The waiting time between eruptions and the duration of the eruption for the Old

Question

The waiting time between eruptions and the duration of the eruption for the Old Faithful geyser in Yellowstone National Park, Wyoming, USA has studied extensively. Suppose we have a dataset consisting of 272 observations of waiting times and eruption duration times. Suppose x = waiting time and y = eruption duration time. A scatterplot of the data suggests that attempting a linear model fit is appropriate. Other summary statistics include: x bar = 70.897, s_x = 13.594, y bar = 3.488 s_y = 1.141 Formula: eruption -1 + waiting Coefficients. Residual standard error: 0.4965 on 270 degrees of freedom Multiple R-squared: 0.8115, Adjusted R-squared: 0.8108 Determine a 95% prediction interval for a new eruption duration time when a waiting times is 70 minutes. Note: 4.025,270 = 1.9688 (a) (-2.189, -1.559) (b) (0.0713, 0.080) (c) (2.44, 4.40) (d) (-1.950, -1.345) (e) (0.0264, 1.012)

Explanation / Answer

First i will complete the regression coefficient table to get the predicted value.

eruption duration time (y) = -1.87367 + 0.075628 * wating time (x)

Here x = 70 minutes

so y^ = -1.87367 + 0.075628 * 70 = 3.42 minutes

95 % prediction interval = y^ +- t0.025,270 * sy sqrt [1/n + (x* - x)2/(n-1)Sx2]

residual standard error = 0.4965

n = 272

95 % prediction interval = 3.42 +- 1.9688 * 0.4965 * sqrt [1+ 1/272 + (70 - 70.897)2 /271 * 13.5942]

Lower Value = 3.42 - 0.98 = 2.44

Upper Value = 3.42 + 0.98 = 4.40

95% prediction interval = (2.44, 4.40) Option C is correct.

Estimate Std. error t - value (Intercept) -1.87367 0.160143 -11.7 Waiting 0.075628 0.002218 34.09

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