Final answer:
The 95% confidence interval for the mean number of rounds of golf played per year by physicians is estimated to be (15.81, 23.69), calculated using the given sample data and the formula for a confidence interval with a known population standard deviation.
Step-by-step explanation:
To estimate the mean number of rounds of golf played per year by physicians with 95% confidence, we can use a confidence interval for the population mean when the population standard deviation is known. The sample mean (μ), denoted as μ, is computed from the given data:
- Sample size (n) = 12
- Sample standard deviation (σ) = 7
- Sample mean = (Σx)/n
The list of rounds played by physicians is: 6, 40, 16, 3, 32, 42, 19, 13, 19, 14, 14, 19. First, calculate the sum of these rounds and then find the mean:
Σx = 6 + 40 + 16 + 3 + 32 + 42 + 19 + 13 + 19 + 14 + 14 + 19 = 237
Calculate the sample mean:
μ = 237 / 12 = 19.75
To compute the 95% confidence interval, we use the formula:
Confidence Interval = μ ± (Z * (σ/√n))
Where Z is the Z-score corresponding to the desired confidence level. For a 95% confidence level, the Z-score is approximately 1.96 (from Z-tables). Substituting the values:
Confidence Interval = 19.75 ± (1.96 * (7/√12))
Calculate the margin of error:
Margin of Error = 1.96 * (7/√12) = approximately 3.94
Finally, compute the lower and upper confidence limits (LCL and UCL) and express the confidence interval:
LCL = 19.75 - 3.94 ≈ 15.81
UCL = 19.75 + 3.94 ≈ 23.69
The 95% confidence interval is estimated to be (15.81, 23.69).