A study of 35 professors showed that the average time they spent creating test questions was 13.5 minutes per question. The standard deviation of the population is 3.8. Which of the following is the 99% confidence interval for the average number of minutes it takes to create a test question?
12.0<u<15.0
To calculate the 99% confidence interval for the average number of minutes it takes to create a test question, we can use the following formula:
Confidence interval = mean ± (Z * (standard deviation / sqrt(n)))
Where:
- mean is the sample mean
- Z is the Z-score corresponding to the desired confidence level (99% in this case)
- standard deviation is the population standard deviation
- n is the sample size
Given:
Sample mean (mean) = 13.5 minutes/question
Population standard deviation (σ) = 3.8
Sample size (n) = 35
First, we need to find the Z-score corresponding to a 99% confidence level. This can be found using a Z-table or a statistical calculator. The Z-score for a 99% confidence level is approximately 2.576.
Now we can calculate the confidence interval:
Confidence interval = 13.5 ± (2.576 * (3.8 / sqrt(35)))
Calculating the expression within the parentheses:
= 2.576 * (3.8 / sqrt(35))
= 2.576 * 0.643 (rounded to three decimal places)
= 1.654 (rounded to three decimal places)
Now we can calculate the confidence interval:
Lower bound = 13.5 - 1.654
Upper bound = 13.5 + 1.654
Lower bound = 11.846
Upper bound = 15.154
Therefore, the 99% confidence interval for the average number of minutes it takes to create a test question is 11.846 minutes to 15.154 minutes.
To calculate the confidence interval for the average number of minutes it takes to create a test question, we can use the formula:
Confidence Interval = X̄ ± Z * (σ / √n)
Where:
X̄ = sample mean
Z = Z-score for the desired confidence level
σ = population standard deviation
n = sample size
In this case, we are given:
X̄ = 13.5 minutes
σ = 3.8 minutes
n = 35 professors
The Z-score for a 99% confidence level can be found using a Z-table or statistical software. For a 99% confidence level, the Z-score is approximately 2.576.
Now we can calculate the confidence interval:
Confidence Interval = 13.5 ± 2.576 * (3.8 / √35)
First, divide the population standard deviation by the square root of the sample size (√35 ≈ 5.92):
Confidence Interval = 13.5 ± 2.576 * (3.8 / 5.92)
Next, calculate the value inside the parentheses:
Confidence Interval = 13.5 ± 2.576 * 0.6419
Multiply the Z-score by the value inside the parentheses:
Confidence Interval = 13.5 ± 1.6549
Finally, calculate the lower and upper bounds of the confidence interval:
Lower bound = 13.5 - 1.6549 ≈ 11.8451
Upper bound = 13.5 + 1.6549 ≈ 15.1549
Therefore, the 99% confidence interval for the average number of minutes it takes to create a test question is approximately 11.8451 to 15.1549 minutes.
99% = mean ± 2.575 SEm
SEm = SD/√n
I'll let you do the calculations.