Final answer:
To construct a 95% confidence interval for the mean weight of newborn elephant calves with a sample mean of 244 lb, a known standard deviation of 15 lb, and a sample of 50 elephants, we calculate a margin of error of approximately 4.157 lb using the z-score of 1.96. This results in a confidence interval of 244 lb ± 4.157 lb.
Step-by-step explanation:
The question pertains to the construction of a 95% confidence interval for the mean weight of newborn elephant calves using the normal distribution. With a sample mean of 244 lb, a known population standard deviation of 15 lb, and a sample size of 50, we use the z-distribution for the calculation since the population standard deviation is known. The formula for a confidence interval is:
Sample Mean ± (z-score * Standard Error)
The standard error (SE) is calculated using the population standard deviation (σ) divided by the square root of the sample size (n):
SE = σ / √n
SE = 15 lb / √50 ≈ 2.121 lb
For a 95% confidence interval, the z-score is 1.96. Hence, the margin of error (ME) is:
ME = z * SE
ME = 1.96 * 2.121 lb ≈ 4.157 lb
The 95% confidence interval is therefore:
244 lb ± 4.157 lb
which means we are 95% confident that the population mean weight of the newborn elephant calves is between 239.843 lb (244 - 4.157) and 248.157 lb (244 + 4.157).
If the sample size were increased to 500, the standard error would decrease, resulting in a narrower confidence interval. This occurs because a larger sample size provides a more precise estimate of the population mean.