Chapter 8, Problem 44SE

A sample survey of 54 discount brokers showed that the mean price charged for a trade of 100 shares at $50 per share was $33.77. The survey is conducted annually. With the historical data available, assume a known population standard deviation of $15.

1. a. Using the sample data, what is the margin of error associated with a 95% confidence interval?
2. b. Develop a 95% confidence interval for the mean price charged by discount brokers for a trade of 100 shares at $50 per share.

To determine

Find the margin of error for 95% confidence level.

Answer to Problem 44SE

The margin of error for 95% confidence level is 4.

Explanation of Solution

Calculation:

The given information is that the sample of 54 discount brokers shows the mean price of the trade is $50 for 100 shares is $33.77 and the population standard deviation is $15.

The formula for margin of error is $z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}}$.

From the “Table 8.1 value of $z_{\frac{\alpha }{2}}$”, the value of $z_{\frac{\alpha }{2}}$ for 95% is 1.96.

The value of margin of error is

$\begin{array}{c}{z}_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}}=1.96\frac{15}{\sqrt{54}}\\ =1.96\times 2.04\\ =4\end{array}$

Thus, the value of margin of error is 4.

To determine

Find the 95% confidence interval for mean price charged by discount brokers for a trade of 100 shares at $50 per share.

Answer to Problem 44SE

The 95% confidence interval for mean price charged by discount brokers for a trade of 100 shares at $50 per share is ($29.77, $37.77).

Explanation of Solution

Calculation:

From part (a), the margin of error is 4.

The mean value for the 54 broker for trade of 100 shares at $50 per share is $33.77.

The value of confidence interval for population mean is

$\begin{array}{c}\overline{x}\pm z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}}=33.77\pm 4\\ =\left(33.77-4,33.77+4\right)\\ =\left(29.77,37.77\right)\end{array}$

Thus, the 99% confidence interval for population mean is ($29.77, $37.77).