Day 25
Hypothesis Test for a Slope



EPSY 5261 : Introductory Statistical Methods

Learning Goals

At the end of this lesson, you should be able to …

  • Conduct a hypothesis test for a slope coefficient.
  • Interpret results from a hypothesis test for a slope coefficient.

Recall: Linear Regression Equation

\[ \hat{y} = \underbrace{\beta_0}_{\text{Intercept}} + \underbrace{\beta_1}_{\text{Slope}}(x) \]

t-Test for a Slope

  • \(H_0: \beta_1=0\)
    • The slope (in the population) equals 0. (i.e., There is no linear relationship between X and Y.)
  • \(H_A: \beta_1\neq0\)
    • The slope (in the population) does not equal 0. (i.e., There is a linear relationship between X and Y.)

Using R Studio

# Fit regression model
lm.a = lm(coffee_sales ~ 1 + temperature, data = coffee)
# See results of hypothesis test (Need broom package)
tidy(lm.a)
# A tibble: 2 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)   491.      28.2       17.4  4.29e-22
2 temperature    -5.34     0.709     -7.54 1.26e- 9

Hypothesis Test for a Slope Activity

Summary

  • If we want to determine if X is a statistically significant predictor of Y we can do a hypothesis test for the slope.
  • A small p-value tells us that there is evidence the slope is different than 0 (i.e., X is a predictor of Y).