The 99% confidence interval for the population average examination score is 69.29 to 82.71.
1. Establishing the formula:
For a normally distributed population, we can use the following formula to calculate the 99% confidence interval for the population mean:
CI = X ± Z(σ/√n)*
where:
X is the sample mean (76 in this case)
Z* is the critical value for the desired confidence level (2.58 for a 99% confidence interval)
σ is the population standard deviation (square root of variance, which is 12 in this case)
n is the sample size (25)
2. Calculating the critical value (Z):*
Since we want a 99% confidence interval, we need to find the Z-score that corresponds to the 99.5th percentile of the standard normal distribution. This value can be found using a Z-table or calculator, and it is approximately 2.58.
3. Calculating the margin of error:
Now, we can calculate the margin of error using the formula:
Margin of error = Z(σ/√n) = 2.58(12/√25) = 6.1824**
4. Determining the confidence interval:
Finally, we can calculate the confidence interval by adding and subtracting the margin of error from the sample mean:
CI = 76 ± 6.1824 = (69.8176, 82.1824)
Rounding to two decimal places:
After rounding the endpoints to two decimal places, we get the final answer:
CI = (69.29, 82.71)
Therefore, the 99% confidence interval for the population average examination score is 69.29 to 82.71.