Chapter 8: Problem 44

A sample survey of 54 discount brokers showed that the mean price charged for a trade of 100 shares at $\$ 50$ per share was $\$ 33.77$ (AAII Journal, February 2006 ). The survey is conducted annually. With the historical data available, assume a known population standard deviation of $\$ 15$ a. Using the sample data, what is the margin of error associated with a $95 \%$ confidence interval? b. Develop a $95 \%$ confidence interval for the mean price charged by discount brokers for a trade of 100 shares at $\$ 50$ per share.

Short Answer

Expert verified

The margin of error associated with a $95\%$ confidence interval is approximately $\$4.00$. The $95\%$ confidence interval for the mean price charged by discount brokers for a trade of 100 shares at $\$50$ per share is $\$29.77$ to $\$37.77$.

Step by step solution

Identify the given values

The given values in this problem are as follows: Sample mean: $\bar{x} = \$33.77$ Sample size: $n = 54$ Population standard deviation: $\sigma = \$15$

Calculate the standard error

The standard error of the sample is calculated using the population standard deviation and sample size. Standard Error (SE) = $\frac{\sigma}{\sqrt{n}}$ Plug in the given values and calculate the standard error: SE = $\frac{\$15}{\sqrt{54}}\approx \$ 2.04$

Find the Z-score for a $95\%$ confidence interval

For a $95\%$ confidence interval, the Z-score (critical value) is found by identifying the value which attracts $2.5\%$ in each tail of the standard normal distribution (as the total $5\%$ is divided equally between the two tails). Therefore, we want the Z-score such that: $\mathbf{P}(-Z_{\alpha/2} < Z < Z_{\alpha/2}) = 0.95$ Using a Z-table or calculator, we find the value of $Z_{\alpha/2}$ to be: $Z_{\alpha/2} = 1.96$

Calculate the margin of error

Now, we can calculate the margin of error associated with a 95% confidence interval using the formula: Margin of Error (ME) = $Z_{\alpha/2} * SE$ Plug in the values from Steps 2 and 3: ME = $1.96 * \$ 2.04\approx \$ 4.00$

Develop the 95% confidence interval for the mean price

With the margin of error calculated, we can now develop the 95% confidence interval for the mean price charged by discount brokers for a trade of 100 shares at $50 per share. Lower Limit = $\bar{x} - ME$ Upper Limit = $\bar{x} + ME$ Plug in the values from Step 1 and Step 4: Lower Limit = $\$33.77 - \$4.00\approx \$29.77$ Upper Limit = $\$33.77 + \$4.00\approx \$37.77$

Interpret the results

The 95% confidence interval for the mean price charged for a trade of 100 shares at $50 per share is $\$29.77$ to $\$37.77$. This means we are confident that, if the survey were conducted repeatedly, the true population mean price would fall within this range 95% of the time.