The U.S. Geological Survey compiled historical data about Old Faithful Geyser (Yellowstone National Park) from 1870 to 1987. Some of these data are published in the book The Story of Old Faithful, by G. D. Marler (Yellowstone Association Press). Let x₁ be a random variable that represents the time interval (in minutes) between Old Faithful’s eruptions for the years 1948 to 1952. Based on 9340 observations, the sample mean interval was x̅₁ = 63.3 minutes. Let x₂ be a random variable that represents the time interval in minutes between Old Faithful’s eruptions for the years 1983 to 1987. Based on 25,111 observations, the sample mean time interval was x̅₂ = 72.1 minutes. Historical data suggest that s₁ = 9.17 minutes and s₂ = 12.67 minutes. Let m1 be the population mean of x₁ and let m² be the population mean of x₂.

(a) Check Requirements Which distribution, normal or Student’s t, do we use to approximate the x̅₁ – x̅₂ distribution? Explain.

(b) Compute a 99% con­fidence interval for m₁ – m₂.

(c) Interpretation Comment on the meaning of the con­fidence interval in the context of this problem. Does the interval consist of positive numbers only? negative numbers only? a mix of positive and negative numbers? Does it appear (at the 99% confi­dence level) that a change in the interval length between eruptions has occurred? Many geologic experts believe that the distribution of eruption times of Old Faithful changed after the major earthquake that occurred in 1959.