Question:Stock market participation and IQ. Refer to The Journal of Finance (December 2011) study of whether the decision to invest in the stock market is dependent on IQ, Exercise 3.46 (p. 182). The researchers found that the probability of a Finnish citizen investing in the stock market differed depending on IQ score. For those with a high IQ score, the probability is .44; for those with an average IQ score, the probability is .26; and for those with a low IQ score, the probability is .14.

a. In a random sample of 500 Finnish citizens with high IQ scores, what is the probability that more than 150 invested in the stock market?

b. In a random sample of 500 Finnish citizens with average IQ scores, what is the probability that more than 150 invest in the stock market?

c. In a random sample of 500 Finnish citizens with low IQ scores, what is the probability that more than 150 invest in the stock market?

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${\sigma }_{\hat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$

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

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

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

$$\begin{gathered} P\left(\hat{p}>0.30\right)=P\left(\frac{\hat{p}-p}{{\sigma }_{\hat{p}}}>\frac{0.30-p}{{\sigma }_{\hat{p}}}\right) \\ =P\left(Z>\frac{0.30-0.26}{\sqrt{\frac{p\left(1-p\right)}{n}}}\right) \\ =P\left(Z>\frac{0.04}{\sqrt{\frac{0.26\times 0.74}{500}}}\right) \\ =P\left(Z>\frac{0.04}{\sqrt{0.0003848}}\right) \\ =P\left(Z>\frac{0.04}{0.0196}\right) \\ =P\left(Z>2.04\right) \\ =1-P\left(Z⩽2.04\right) \\ =1-0.9793 \\ =0.0207 \end{gathered}$$

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

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

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

Therefore, the required probability is 0.0.