3. Eruptions from the Old Faithful geyser in Yellowstone National Park do not oc

Question

3. Eruptions from the Old Faithful geyser in Yellowstone National Park do not occur at uniform intervals of time, nor do they all have the same duration. Can we improve our chances of predicting the time till the next eruption if we know how long the previous one lasted? Scientists collected data on the duration (in minutes) of 222 eruptions of Old Faithful, and for each one, they noted the time interval (in minutes) till the next eruption. They then used statistical software to create a linear regression model. Below is some of the output. Coefficients Term Coef SE Coef T-Value P-Value Constant 33.9668 1.428 23.8 < 0.0001 Duration 10.3582 0.3822 27.1 < 0.0001 (a) What is the predictor variable in this situation, and what is the response variable? (b) Write down the equation for the linear regression model derived from these data. What is the slope of the regression line? (c) According to the linear regression model from these data, do the predictor and response variables have a positive, negative or zero association? Please explain. (d) According to the linear regression model, what is the time interval until the next eruption, after an eruption that lasts two minutes? after four minutes? (e) The scientists claim with 95% confidence that the mean length of time that will elapse after a 4-minute eruption, until the next one, is between 74.01 and 76.89 minutes. However, they say that if you arrive at Old Faithful just as a 4-minute eruption is ending, then they can predict with 95% confidence only that you will have to wait somewhere from 63.18 to 87.62 minutes until the next one. Why is their 95% confidence interval for the predicted time that you will have to wait so much wider than their 95% confidence interval for the mean length of time between a 4-minute eruption and the next one? Please explain.

Explanation / Answer

(a)

Response variable: time interval

Independent variable: duration

(b)

The equation is

time-interval = 33.9668+ 10.3582Duration

(c)

Since slope is posiitve so correlation between the variables is positive.

(d)

For duration = 2 predicted time interval is

time-interval = 33.9668+ 10.3582*2 = 54.6832

For duration = 4 predicted time interval is

time-interval = 33.9668+ 10.3582*4 = 75.3996

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