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?
To calculate the 99% confidence interval for the mean yearly milk consumption, you will need to use the formula:
Confidence Interval = Mean ± (Z * (Standard Deviation / √Sample Size))
Here, Z represents the Z-score, which determines the level of confidence. For a 99% confidence level, the Z-score is approximately 2.576 (using a standard normal distribution table or a statistical software).
Given information:
Mean (x̄) = 72 gallons
Standard Deviation (σ) = 15 gallons
Sample Size (n) = 20
Now, let's substitute these values into the formula:
Confidence Interval = 72 ± (2.576 * (15 / √20))
To calculate the square root of 20 (√20), it equals approximately 4.472.
Confidence Interval = 72 ± (2.576 * (15 / 4.472))
Now, we can simplify the expression:
Confidence Interval = 72 ± (2.576 * 3.355)
Confidence Interval = 72 ± 8.633
To find the lower bound of the confidence interval, subtract the value obtained from 72:
Lower Bound = 72 - 8.633
Lower Bound ≈ 63.367
To find the upper bound of the confidence interval, add the value obtained to 72:
Upper Bound = 72 + 8.633
Upper Bound ≈ 80.633
Therefore, the 99% confidence interval for the mean yearly milk consumption is approximately 63.367 to 80.633 gallons.