Consider this small study of human height versus shoe size at (https://www.statcrunch.com/5.0/viewreport.php?reportid=35115), summarized in Figure 1 below, which found the following best fit SLR regression model correlating U.S. shoe size and height in inches: Height = 50.87 + 1.657 ShoeSize with ?b = 3.78 which yielded an R2 value of 0.387. This analysis is repeated in the R script hw3-shoesize-height.R using the data in hw3-sullivan-statistical-survey-data.xlsx.

a) What is the basic definition of R2 in this context, and how should we interpret R2 values?

b) How expected or unexpected is this level of explanatory power (i.e., the R2 = 0.387 value) under the assumption that this best fit model is true, and given these sample sizes? To answer that question, use the following number of individuals with each shoe size in the data set to simulate synthetic data under their best fit linear model given above. Repeat this data simulation 50,000 times using a for loop (for(i in 1:50000) { ...) and each time store the R2 value using something like fit=lm(...); fitsum

c. Discuss what your results mean in terms of how well this particular data set meets the assumptions of the SLR model.

Figure 1: Source: https://www.statcrunch.com/5.0/viewreport.php?reportid=35115 based on data from the Sullivan Statistics Survey at https://www.statcrunch.com/app/ index.php?dataid=450205.

Simple linear regression results Dependent Variable: Height Independent Variable: Foot Height 50.8747981.6565183 Foot Sample size: 199 R (correlation coefficient)-0.6222 R-sq-0.38713068 Estimate of error standard deviation: 3.7840705 Scatterplot 1: Correlation of Shoe Size and Height Height (inches) 80 75 Paramcter estimates: Parameter Estimate Std. Err. Alternative DF T-Stat P-Value 0 197 33.22651