1-0.252)≈37.7$

And

$Var\left[W\right]=200(0.236)(1-0.236)≈36.1$

Thus,

We need to approximate

M by a normal random variable

With

$\mathrm{μ}=50.4\text{and}{\mathrm{σ}}^2≈37.7$

W by a normal random variable

With

$\mathrm{μ}=47.2\text{and}{\mathrm{σ}}^2≈36.1$

Now,

We need to compute $P\{M≤W\}=P\{M-W≤0\}$.

Approximate $M-W$by a normal distribution

With

$\mathrm{μ}=50.4-47.2=3.2$

And

${\mathrm{σ}}^2≈37.7+36.1=73.8$

Thus,

$P\{M≤W\}=0.3557$