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 the90% 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% confidenceinterval, 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.

C. You can be 90% confident that the population mean price of the stock is between the bounds of the 90% confidenceinterval, and 95% confident for the 95% interval.

Step-by-step explanation:

90% confidence interval:

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

[tex]\alpha = \frac{1 - 0.9}{2} = 0.05[/tex]

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

That is z with a pvalue of [tex]1 - 0.05 = 0.95[/tex], so Z = 1.645.

Now, find the margin of error M as such

[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]

In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.

[tex]M = 1.645\frac{11.36}{\sqrt{40}} = 2.95[/tex]

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 [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:

[tex]\alpha = \frac{1 - 0.95}{2} = 0.025[/tex]

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

That is z with a pvalue of [tex]1 - 0.025 = 0.975[/tex], so Z = 1.96.

Now, find the margin of error M as such

[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]

In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.

[tex]M = 1.96\frac{11.36}{\sqrt{40}} = 3.52[/tex]

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% confidenceinterval, and 95% confident for the 95% interval.