A random sample of six CEOs reported how many times per year they play tennis. The data follows.
3 41 96 32 52 19
Assuming the number of games played is normally distributed and the population standard deviation σ = 10, a 95% confidence interval for μ is:
95% = mean ± 1.96 SEm
SEm = SD/√n
28.5
10
28.5, 44.5
28.5,44.5
96
3 96
cap
To calculate the 95% confidence interval for the population mean (μ) in this case, we can use the formula:
Confidence Interval = Sample Mean ± (Critical Value) × (Standard Error)
1. Calculate the sample mean (x̄):
- Add up all the values in the sample: 3 + 41 + 96 + 32 + 52 + 19 = 243
- Divide the sum by the number of observations: 243 / 6 = 40.5
The sample mean is 40.5 (x̄ = 40.5).
2. Determine the critical value for a 95% confidence interval:
Since we have a small sample size (n = 6), we need to use a t-distribution. With a 95% confidence level and 5 degrees of freedom (n - 1), the critical value is approximately 2.571.
3. Calculate the standard error (SE):
The standard error measures the variability of the sample mean.
- Divide the population standard deviation (σ = 10) by the square root of the sample size (n = 6).
- SE = σ / √n = 10 / √6 ≈ 4.08
4. Plug in the values into the formula and calculate the confidence interval:
Confidence Interval = 40.5 ± (2.571) × (4.08)
To find the lower bound:
Lower Bound = 40.5 - (2.571) × (4.08) ≈ 29.06
To find the upper bound:
Upper Bound = 40.5 + (2.571) × (4.08) ≈ 51.94
Therefore, the 95% confidence interval for the true population mean (μ) is approximately 29.06 to 51.94.