91Ó°ÊÓ

The probability of Finnish citizens investing in the stock market depends on their IQ scores.

For high IQ score p=0.44, average IQ score p=0.26, and low IQ score=0.14 .

A random sample of size 500 is selected from each IQ score.

For each IQ score, the sample proportion who invests in the stock market is:

$$\begin{gathered} \hat{p}=\frac{150}{500} \\ =0.30 \end{gathered}$$

.

The mean of the sample proportion is .$E\left(\hat{p}\right)=p$

The standard deviation of the sampling distribution $\hat{p}$is${\mathrm{σ}}_{\hat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$

a.For high IQ scores, the probability that more than 150 invested in the stock market, that is, sample proportion greater than 0.30, is obtained as

$$\begin{gathered} P\left(\hat{p}>0.30\right)=P\left(\frac{\hat{p}-p}{{\mathrm{σ}}_{\hat{p}}}>\frac{0.30-p}{{\mathrm{σ}}_{\hat{p}}}\right) \\ =P\left(Z>\frac{0.30-0.44}{\sqrt{\frac{p\left(1-p\right)}{n}}}\right) \\ =P\left(Z>\frac{-0.14}{\sqrt{\frac{0.44×0.56}{500}}}\right) \\ =P\left(Z>\frac{-0.14}{\sqrt{0.0004928}}\right) \\ =P\left(Z>\frac{-0.14}{0.0222}\right) \\ =P\left(Z>-6.31\right) \\ ≈1.00 \end{gathered}$$

Since by using the z-table, the required probability is approximately 1.

Therefore, the required probability is 1.0.

b.

For average IQ scores, the probability that more than 150 invested in the stock market, that is, sample proportion greater than 0.30, is obtained as

$$\begin{gathered} P\left(\hat{p}>0.30\right)=P\left(\frac{\hat{p}-p}{{\mathrm{σ}}_{\hat{p}}}>\frac{0.30-p}{{\mathrm{σ}}_{\hat{p}}}\right) \\ =P\left(Z>\frac{0.30-0.26}{\sqrt{\frac{p\left(1-p\right)}{n}}}\right) \\ =P\left(Z>\frac{0.04}{\sqrt{\frac{0.26×0.74}{500}}}\right) \\ =P\left(Z>\frac{0.04}{\sqrt{0.0003848}}\right) \\ =P\left(Z>\frac{0.04}{0.0196}\right) \\ =P\left(Z>2.04\right) \\ =1-P\left(Zâ\copyright ½2.04\right) \\ =1-0.9793 \\ =0.0207 \end{gathered}$$

Using the z-table, the value at 2.00 and 0.04 is the probability of a z-score less than or equal to 2.04.

Therefore, the required probability is 0.0207.

For low IQ scores, the probability that more than 150 invested in the stock market, that is, sample proportion greater than 0.30, is obtained as

$$\begin{gathered} P\left(\hat{p}>0.30\right)=P\left(\frac{\hat{p}-p}{{\mathrm{σ}}_{\hat{p}}}>\frac{0.30-p}{{\mathrm{σ}}_{\hat{p}}}\right) \\ =P\left(Z>\frac{0.30-0.14}{\sqrt{\frac{p\left(1-p\right)}{n}}}\right) \\ =P\left(Z>\frac{0.16}{\sqrt{\frac{0.14×0.86}{500}}}\right) \\ =P\left(Z>\frac{0.16}{\sqrt{0.0002408}}\right) \\ =P\left(Z>\frac{0.16}{0.0155}\right) \\ =P\left(Z>10.32\right) \\ ≈0.00 \end{gathered}$$

Using the z-table, the value is approximately 0.0.

Therefore, the required probability is 0.0.