Question

Listed below are altitudes (in thousands of feet) and outside air temperatures (in °F) recorded by a curious passenger during Delta Flight 1053 from New Orleans to Atlanta.

Altitude	3	10	14	22	28	31	33
Temperature	57	37	24	-5	-30	-41	-54

Below is the scatterplot of this data with the best-fit line.

Correlation coefficient (r) = -0.997

Mean_altitude = 20.14

Mean_temperature = -1.71

Standard deviation_altitude (s_x) = 11.423

Standard deviationtemperature (s_y) = 42.221

Write down the equation for the best-fit line:

temperature =                            [ Select ]                       ["2.5", "-3.2", "-3.7", "3.7"]         altitude +                            [ Select ]                       ["65.3", "72.8", "77.9", "-69.4"]      

What is r² of the best-fit line?

r² =                            [ Select ]                       ["-0.994", "0.994", "-0.999", "0.999"]      

The regression equation predicts the outside temperature to be 61.7 when the altitude is 3. What is the residual when the altitude is 3?

Residual_altitude=3 =                            [ Select ]                       ["-6.3", "-4.7", "4.7", "6.3"]      

Answer 1

Using R code as follow we get

> Altitude=c(3,10,14,22,28,31,33)
> Temperature=c(57,37,24,-5,-30,-41,-54)
> cor(Altitude,Temperature)
[1] -0.9967768
> fit=lm(Temperature~Altitude)

> fit

Call:
lm(formula = Temperature ~ Altitude)

Coefficients:
(Intercept) Altitude
72.498 -3.684

> summary(fit)

Call:
lm(formula = Temperature ~ Altitude)

Residuals:
1 2 3 4 5 6 7
-4.4453 1.3449 3.0821 3.5566 0.6624 0.7153 -4.9161

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 72.4982 3.0169 24.03 2.33e-06 ***
Altitude -3.6843 0.1326 -27.78 1.13e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.71 on 5 degrees of freedom
Multiple R-squared: 0.9936,   Adjusted R-squared: 0.9923
F-statistic: 771.9 on 1 and 5 DF, p-value: 1.131e-06

Equation of best line fit is

temperature = - 3.7 * altitude + 72.8

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r2=0.999

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Residual altitude of 3 = actual vaule of 3 - fitted value of 3

= 57 - 61.7

Residual altitude of 3 = - 4.7