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 40 business​ days, the mean closing price of a certain stock was ​$108.50 . Assume the population standard deviation is​$11.36 . The​ 90% confidence interval is ​(nothing ​,nothing ​).​(Round to two decimal places as​ needed.) The​ 95% confidence interval is ​(nothing ​,nothing ​). ​(Round to two decimal places as​ needed.) 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 outside 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 between 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 36 of the 40 ​days, and was within the​ 95% confidence interval for approximately 38 of the 40 days.

Answer 1

Answer:

The​ 90% confidence interval is ($105.55,$111.45)

The 95% confidence interval is ($104.98,$112.02)

The 95% confidence interval is wider.

C. 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.

Step-by-step explanation:

90% confidence interval:

We have that to find our $\alpha$ level, that is the subtraction of 1 by the confidence interval divided by 2. So:

$\alpha = \frac{1 - 0.9}{2} = 0.05$

Now, we have to find z in the Ztable as such z has a pvalue of $1 - \alpha$.

That is z with a pvalue of $1 - 0.05 = 0.95$, so Z = 1.645.

Now, find the margin of error M as such

$M = z\frac{\sigma}{\sqrt{n}}$

In which $\sigma$ is the standard deviation of the population and n is the size of the sample.

$M = 1.645\frac{11.36}{\sqrt{40}} = 2.95$

The lower end of the interval is the sample mean subtracted by M. So it is 108.50 - 2.95 = $105.55

The upper end of the interval is the sample mean added to M. So it is 108.50 + 2.95 = $111.45

The​ 90% confidence interval is ($105.55,$111.45)

95% confidence interval:

We have that to find our $\alpha$ level, that is the subtraction of 1 by the confidence interval divided by 2. So:

$\alpha = \frac{1 - 0.95}{2} = 0.025$

Now, we have to find z in the Ztable as such z has a pvalue of $1 - \alpha$.

That is z with a pvalue of $1 - 0.025 = 0.975$, so Z = 1.96.

Now, find the margin of error M as such

$M = z\frac{\sigma}{\sqrt{n}}$

In which $\sigma$ is the standard deviation of the population and n is the size of the sample.

$M = 1.96\frac{11.36}{\sqrt{40}} = 3.52$

The lower end of the interval is the sample mean subtracted by M. So it is 108.50 - 3.52 = $104.98

The upper end of the interval is the sample mean added to M. So it is 108.50 + 3.52 = $112.02

The 95% confidence interval is ($104.98,$112.02)

Which interval is​ wider?

The 95% confidence interval has a larger margin of error, so it is wider.

x% confidence interval:

A confidence interval is built from a sample, has bounds a and b, and has a confidence level of x%. It means that we are x% confident that the population mean is between a and b.

So the correct interpretation is:

C. 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.