Performance of stock screeners. In Exercise 2.44 (p. 95) you learned that stock screeners are automated tools used by investment companies to help clients select a portfolio of stocks to invest in. The table below lists the annualized percentage return on investment (as compared to the Standard & Poor’s 500 Index) for 13 randomly selected stock screeners provided by the American Association of Individual Investors (AAII)

9.0, -.1, -1.6, 14.6, 16.0, 7.7, 19.9, 9.8, 3.2, 24.8, 17.6, 10.7, 9.1

a. Find a 90% confidence interval for the average annualized percentage return on investment of all stock screeners provided by AAII. Interpret the result.

b. Recall that a negative annualized return reflects a stock portfolio that performed worse than the S&P 500. On average, do the AAII stock screeners perform worse or better than the S&P 500? Explain.

c. What assumption about the distribution of the annualized percentage returns on investment is required for the inference, part b, to be valid? Is this assumption reasonably satisfied?

Referring to exercise 2.44, the stock screeners are automated tools used by learned that stock screeners are automated tools used by investment companies to help clients select a portfolio of stocks to invest in.

a.

The sample mean is,

\(\begin{aligned}\bar x &= \frac{{\sum\limits_{i = 1}^n {{x_i}} }}{n}\\ &= \frac{1}{{13}}\left( {9 + \left( { - 0.1} \right) + \left( { - 1.6} \right) + ... + 9.1} \right)\\ &= \frac{{140.7}}{{13}}\\ &= 10.823076\\ \approx 10.82\end{aligned}\)

The sample standard deviation is,

\(\begin{aligned}s &= \frac{1}{{n - 1}}\left( {{{\sum\limits_{i = 1}^n {\left( {{x_i} - \bar x} \right)} }^2}} \right)\\ &= 7.7115\\ \approx 7.71\end{aligned}\)

The 90% confidence interval is,

\(\begin{aligned}100\left( {1 - \alpha } \right)\% &= 90\% \\\left( {1 - \alpha } \right) &= 0.90\end{aligned}\)

So,

\(\begin{aligned}\alpha &= 0.1\\\frac{\alpha }{2& = 0.05\end{aligned}\)

Therefore, the 90% confidence interval is,

\(\begin{aligned}C.I &= \bar x \pm {t_{n - 1,\frac{\alpha }{2}}}\frac{s}{{\sqrt n }}\\ &= 10.82 \pm {t_{12,0.05}}\frac{{7.71}}{{\sqrt {13} }}\\ &= 10.82 \pm \left( {1.782 \times 2.138} \right)\\ &= 10.82 \pm 3.810\\ &= \left( {7.01,14.63} \right)\end{aligned}\)

Hence, a 90% confidence interval for the average annualized percentage return on investment of all stock screens provided by AAII is\(\left( {7.01,14.63} \right)\).

90% confidence that the average annualized percentage return on investment of all stock screeners provided by AAII will lie either in the interval \(\left( {7.01,14.63} \right)\) or it is not.

b.

For the confidence interval \(\left( {7.01,14.63} \right)\), there is a 90% chance to AAII stock screeners perform better than the S&P 500.

c.

It can be concluded that the assumption about the distribution of annualized percentage returns from investment is required for inference part b to be valid is that the distribution is normal.

From the above graph, it can be concluded that the assumption is reasonably satisfied.