
When discussing whether there is a relationship between the response and the explanatory variable you must explicitly note that you are referring to the slope pvalue. You cannot just refer the reader to the “pvalue” in the table from
summary()
because there are three pvalues in that table. Be precise with your language!!  Remember to use CI when describing rates of change (i.e., slopes) or predictions; don’t just use the best estimate.
Ashland Climate
Yes, there is a signficant relationship between average precipitation and average high temperature for Ashland, WI (p<0.00005; Table 1). Specifically, as the average high temperature increases by 1^{o}C the mean average precipitation increases between 2.41 and 3.32 mm Table 2.
The yintercept value suggests that the mean average monthly preciptation for all months with an average high temperature of 0^{o}C is between 25.0 and 39.2 mm Table 2. It makes sense to interpret this intercept because a temperature of zero is possible.
Mean average preciptation for all months with an average high of 10^{o}C is between 55.7 and 65.8 mm.
The average preciptation for a month with an average high of 10^{o}C is between 42.6 and 79.0 mm.
The prediction interval for the individual (question 4) is wider than the confidence interval for the mean (question 3) because there is more variability in predicting an individual as compared to a mean. Variability for predicting an individual includes both sampling and natural variability, whereas variability for the mean includes only sampling variability.
The relationship between average precipitation and average high temperature is illustrated in Figure 1.
Table 1: Summary of the linear regression of proportion of average monthly preciptation on average high temperature for Ashland, WI.
Estimate Std. Error t value Pr(>t)
(Intercept) 32.0887 3.1782 10.10 1.46e06
ahi 2.8676 0.2053 13.97 6.91e08

Residual standard error: 7.842 on 10 degrees of freedom
Multiple Rsquared: 0.9513, Adjusted Rsquared: 0.9464
Fstatistic: 195.2 on 1 and 10 DF, pvalue: 6.913e08
Table 2: Confidence intervals for coefficients of the linear regression of average monthly preciptation on average high temperature for Ashland, WI.
2.5 % 97.5 %
(Intercept) 25.007285 39.170170
ahi 2.410234 3.324969
Figure 1: Scatterplot of average monthly preciptation and average high temperature for Ashland, WI with the bestfit line.
R Appendix.
d2 < read.csv("Ashland.csv")
lm2 < lm(aprecip~ahi,data=d2)
summary(lm2)
confint(lm2)
predict(lm2,data.frame(ahi=10),interval="confidence")
predict(lm2,data.frame(ahi=10),interval="prediction")
fitPlot(lm2,xlab="Average High Temperature (C)",ylab="Average Preciptation (mm)")