Question

OLS Mechanics

2. Describe the difference in interpretation, if there is any, between the population parameters (β0 and β1) and estimated coefficients (βˆ 0 and βˆ 1).

3. Describe what information βˆ 1 and ρ can reveal. What information do they both share? And what distinct information do they provide?

Answer 1

that we are ultimately always interested in drawing conclusions about the population not the particular sample we observed. In the simple regression setting, we are often interested in learning about the population intercept β0 and the population slope β1. As you know, confidence intervals and hypothesis tests are two related, but different, ways of learning about the values of population parameters. Here, we will learn how to calculate confidence intervals and conduct hypothesis tests for both β0 and β1.

Let's revisit the example concerning the relationship between skin cancer mortality and state latitude. The response variable y is the mortality rate (number of deaths per 10 million people) of white males due to malignant skin melanoma from 1950-1959. The predictor variable x is the latitude (degrees North) at the center of each of 49 states in the United States. A subset of the data look like:

#	State	Latitude	Mortality
1	Alabama	33.0	219
2	Arizona	34.5	160
3	Arkansas	35.0	170
4	California	37.5	182
5	Colorado	39.0	149

2.

P-values and coefficients in regression analysis work together to tell you which relationships in your model are statistically significant and the nature of those relationships. The coefficients describe the mathematical relationship between each independent variable and the dependent variable. The p-values for the coefficients indicate whether these relationships are statistically significant.

After fitting a regression model, check the residual plots first to be sure that you have unbiased estimates. After that, it’s time to interpret the statistical output. Linear regression analysis can produce a lot of results, which I’ll help you navigate. In this post, I cover interpreting the p-values and coefficients for the independent variables.