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 9540 observations, the sample mean interval was x1 = 61.8 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 23,585 observations, the sample mean time interval was x2 = 72.2 minutes. historical data suggest that 1 = 8.14 minutes and 2 = 12.41 minutes. let 1 be the population mean of x1 and let 2 be the population mean of x2. (a) compute a 95% confidence interval for 1 – 2. (use 2 decimal places.) lower limit upper limit incorrect: your answer is incorrect.

Answer 1

The 95% confidence interval for the difference between the population means of the time intervals between Old Faithful eruptions for the specified years is approximately −10.63 to −10.18 minutes.

To compute the 95% confidence interval for the difference between the population means (μ_1 −μ_2 ) of the time intervals between Old Faithful eruptions for the given years, you can use the formula for the confidence interval for the difference between two means:

Confidence interval=( x1 − x2 )±Z× √s1/ n1+ s2/n2

are the sample means for x1 and x2 respectively.

s_1 and s_2 are the sample standard deviations for x1 and x2 respectively.n_1 and n_2 are the sample sizes for x1 and x2 respectively.

Z is the critical value from the standard normal distribution corresponding to the desired confidence level. For a 95% confidence interval, Z is approximately 1.96.

Given:

Sample mean for x1 (n_1) = 61.8 minutes

Sample mean for x2 (n_2 ) = 72.2 minutes

Population standard deviation for x1 (σ_1 ) = 8.14 minutes

Population standard deviation for x2 (σ^2 ) = 12.41 minutes

Sample size for x1 (n_1 ) = 9540

Sample size for x2 (n_2 ) = 23585

Let's calculate the confidence interval:

Standard Error= σ^2/ n1 + σ^2/ n^2

​Standard Error= 8.14^2/ 9540 + 12.41^2/ 23585

Standard Error≈ √0.006945+0.006286 ≈ √0.013231 ≈0.115 (approximately)

Now, using the formula for the confidence interval:

Confidence interval=(61.8−72.2)±1.96×0.115

Confidence interval=−10.4±1.96×0.115

Calculating the interval:

Lower Limit:

−10.4−1.96×0.115≈−10.4−0.225=−10.625≈−10.63

Upper Limit:

−10.4+1.96×0.115≈−10.4+0.225=−10.175≈−10.18

Therefore, the 95% confidence interval for the difference between the population means of the time intervals between Old Faithful eruptions for the specified years is approximately −10.63 to −10.18 minutes.