Final answer:
To find the sample size needed to estimate the mean with a specified margin of error, we use the formula E = z* σ/√n)with the known standard deviation and the z-score for the desired confidence level. The z-score for a 95% confidence interval is approximately 1.96, and the given margin of error is 7.5 pounds, but we cannot compute the actual confidence interval without the sample mean.
a. 100
b. 7.5
c. 1.96
d. The 95% confidence interval for the average weight is x - E, x + E, where x is the sample mean, and E is the margin of error.
Step-by-step explanation:
The student's question relates to finding the sample size needed for an interval estimate, the margin of error, the critical value, and constructing a 95% confidence interval for the average weight of adult gorillas. Given the standard deviation of 15.4 pounds and the desired margin of error of 7.5 pounds, we can use the formula for the margin of error of a confidence interval to find the necessary sample size.
The standard formula for the margin of error when estimating a population mean with known standard deviation is E = z σ/√n , where E is the margin of error, z is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.
To identify the critical value z-score for a 95% confidence interval, we refer to the standard normal distribution table or use statistical software to find the z-score that corresponds to the area to the left of the critical value being 0.975 since 95% confidence is the middle area, we need the tail areas to add up to 5%.
This value is typically around 1.96. The margin of error E is already given as 7.5 pounds. Using the margin of error formula, we can solve for n to calculate the sample size needed for the interval estimate.
We can then apply these findings to construct the 95% confidence interval for the average weight by using the formula mean ± margin of error. However, we need the sample mean to do this, which is not provided in the question; thus we cannot compute the confidence interval without it.
a. 100
b. 7.5
c. 1.96
d. The 95% confidence interval for the average weight is (x- E, x + E), where x is the sample mean, and E is the margin of error