Question

a prominent medical group claims that the population mean of the surgery durations for all brain tumor patients is 4.71 hours. you are a data analyst for a health insurance company and want to test that claim. to do so, you select a random sample of 32 brain tumor surgery patients, and you record the surgery duration for each. assume it is known that the population standard deviation of the durations of all brain tumor surgeries is 1.68 hours. based on your sample, follow the steps below to construct a 99% confidence interval for the population mean of the surgery durations for all brain tumor patients. then state whether the confidence interval you construct contradicts the medical group’s claim. (if necessary, consult a list of formulas.) (a) click on “take sample” to see the results from your random sample of 32 brain tumor patients. number of patients 32 sample mean 3.21 sample standard deviation 1.39 population standard deviation 1.68 enter the values of the sample size, the point estimate for the population mean, the population standard deviation, and the critical value you need for your 99% confidence interval. (choose the correct critical value from the table of critical values provided.) when you are done, select “compute”. sample size: point estimate: population standard deviation: critical value: standard error: margin of error: 99% confidence interval: ( z_{0.005} approx 2.576 ) ( z_{0.010} approx 2.326 ) ( z_{0.025} approx 1.960 ) ( z_{0.050} approx 1.645 ) ( z_{0.100} approx 1.282 ) (b) based on your sample, graph the 99% confidence interval for the population mean of the surgery durations for all brain tumor patients. - enter the lower and upper limits on the graph to show your confidence interval. - for the point (( \bullet )), enter the medical group’s claim of 4.71 hours. 99% confidence interval: (c) does the 99% confidence interval you constructed contradict the medical group’s claim? choose the best answer from the choices below. - no, the confidence interval does not contradict the claim. the medical group’s claim of 4.71 hours is inside the 99% confidence interval. - no, the confidence interval does not contradict the claim. the medical group’s claim of 4.71 hours is outside the 99% confidence interval. - yes, the confidence interval contradicts the claim. the medical group’s claim of 4.71 hours is inside the 99% confidence interval. - yes, the confidence interval contradicts the claim. the medical group’s claim of 4.71 hours is outside the 99% confidence interval.

Explanation:

Part (a)

Step1: Identify Sample Size

The sample size \( n \) is the number of patients, which is 32.

Step2: Identify Point Estimate

The point estimate for the population mean is the sample mean, which is 3.21.

Step3: Identify Population Standard Deviation

The population standard deviation \( \sigma \) is given as 1.68.

Step4: Identify Critical Value

For a 99% confidence interval, the significance level \( \alpha = 1 - 0.99 = 0.01 \), so \( \alpha/2 = 0.005 \). The critical value \( z_{\alpha/2} \) is \( z_{0.005} \approx 2.576 \).

Step5: Calculate Standard Error

The formula for standard error (SE) is \( SE = \frac{\sigma}{\sqrt{n}} \). Substituting \( \sigma = 1.68 \) and \( n = 32 \):
\( SE = \frac{1.68}{\sqrt{32}} \approx \frac{1.68}{5.65685} \approx 0.297 \)

Step6: Calculate Margin of Error

The margin of error (ME) is \( ME = z_{\alpha/2} \times SE \). Substituting \( z_{\alpha/2} = 2.576 \) and \( SE \approx 0.297 \):
\( ME = 2.576 \times 0.297 \approx 0.765 \)

Step7: Calculate Confidence Interval

The confidence interval is \( \text{Point Estimate} \pm ME \), so:
Lower limit: \( 3.21 - 0.765 = 2.445 \)
Upper limit: \( 3.21 + 0.765 = 3.975 \)

Part (b)

The 99% confidence interval is from approximately 2.445 to 3.975. The medical group's claim is 4.71, which is plotted as a point.

Part (c)

To determine if the confidence interval contradicts the claim, we check if 4.71 is inside or outside the interval (2.445, 3.975). Since 4.71 > 3.975, it is outside the interval. So the confidence interval contradicts the claim because the claim's value is outside the interval.

Answer:

Yes, the confidence interval contradicts the claim. The medical group’s claim of 4.71 hours is outside the 99% confidence interval.