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 9260 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 25,340 observations, the sample mean time interval was x2 = 70.6 minutes. Historical data suggest that σ1 = 8.14 minutes and σ2 = 12.97 minutes. Let μ1 be the population mean of x1 and let μ2 be the population mean of x2.

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 9260 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 25,340 observations, the sample mean time interval was x2 = 70.6 minutes. Historical data suggest that σ1 = 8.14 minutes and σ2 = 12.97 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 _________

Answer 1

To compute a 95% confidence interval for μ1 - μ2, we can use the formula:

Confidence Interval = (x1 - x2) +/- (Z * √((σ1^2 / n1) + (σ2^2 / n2)))

Where:
- x1 and x2 are the sample means
- σ1 and σ2 are the population standard deviations
- n1 and n2 are the sample sizes
- Z is the critical value corresponding to the desired confidence level (95% in this case)

Given:
- x1 = 62.6 minutes
- x2 = 70.6 minutes
- σ1 = 8.14 minutes
- σ2 = 12.97 minutes
- n1 = 9260 observations
- n2 = 25340 observations

First, let's calculate (σ1^2 / n1) and (σ2^2 / n2):

(σ1^2 / n1) = (8.14^2 / 9260) ≈ 0.00714
(σ2^2 / n2) = (12.97^2 / 25340) ≈ 0.006634

Next, we need to find the critical value Z for a 95% confidence level. We can look up this value in a standard normal distribution table or use a calculator. For a 95% confidence level, the Z value is approximately 1.96.

Now we can calculate the confidence interval:

Confidence Interval = (62.6 - 70.6) +/- (1.96 * √(0.00714 + 0.006634))
= -8 +/- (1.96 * √0.013774)
= -8 +/- (1.96 * 0.117341)
= -8 +/- 0.2294442

Finally, we can compute the lower and upper limits of the confidence interval:

lower limit = -8 - 0.2294442 ≈ -8.23
upper limit = -8 + 0.2294442 ≈ -7.77

Therefore, the 95% confidence interval for μ1 - μ2 is approximately (-8.23, -7.77).