Question

The weights at birth of five randomly chosen baby Orca whales were 425, 454, 380, 405, and 395 pounds. Assume the distribution of weights is normally distributed. Find a 98 % confidence interval for the mean weight of all baby Orca whales. Use technology for your calculations. Give the confidence interval in the form "estimate + margin of error." Round to the nearest tenth of a pound.

Answer 1

To find the confidence interval for the mean weight of all baby Orca whales, we can use the following formula:

Confidence interval = estimate ± (critical value) × (standard error)

where:

estimate = sample mean

critical value = the value from the t-distribution that corresponds to our desired confidence level and degrees of freedom

standard error = standard deviation of the sample divided by the square root of the sample size

First, we need to calculate the sample mean and standard deviation:

Sample mean:

x = (425 + 454 + 380 + 405 + 395) / 5 = 411.8 pounds

Sample standard deviation:

s = sqrt(((425 - 411.8)^2 + (454 - 411.8)^2 + (380 - 411.8)^2 + (405 - 411.8)^2 + (395 - 411.8)^2) / 4) = 28.6 pounds

Next, we need to find the critical value from the t-distribution with degrees of freedom (n - 1) and our desired confidence level of 98%. With a sample size of 5, our degrees of freedom is 4. Using a t-table or a calculator, we find the critical value to be 3.747.

Finally, we can calculate the confidence interval:

Confidence interval = 411.8 ± 3.747 × (28.6 / sqrt(5)) = 411.8 ± 45.1

Rounded to the nearest tenth of a pound, the confidence interval is:

Confidence interval = 411.8 ± 45.1 = [366.7, 456.9]

Therefore, we can say with 98% confidence that the true mean weight of all baby Orca whales is between 366.7 and 456.9 pounds