Question

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 8800 observations, the sample mean interval was x₁ = 61.2 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 24,404 observations, the sample mean time interval was x₂ = 71.2 minutes. Historical data suggest that σ₁ = 8.49minutes and σ₂ = 12.62 minutes. Let μ₁ be the population mean of x₁ and let μ₂ be the population mean of x₂.

(a) Compute a 95% confidence interval for μ₁ – μ₂. (Use 2 decimal places.)

lower limit
upper limit

(b) 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 95% 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.

Because the interval contains only positive numbers, we can say that the interval length between eruptions has gotten shorter.Because the interval contains both positive and negative numbers, we can not say that the interval length between eruptions has gotten longer.     We can not make any conclusions using this confidence interval.Because the interval contains only negative numbers, we can say that the interval length between eruptions has gotten longer.

Answer 1

Solution :

For sample 1 :x̅1 = 61.2, σ1 = 8.49, n1 = 8800

For sample 2 :x̅2 = 71.2, σ2 = 12.62, n2 = 24404

(a) 95% Confidence interval :  

At α = 0.05, two tailed critical value, z_c = ABS(NORM.S.INV(0.05/2)) = 1.96

Lower Bound = (x̅1 - x̅2) - z_c*√(σ1²/n1 +σ2²/n2)

= (61.2 - 71.2) - 1.96*√(8.49²/8800 + 12.62²/24404) =-10.24

Upper Bound = (x̅1 - x̅2) + z_c*√(σ1²/n1 +σ2²/n2)

= (61.2 - 71.2) + 1.96*√(8.49²/8800 + 12.62²/24404) =-9.76

(b) Because the interval contains only negative numbers, we can say that the interval length between eruptions has gotten longer.

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