Question

based on your sample, follow the steps below to construct a 95% 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 groups 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: 4.65, sample standard deviation: 1.44, 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 95% 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:
95% confidence interval:

z_{0.005}=2.576
z_{0.010}=2.326
z_{0.025}=1.960
z_{0.050}=1.645
z_{0.100}=1.282

(b) based on your sample, graph the 95% 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 (•), enter the medical groups claim of 2.77 hours.

95% confidence interval:
0.00 to 10.00 on the axis (0.00, 2.00, 4.00, 5.00, 6.00, 8.00, 10.00)

(c) does the 95% confidence interval you constructed contradict the medical groups claim? choose the best answer from the choices below.

• no, the confidence interval does not contradict the claim. the medical groups claim of 2.77 hours is inside the 95% confidence interval.
• no, the confidence interval does not contradict the claim. the medical groups claim of 2.77 hours is outside the 95% confidence interval.
• yes, the confidence interval contradicts the claim. the medical groups claim of 2.77 hours is inside the 95% confidence interval.
• yes, the confidence interval contradicts the claim. the medical groups claim of 2.77 hours is outside the 95% confidence interval.

Explanation:

Step1: Identify key values

Sample size \( n = 32 \), point estimate \( \bar{x} = 4.65 \), population standard deviation \( \sigma = 1.68 \), critical value \( z_{0.025} = 1.960 \) (for 95% confidence).

Step2: Calculate standard error

Standard error \( SE = \frac{\sigma}{\sqrt{n}} = \frac{1.68}{\sqrt{32}} \approx \frac{1.68}{5.6568} \approx 0.297 \)

Step3: Calculate margin of error

Margin of error \( ME = z_{0.025} \times SE = 1.960 \times 0.297 \approx 0.582 \)

Step4: Compute confidence interval

Lower limit: \( 4.65 - 0.582 = 4.068 \); Upper limit: \( 4.65 + 0.582 = 5.232 \)

Step5: Check claim contradiction

Medical group’s claim is 2.77, which is outside the interval (4.068, 5.232).

Answer:

(a) Sample size: 32; Point estimate: 4.65; Population standard deviation: 1.68; Critical value: 1.960
(b) Lower limit ≈ 4.07, Upper limit ≈ 5.23
(c) Yes, the confidence interval contradicts the claim. The medical group’s claim of 2.77 hours is outside the 95% confidence interval.