The US Geological Survey compiled historical data about Old

The U.S. Geological Survey compiled historical data about Old Faithful Geyser (Yellowstone National Park) from 1870 to 1987. Let x₁ be a random variable that represents the time interval (in minutes) between Old Faithful eruptions for the years 1948 to 1952. Based on 9100 observations, the sample mean interval was x₁ = 62.6 minutes. Let x₂ be a random variable that represents the time interval in minutes between Old Faithful eruptions for the years 1983 to 1987. Based on 23,936 observations, the sample mean time interval was x₂ = 72.8 minutes. Historical data suggest that ?₁ = 8.70 minutes and ?₂ = 11.85 minutes. Let ?₁ be the population mean of x₁ and let ?₂ be the population mean of x₂.

(a) Compute a 99% confidence interval for ?₁

Solution

Given a=1-0.99=0.01, Z(0.005) = 2.58 (from standard normal table)

So Lower Limit:

(xbar1-xbar2) - Z*sqrt(s1^2/n1+s2^2/n2)

=(62.6-72.8)-2.58*sqrt(8.7^2/9100+11.85^2/23936)

=-10.51

So Upper Limit:

(xbar1-xbar2) + Z*sqrt(s1^2/n1+s2^2/n2)

=(62.6-72.8) + 2.58*sqrt(8.7^2/9100+11.85^2/23936)

=-9.89