The U.S. Dairy Industry wants to estimate the mean yearly milk consumption. A sample of 20 people reveals the mean yearly consumption to be 72 gallons with a standard deviation of 15 gallons. What is the 99% confidence interval for the population mean?
To find the 99% confidence interval for the population mean, we can use the formula:
Confidence Interval = sample mean ± (Z * (standard deviation / √n))
Where:
- Sample mean is the mean yearly consumption found in the sample, which is 72 gallons.
- Z is the Z-score corresponding to the desired confidence level. For a 99% confidence level, Z is 2.576. You can find the Z-score from a standard normal distribution table or use a calculator.
- Standard deviation is the standard deviation of the sample, which is 15 gallons.
- √n is the square root of the sample size, which is the square root of 20.
Let's calculate the confidence interval step by step:
1. Calculate the square root of the sample size: √20 = 4.47
2. Calculate the margin of error: Z * (standard deviation / √n) = 2.576 * (15 / 4.47) = 8.66
3. Calculate the lower bound of the confidence interval: 72 - 8.66 = 63.34
4. Calculate the upper bound of the confidence interval: 72 + 8.66 = 80.66
Therefore, the 99% confidence interval for the population mean yearly milk consumption is [63.34, 80.66] gallons. This means we can be 99% confident that the true population mean falls within this range.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion of .005 to get Z value.
99% = mean ± 2.575 SD