For 30 randomly selected Rolling Stones concerts, the mean gross earnings is 2.37 million dollars. Assuming a population standard deviation gross earnings of 0.52 million dollars, obtain a 99% confidence interval for the mean gross earnings of all Rolling Stones concerts (in millions).
To obtain the confidence interval for the mean gross earnings of all Rolling Stones concerts, we can use the formula for calculating a confidence interval for a population mean.
The formula is:
Confidence interval = Sample mean ± (Critical value) * (Standard deviation / √n)
Given information:
Sample mean (x̄) = 2.37 million dollars
Population standard deviation (σ) = 0.52 million dollars
Sample size (n) = 30
Confidence level = 99%
To find the critical value, we need to refer to the Z-table. Since the sample size is greater than 30, we can use the standard normal distribution values.
For a 99% confidence level, we need to find the z-value that corresponds to the area (probability) of 0.995 in the standard normal distribution.
Using the Z-table or a statistical software, we find that the critical value is approximately 2.576.
Now we can calculate the confidence interval using the formula:
Confidence interval = 2.37 ± (2.576) * (0.52 / √30)
Let's compute the values:
Step 1: Calculate the standard error = standard deviation / √n
Standard error = 0.52 / √30 ≈ 0.095
Step 2: Calculate the margin of error = (Critical value) * (Standard error)
Margin of error = 2.576 * 0.095 ≈ 0.245
Step 3: Calculate the lower and upper bounds of the confidence interval
Lower bound = Sample mean - Margin of error = 2.37 - 0.245 ≈ 2.125
Upper bound = Sample mean + Margin of error = 2.37 + 0.245 ≈ 2.615
Therefore, the 99% confidence interval for the mean gross earnings of all Rolling Stones concerts is approximately 2.125 million dollars to 2.615 million dollars.