What is the regression equation?

Question

The data show the time intervals after an eruption​ (to the next​ eruption) of a certain geyser. Find the regression​ equation, letting the first variable be the independent​ (x) variable. Find the best predicted time of the interval after an eruption given that the current eruption has a height of 77 feet. Use a significance level of 0.05.

Height (ft) Interval after (min)

128 84

129 90

82 62

95 69

99 84

91 66

122 86

106 79

Answer 1

We are using a simplified formula made for hand calculations of the B1 and B2 estimates with one pass

(no differences of observed and mean need be calculated). Sum of xx or Sxx is sum of x²

hgt(x) min(y) xy x2

128 84 10752 16384

129 90 11610 16641

82 62 5084 6724

95 69 6555 9025 B1= n*Sxy - Sx*Sy

99 84 8316 9801 n*Sxx - (Sx)2

91 66 6006 8281

122 86 10492 14884 B0 = (Sy/n –B1*Sx/n)

106 79 8374 11236

∑ 852 620 67189 92976 B1 = 0.5179

N = 8 B0 = 22.3465

Interval = 22.347+ 0.5179 * Height for height = 77 ft

est Time = 62.2248 min

Much of this can be done automatically with a regular graphing calculator or programable scientific calculator.

This manual calculation does NOT contain estimators for error on the parameters and it would be senseless to repeat the formulae for calculating them here.

USING R language we can calculate a regression and make a prediction with a confidence interval. But next time, state the program that you are trying to use in your question.

note: following code assumes the data are in a data frame labeled geyser with variables height and time.

The model assumption includes an intercept. an alternate model with intercept set to zero would be more appropriate.

load("geyser.rdat")

attach(geyser)

fit<-lm(time~height)

summary(fit)

newdat<-data.frame(height=77,time=0)

predict.lm(fit,newdat,se.fit = TRUE, interval="prediction", level = 0.95)

results::

Call> lm(formula = time ~ height)

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 22.3465 11.7599 1.900 0.10614

height 0.5179 0.1091 4.747 0.00317 **

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.161 on 6 degrees of freedom

Multiple R-squared: 0.7898, Adjusted R-squared: 0.7547

F-statistic: 22.54 on 1 and 6 DF, p-value: 0.003167

PREDICTION WITH CI

fit lwr upr

1 62.22274 46.68627 77.75922