Note:
• 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 p-value. You cannot just refer the reader to the “p-value” in the table from summary() because there are three p-values 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

1. 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 1oC the mean average precipitation increases between 2.41 and 3.32 mm Table 2.

2. The y-intercept value suggests that the mean average monthly preciptation for all months with an average high temperature of 0oC is between 25.0 and 39.2 mm Table 2. It makes sense to interpret this intercept because a temperature of zero is possible.

3. Mean average preciptation for all months with an average high of 10oC is between 55.7 and 65.8 mm.

4. The average preciptation for a month with an average high of 10oC is between 42.6 and 79.0 mm.

5. 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.

6. 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 1.46e-06
ahi           2.8676     0.2053   13.97 6.91e-08
---
Residual standard error: 7.842 on 10 degrees of freedom
Multiple R-squared: 0.9513, Adjusted R-squared: 0.9464
F-statistic: 195.2 on 1 and 10 DF,  p-value: 6.913e-08 

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 best-fit 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)")