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
99% = mean ± 2.575 SD
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.