A sample of 6 adult elephants had an average weight of 12,200 pounds, with a sample standard deviation of 200 pounds. We know the sample is from a normal distribution. The 95% confidence interval of the true mean is:
To calculate the confidence interval, we use the formula:
Confidence interval = Sample mean ± (Critical value * Standard error)
First, let's calculate the standard error:
Standard error = Sample standard deviation / √(Sample size)
Standard error = 200 pounds / √(6)
Standard error ≈ 200 pounds / 2.4495
Standard error ≈ 81.63 pounds
Next, we need to determine the critical value. For a 95% confidence level with a sample size of 6, the degree of freedom is 5 (n-1) and the critical value is 2.571 (looked up from the t-distribution table).
Confidence interval = 12,200 pounds ± (2.571 * 81.63 pounds)
Lower confidence limit = 12,200 pounds - (2.571 * 81.63 pounds)
Lower confidence limit ≈ 12,200 pounds - 209.98 pounds
Lower confidence limit ≈ 11,990.02 pounds
Upper confidence limit = 12,200 pounds + (2.571 * 81.63 pounds)
Upper confidence limit ≈ 12,200 pounds + 209.98 pounds
Upper confidence limit ≈ 12,409.98 pounds
The 95% confidence interval of the true mean weight is approximately 11,990.02 pounds to 12,409.98 pounds.