Question

2.6 Collins temperature data (Data file: ftcollinstemp) The data file gives the mean temperature in the fall of each year, defined as Sep- tember 1 to November 30, and the mean temperature in the following winter, defined as December 1 to the end of February in the following calendar year, in degrees Fahrenheit, for Ft. Collins, CO (Colorado Climate Center, 2012). These data cover the time period from 1900 to 2010. The question of interest is: Does the average fall temperature predict the average winter temperature?

2.6.1 Draw a scatterolot of the response versus the predictor and describe any pattern you might see in the plot.

2.6.2 Use statistical software to fit the regression of the response on the predictor. Add the fitted line to your graph. Test the slope to be 0 against a two-sided alternative, and summarize your results.

2.6.3 Compute or obtain from your computer output the value of the variability in winter explained by fall and explain what this means.

2.6.4 Divide the data into 2 time periods, an early period from 1900 to 1989 , and a late period from 1990 to 2010. You can do this using the variable year in the data file. Are the results different in the two time periods?

Answer 1

# this code is done in R statistical software
library(alr4)

data = ftcollinstemp

#### 2.6.1 ####
plot(data$fall, data$winter)

# plot is in 2.6.2 part
# from the plot we can see that their is
# no linear relationship between average fall tempreature
# and average winter temprature


#### 2.6.2 ####
model = lm(winter ~ fall, data = data)
summary(model)

##
## Call:
## lm(formula = winter ~ fall, data = data)
##
## Residuals:
##     Min      1Q Median      3Q     Max
## -7.8186 -1.7837 -0.0873 2.1300 7.5896
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.7843     7.5549   1.825   0.0708 .
## fall          0.3132     0.1528   2.049   0.0428 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.179 on 109 degrees of freedom
## Multiple R-squared: 0.0371, Adjusted R-squared: 0.02826
## F-statistic:   4.2 on 1 and 109 DF, p-value: 0.04284

abline(model)

# in the above summary we can see that the P-value corresponding to
# fall is 0.0428 which is not that significant. That is if we take alpha = 0.01
# we will fail to reject H0 that implies that the predictor fall is not significant

#### 2.6.3 ####
R_square = summary(model)$r.squared;R_square

## [1] 0.03709854

# the R_square value is 0.03709854 that implies that approximately 3.7%
# of the variablity of average winter temprature is explained by average
# fall temprature

#### 2.6.4 ####
data1 = data[1:90, ]
data2 = data[90:length(data$year), ]
model1 = lm(winter ~ fall, data = data1)
summary(model1)

##
## Call:
## lm(formula = winter ~ fall, data = data1)
##
## Residuals:
##     Min      1Q Median      3Q     Max
## -6.8976 -1.6349 0.0118 2.0079 7.3387
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept) 22.7079     8.2600   2.749 0.00725 **
## fall          0.1209     0.1681   0.719 0.47397  
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.057 on 88 degrees of freedom
## Multiple R-squared: 0.005842,   Adjusted R-squared: -0.005455
## F-statistic: 0.5171 on 1 and 88 DF, p-value: 0.474

model2 = lm(winter ~ fall, data = data2)
summary(model2)

##
## Call:
## lm(formula = winter ~ fall, data = data2)
##
## Residuals:
##     Min      1Q Median      3Q     Max
## -5.4731 -1.4740 0.0786 1.9299 4.9411
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) 25.9511    16.9685   1.529    0.142
## fall          0.1161     0.3341   0.348    0.732
##
## Residual standard error: 2.636 on 20 degrees of freedom
## Multiple R-squared: 0.006004,   Adjusted R-squared: -0.0437
## F-statistic: 0.1208 on 1 and 20 DF, p-value: 0.7318

# we can see from the summary there is not much difference in the R_square also
# the variable fall is not significant to explain winter temprature