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 9520 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 25,340 observations, the sample mean time interval was x₂ = 71.2 minutes. Historical data suggest that σ₁ = 8.35 minutes and σ₂ = 12.41 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

As per the provided information is:

Let x1 be a random variable that represents the time interval (in minutes) between Old Faithful eruptions for the years 1948 to 1952.

Let x2 be a random variable that represents the time interval in minutes between Old Faithful eruptions for the years 1983 to 1987.

Let μ1 be the population mean of x1.

let μ2 be the population mean of x2.

Sample 1:

Sample 2:

(a):

At the significance level 0.05 the two tailed critical value obtained from the z-table is +/- 1.96.

The 95% confidence interval for μ1 – μ2 can be calculated as,

Therefore, the 95% confidence interval for μ1 – μ2 is (-10.23, -9.77).

Lower limit = -10.23

Upper limit = -9.77

(b):

*Meaning: We are 95% confident that the true mean difference for time interval (in minutes) between Old Faithful eruptions for the years 1948 to 1952 and years 1983 to 1987 lies between the values -10.23 and -9.77.

*The interval contains only negative numbers.

*Since, the null value zero does not lies between the estimated range (-10.23 and -9.77), so it can be said that there is a significant change in the interval length between eruptions has occurred.

*We cannot make any conclusions using this confidence interval in respect to the believe of geologic experts that the distribution of eruption times of Old Faithful changed after the major earthquake that occurred in 1959.