Questions: 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. If convenient, use technology to construct the confidence intervals. A random sample of 40 home theater systems has a mean price of 135.00. Assume the population standard deviation is 17.80. The 90% confidence interval is (130.37, 139.63). (Round to two decimal places as needed.) Construct a 95% confidence interval for the population mean. The 95% confidence interval is (129.48, 140.52). (Round to two decimal places as needed.) Interpret the results. Choose the correct answer below. A. With 90% confidence, it can be said that the population mean price lies in the first interval. With 95% confidence, it can be said that the population mean price lies in the second interval. The 95% confidence interval is narrower than the 90%. B. With 90% confidence, it can be said that the population mean price lies in the first interval. With 95% confidence, it can be said that the population mean price lies in the second interval. The 95% confidence interval is wider than the 90%. C. With 90% confidence, it can be said that the sample mean price lies in the first interval. With 95% confidence, it can be said that the sample mean price lies in the second interval. The 95% confidence interval is wider than the 90%.

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. If convenient, use technology to construct the confidence intervals. A random sample of 40 home theater systems has a mean price of 135.00. Assume the population standard deviation is 17.80. The 90% confidence interval is (130.37, 139.63). (Round to two decimal places as needed.) Construct a 95% confidence interval for the population mean. The 95% confidence interval is (129.48, 140.52). (Round to two decimal places as needed.) Interpret the results. Choose the correct answer below. A. With 90% confidence, it can be said that the population mean price lies in the first interval. With 95% confidence, it can be said that the population mean price lies in the second interval. The 95% confidence interval is narrower than the 90%. B. With 90% confidence, it can be said that the population mean price lies in the first interval. With 95% confidence, it can be said that the population mean price lies in the second interval. The 95% confidence interval is wider than the 90%. C. With 90% confidence, it can be said that the sample mean price lies in the first interval. With 95% confidence, it can be said that the sample mean price lies in the second interval. The 95% confidence interval is wider than the 90%.
Transcript text: 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. If convenient, use technology to construct the confidence intervals. A random sample of 40 home theater systems has a mean price of $\$ 135.00$. Assume the population standard deviation is $\$ 17.80$. The $90 \%$ confidence interval is ( $130.37,139.63$ ). (Round to two decimal places as needed.) Construct a $95 \%$ confidence interval for the population mean. The 95\% confidence interval is (129.48, 140.52). (Round to two decimal places as needed.) Interpret the results. Choose the correct answer below. A. With $90 \%$ confidence, it can be said that the population mean price lies in the first interval. With $95 \%$ confidence, it can be said that the population mean price lies in the second interval. The $95 \%$ confidence interval is harrower than the $90 \%$. B. With $90 \%$ confidence, it can be said that the population mean price lies in the first interval. With $95 \%$ confidence, it can be said that the population mean price lies in the second interval. The $95 \%$ confidence interval is wider than the $90 \%$. C. With $90 \%$ confidence, it can be said that the sample mean price lies in the first interval. With $95 \%$ confidence, it can be said that the sample mean price lies in the second interval. The $95 \%$ confidence interval is wider than the $90 \%$.
failed

Solution

failed
failed

Solution Steps

Step 1: Given Information

We are provided with the following data:

  • Sample mean (\(\bar{x}\)): \(135.00\)
  • Population standard deviation (\(\sigma\)): \(17.80\)
  • Sample size (\(n\)): \(40\)
Step 2: Calculate the 95% Confidence Interval

To calculate the 95% confidence interval for the population mean, we use the formula:

\[ \bar{x} \pm z \frac{\sigma}{\sqrt{n}} \]

For a 95% confidence level, the critical value \(z\) is approximately \(1.96\). Thus, we can compute the confidence interval as follows:

\[ 135.00 \pm 1.96 \cdot \frac{17.80}{\sqrt{40}} \]

Calculating the margin of error:

\[ \text{Margin of Error} = 1.96 \cdot \frac{17.80}{\sqrt{40}} \approx 5.52 \]

Therefore, the confidence interval is:

\[ (135.00 - 5.52, 135.00 + 5.52) = (129.48, 140.52) \]

Step 3: Interpretation of Results

The interpretation of the confidence intervals is as follows:

  • With \(90\%\) confidence, it can be said that the population mean price lies in the first interval \((130.37, 139.63)\).
  • With \(95\%\) confidence, it can be said that the population mean price lies in the second interval \((129.48, 140.52)\).
  • The \(95\%\) confidence interval is wider than the \(90\%\) confidence interval.

Final Answer

The \(95\%\) confidence interval is \((129.48, 140.52)\) and the interpretation leads us to conclude that the answer is:

\(\boxed{B}\)

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Let sin A = -3/5 with A in QIII and find sin (2A) =
failedAnswered
Use the spinner shown to answer the question. Assume that it is equally probable that the pointer will land on any one of the colored regions. If the pointer lands on a borderline, spin again. If the spinner is spun once, find the probability that the pointer lands in a region that is brown of red. The probability that the pointer lands in a region that is brown or red is (Type an integer or a simplified fraction.)
failedAnswered
What is the primary concern of a board of directors today? (Page 76) To make decisions about how to run the day-to-day affairs of the business To monitor the decisions made by managers on behalf of the company To determine whether or not to pay dividends to stockholders To watch out for the interests of the employees
failedAnswered
In your new job, you will be responsible for upstream management of the firm's supply chain. This means you will be managing all of the following except: outbound logistics inbound logistics raw materials warehouse facilities storage facilities
failedAnswered
A charged particle ( q=7.5 × 10^-10 C) experiences a force of F=1.25 i-4.2 j N in an electric field. - Part (a) V Write an expression for the electric field vector E to which the charge is subject, in terms of the force F. E=F / q Part (b) Assume this field is generated by a point charge of Q=5 × 10^-9 C. How far away is this charge located? Give your answer in meters.