Question

i am a data analyst for a health - insurance company and want to estimate the population mean of the surgery durations for all brain tumor patients. i select a random sample of 32 brain tumor surgery patients, and i 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. (if necessary, consult a list of formulas.) click on \take sample\ to see the results from your random sample of 32 brain tumor patients. enter the values of the sample size, the point estimate for the population mean, the population standard deviation, and the critical value for your 99% confidence interval. (choose the correct critical value from the table of critical values provided.) when you are done, select sample size: standard error: point estimate: population standard deviation: margin of error: critical value: 99% confidence interval: confidence level critical value 99% $z_{0.005}=2.576$ 95% $z_{0.025}=1.960$

Explanation:

Step1: Identify sample - size

The number of patients in the sample is the sample - size. Given $n = 32$.

Step2: Determine point - estimate

The sample mean is the point - estimate for the population mean. Given $\bar{x}=3.29$.

Step3: Find population standard deviation

Given $\sigma = 1.68$.

Step4: Identify critical value

For a 99% confidence interval, the critical value from the table is $z_{\alpha/2}=z_{0.005}=2.576$.

Step5: Calculate standard error

The formula for the standard error of the mean is $SE=\frac{\sigma}{\sqrt{n}}$. Substituting $\sigma = 1.68$ and $n = 32$, we have $SE=\frac{1.68}{\sqrt{32}}\approx\frac{1.68}{5.657}\approx0.297$.

Step6: Calculate margin of error

The margin of error $E = z_{\alpha/2}\times SE$. Substituting $z_{\alpha/2}=2.576$ and $SE\approx0.297$, we get $E=2.576\times0.297\approx0.765$.

Step7: Construct confidence interval

The 99% confidence interval for the population mean $\mu$ is $\bar{x}-E<\mu<\bar{x} + E$. Substituting $\bar{x}=3.29$ and $E = 0.765$, we have $3.29-0.765<\mu<3.29 + 0.765$, or $2.525<\mu<4.055$.

Answer:

Sample size: 32
Point estimate: 3.29
Population standard deviation: 1.68
Critical value: 2.576