Question

you are given the sample mean and the population standard deviation. use this information to construct the 90% and 95% confidence intervals for the population mean. interpret the results and compare the widths of the confidence intervals. from a random sample of 43 business days, the mean closing price of a certain stock was $121.57. assume the population standard deviation is $9.87. which interval is wider? choose the correct answer below. the 95% confidence interval the 90% confidence interval interpret the results. a. you can be 90% confident that the population mean price of the stock is between the bounds of the 90% confidence interval, and 95% confident for the 95% interval. b. you can be certain that the population mean price of the stock is either between the lower bounds of the 90% and 95% confidence intervals or the upper bounds of the 90% and 95% confidence intervals. c. you can be 90% confident that the population mean price of the stock is outside the bounds of the 90% confidence interval, and 95% confident for the 95% interval. d. you can be certain that the closing price of the stock was within the 90% confidence interval for approximately 39 of the 43 days, and was within the 95% confidence interval for approximately 41 of the 43 days.

Explanation:

Step1: Recall confidence - interval formula

The formula for a confidence interval for the population mean when the population standard - deviation $\sigma$ is known is $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $z_{\alpha/2}$ is the z - score, $\sigma$ is the population standard deviation, and $n$ is the sample size. Given $\bar{x} = 121.57$, $\sigma=9.87$, and $n = 43$.

Step2: Find z - scores for 90% and 95% confidence intervals

For a 90% confidence interval, $\alpha=1 - 0.90=0.10$, and $\alpha/2 = 0.05$. The $z$ - score $z_{\alpha/2}=z_{0.05}\approx1.645$. For a 95% confidence interval, $\alpha = 1-0.95 = 0.05$, and $\alpha/2=0.025$. The $z$ - score $z_{\alpha/2}=z_{0.025}\approx1.96$.

Step3: Calculate the margin of error for 90% confidence interval

The margin of error $E_1=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$. Substituting the values, we have $E_1 = 1.645\times\frac{9.87}{\sqrt{43}}\approx1.645\times\frac{9.87}{6.5574}\approx1.645\times1.5052\approx2.47$. The 90% confidence interval is $\bar{x}\pm E_1=121.57\pm2.47=(119.10,124.04)$.

Step4: Calculate the margin of error for 95% confidence interval

The margin of error $E_2=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$. Substituting the values, we have $E_2 = 1.96\times\frac{9.87}{\sqrt{43}}\approx1.96\times\frac{9.87}{6.5574}\approx1.96\times1.5052\approx2.95$. The 95% confidence interval is $\bar{x}\pm E_2=121.57\pm2.95=(118.62,124.52)$.

Step5: Compare the widths and interpret

The width of a confidence interval is $2E$. The width of the 90% confidence interval is $2E_1\approx4.94$, and the width of the 95% confidence interval is $2E_2\approx5.90$. The 95% confidence interval is wider.
The interpretation of a confidence interval is that you can be 90% confident that the population mean price of the stock is between the bounds of the 90% confidence interval, and 95% confident for the 95% interval.

Answer:

Which interval is wider? The 95% confidence interval
Interpret the results: A. You can be 90% confident that the population mean price of the stock is between the bounds of the 90% confidence interval, and 95% confident for the 95% interval.