Porsche Price (3)

library(tidyverse)

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## ✔ readr   1.3.1     ✔ forcats 0.4.0

## ── Conflicts ──────────────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()

library(Stat2Data)
library(broom)

data("PorschePrice")

Fit a simple linear regression predicting Price from Mileage in the PorschePrice dataset

model <- lm(Price ~ Mileage, data = PorschePrice)
model

## 
## Call:
## lm(formula = Price ~ Mileage, data = PorschePrice)
## 
## Coefficients:
## (Intercept)      Mileage  
##     71.0905      -0.5894

Use the “tidy” function to get the p-value of the slope

model %>%
  tidy()

## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)   71.1      2.37        30.0 7.87e-23
## 2 Mileage       -0.589    0.0566     -10.4 3.98e-11

obs_stat <- model %>%
  tidy() %>%
  filter(term == "Mileage") %>%
  pull(statistic)

pt(obs_stat, df = nrow(PorschePrice) - 2) * 2

## [1] 3.981734e-11

Use the “tidy” function to get the confidence interval of a slope

model %>%
  tidy(conf.int = TRUE)

## # A tibble: 2 x 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)   71.1      2.37        30.0 7.87e-23   66.2      75.9  
## 2 Mileage       -0.589    0.0566     -10.4 3.98e-11   -0.705    -0.473

t_star <- qt(0.025, df = nrow(PorschePrice) - 2, lower.tail = FALSE)

b1 <- model %>%
  tidy() %>%
  filter(term == "Mileage") %>%
  pull(estimate)

stderr <- model %>%
  tidy() %>%
  filter(term == "Mileage") %>%
  pull(std.error)

b1 + t_star * stderr

## [1] -0.4733618

b1 - t_star * stderr

## [1] -0.7054401

What is the null hypothesis? What is the alternative hypothesis? What is the result of the hypothesis test if \(\alpha = 0.05\)?

\(\beta_1 = 0\) \(\beta_1 \neq 0\)

We reject the null hypothesis!