To compute a 95% confidence interval for the population mean length of all term papers, we can use the t-distribution since the sample size is small (<30) and the population standard deviation is unknown.
Step 1: Calculate the sample mean:
- Add up all the values: 14 + 20 + 25 + 10 + 16 + 8 + 15 + 12 + 18 + 9 = 147
- Divide by the sample size: 147 / 10 = 14.7
Step 2: Calculate the sample standard deviation:
- Subtract the sample mean from each value and square the result:
(14 - 14.7)^2, (20 - 14.7)^2, (25 - 14.7)^2, (10 - 14.7)^2, (16 - 14.7)^2, (8 - 14.7)^2, (15 - 14.7)^2, (12 - 14.7)^2, (18 - 14.7)^2, (9 - 14.7)^2
- Add up all these squared differences: 36.1 + 21.61 + 113.29 + 21.61 + 1.69 + 40.89 + 0.09 + 7.29 + 10.89 + 27.09 = 280.47
- Divide by (n-1) (where n is the sample size): 280.47 / 9 = 31.16
- Take the square root of the result: sqrt(31.16) ≈ 5.58
Step 3: Calculate the standard error:
- Divide the sample standard deviation by the square root of the sample size: 5.58 / sqrt(10) ≈ 1.76
Step 4: Find the critical value from the t-distribution:
- The degrees of freedom will be (n-1), where n is the sample size: (10 - 1) = 9
- Calculate the critical value using a t-table or software. For 95% confidence level and 9 degrees of freedom, the critical value is approximately 2.262.
Step 5: Calculate the margin of error:
- Multiply the standard error by the critical value: 1.76 * 2.262 ≈ 3.98
Step 6: Calculate the confidence interval:
- Subtract the margin of error from the sample mean: 14.7 - 3.98 ≈ 10.72
- Add the margin of error to the sample mean: 14.7 + 3.98 ≈ 18.68
The 95% confidence interval for the population mean length of all term papers is approximately 10.72 to 18.68 pages.
Interpretation: Based on the sample of 10 term papers, we are 95% confident that the population mean length of all term papers for Mr. Crandall's class falls between 10.72 and 18.68 pages. This means if we were to repeat this sampling process many times, 95% of the resulting confidence intervals would contain the true population mean length.
Assumptions: In order to use the t-distribution and perform a confidence interval analysis, we assume that the sample is random, the sample size is small enough, the population standard deviation is unknown, and the variable being measured (length of term papers) is approximately normally distributed in the population.