Question

The Appalachian Bear Center (ABC) is a not-for-profit organization located near the Great Smoky Mountains National Park. ABC’s programs include the rehabilitation of orphaned and injured black bears, as well as research and education about Appalachian black bears. ABC provides the most natural environment possible for rehabilitating black bears before their release back into the wild. Katie Settlage performed a study to learn more about the Appalachian black bear population in the Great Smoky Mountains National Park. She and a team of researchers used a sample of 68 black bears in the park and took measurements such as paw size, weight, and shoulder height.

Answer the following questions based on this data. As always, you must show all work and formulas used in order to receive full credit. Round all decimals to three places unless otherwise noted.

1. In the sample of 68 bears, 40 were males. Construct an 80% confidence interval for the population proportion of bears that are males and write a statement interpreting the interval.

Answer 1

sample proportion, pcap = 40/68 = 0.5882
sample size, n = 68
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.5882 * (1 - 0.5882)/68) = 0.0597

Given CI level is 80%, hence α = 1 - 0.8 = 0.2
α/2 = 0.2/2 = 0.1, Zc = Z(α/2) = 1.28

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.5882 - 1.28 * 0.0597 , 0.5882 + 1.28 * 0.0597)
CI = (0.5118 , 0.6646)

Therefore, based on the data provided, the 80% confidence interval for the population proportion is 0.5118 < p < 0.6646 , which indicates that we are 80% confident that the true population proportion p is contained by the interval (0.5118 , 0.6646)