Answer to Question #238261 in Statistics and Probability for suvi

Question

3.1 The average dividend yield of a random sample of 25 JSE-listed companies this year was found
to be 14.5%, with a sample standard deviation of 3.4%. Assume that dividend yields are
normally distributed.
3.1.1 Calculate, with 90% confidence, the actual mean dividend yield of all JSE-listed
companies this year. Interpret the finding. (6)
3.1.2 Calculate, with 95% confidence, the actual mean dividend yield of all JSE-listed
companies this year. Compare the interval with the one calculated in 3.1.1.
(12

Answer 1

3.1.1

The critical value for $\alpha=0.1$ is $z_c=z_{1-\alpha/2}=1.6449.$

The corresponding confidence interval is computed as shown below:

$$CI=(\bar{x}-z_c\times\dfrac{\sigma}{\sqrt{n}}, \bar{x}-z_c\times\dfrac{\sigma}{\sqrt{n}})$$

$$=(0.145-1.6449\times\dfrac{0.034}{\sqrt{25}}, 0.145+1.6449\times\dfrac{0.034}{\sqrt{25}})$$

$$=(0.1338, 0.1562)$$

Therefore, based on the data provided, the 90 % confidence interval for the population mean is $0.1338<\mu<0.1562,$ which indicates that we are 90% confident that the true population mean $\mu$ is contained by the interval $(0.1338, 0.1562),$

$(13.38\%, 15.62\%)$

3.1.2

The critical value for $\alpha=0.05$ is $z_c=z_{1-\alpha/2}=1.96.$

The corresponding confidence interval is computed as shown below:

$$CI=(\bar{x}-z_c\times\dfrac{\sigma}{\sqrt{n}}, \bar{x}-z_c\times\dfrac{\sigma}{\sqrt{n}})$$

$$=(0.145-1.95\times\dfrac{0.034}{\sqrt{25}}, 0.145+1.96\times\dfrac{0.034}{\sqrt{25}})$$

$$=(0.1317, 0.1583)$$

Therefore, based on the data provided, the 95 % confidence interval for the population mean is $0.1317<\mu<0.1583,$ which indicates that we are 95% confident that the true population mean $\mu$ is contained by the interval $(0.1317, 0.1583),$

$(13.17\%, 15.83\%)$

We find that the 95% confidence level has a wider interval as compared to the 90% confidence interval.

Increasing the confidence level increases the error bound, making the confidence interval wider. Decreasing the confidence level decreases the error bound, making the confidence interval narrower.