Respuesta :
Answer:
1. P(X≥35) = 0.0183
2. P(X≤21) = 0.0183
3. P(0.18<p<0.25) = 0.7915
Step-by-step explanation:
We have the proportion for women: pw=0.22, and the proportion for men: pm=0.19.
1. We have a sample of 140 woman and we have to calculate the probability of getting 35 or more who do volunteer work.
This is equivalent to a proportion of
[tex]p=X/n=35/140=0.25[/tex]
The standard error of the proportion is:
[tex]\sigma_p=\sqrt{\dfrac{p(1-p)}{n}}=\sqrt{\dfrac{0.22*0.78}{300}}\\\\\\ \sigma_p=\sqrt{0.0006}=0.0239[/tex]
We calculate the z-score as:
[tex]z=\dfrac{p-p_w}{\sigma_p}=\dfrac{0.25-0.22}{0.0239}=\dfrac{0.03}{0.0239}=0.8198[/tex]
Then, the probability of having 35 women or more who do volunteer work in this sample of 140 women is:
[tex]P(X>35)=P(p>0.25)=P(z>2.0906)=0.0183[/tex]
2. We have to calculate the probability of having 21 or fewer women in the group who do volunteer work.
The proportion is now:
[tex]p=X/n=21/140=0.15[/tex]
We can calculate then the z-score as:
[tex]z=\dfrac{p-p_w}{\sigma_p}=\dfrac{0.15-0.2}{0.0239}=\dfrac{-0.05}{0.0239}=-2.0906[/tex]
Then, the probability of having 21 women or less who do volunteer work in this sample of 140 women is:
[tex]P(X<21)=P(p<0.15)=P(z<-2.0906)=0.0183[/tex]
3. For the sample with men and women, we use the proportion for both, which is π=0.2.
The sample size is n=300.
Then, the standard error of the proportion is:
[tex]\sigma_p=\sqrt{\dfrac{p(1-p)}{n}}=\sqrt{\dfrac{0.2*0.8}{300}}\\\\\\ \sigma_p=\sqrt{0.0005}=0.0231[/tex]
We can calculate the z-scores for p1=0.18 and p2=0.25:
[tex]z_1=\dfrac{p_1-\pi}{\sigma_p}=\dfrac{0.18-0.2}{0.0231}=\dfrac{-0.02}{0.0231}=-0.8660\\\\\\z_2=\dfrac{p_2-\pi}{\sigma_p}=\dfrac{0.25-0.2}{0.0231}=\dfrac{0.05}{0.0231}=2.1651[/tex]
We can now calculate the probabilty of having a proportion within 0.18 and 0.25 as:
[tex]P=P(0.18<p<0.25)=P(-0.8660<z<2.1651)\\\\P=P(z<2.1651)-P(z<-0.8660)\\\\P=0.9848-0.1933\\\\P=0.7915[/tex]
