7  t-tests and t CIs

Author

Affiliation

Dr. Devan Becker

Wilfrid Laurier University

8 The \(t\) distribution in R

8.1 Critical Values

In order to find the critical value of a CI: For a \((1-\alpha)\) CI, there should \((1-\alpha)\) in the middle of the sampling distribution. This means that there is \(\alpha/2\) is to the left of the lower value (\(-t^*\)) and \(\alpha/2\) to the right.

If we have a sample of, say, 20 people, then the critical value of a 95% CI is:

Verify these values with the t-table!

8.2 \(P\)-values

Just like with the normal distribution and the binomial distribution, we use the p*() functions to calculate probabilities.

If we have a t-statistic of, say, 2.093024 with a sample size of 20, then the \(P\)-value still depends on what the alternative hypothesis:

Notice that the p-value for 2.093024 is exactly 0.05 (within rounding errors) for a two-sided test, and that the critical value for a 95% CI is 2.093024. This is not a coincidence!

9 Example: Keystrokes to detect unauthorized usage

The following data represent the time between keystrokes when entering a password. What’s a 95% CI?

First, if the histogram of the data looks reasonable normal then we can guess that the population is reasonably normal:

This is not horrifically non-normal, so we’ll go with it.

R can make a CI with a built-in function that we’ll see later, but let’s use R as a calculator for now.

We are 95% confident that the true mean keystroke time is in the interval above.

Now let’s test the hypothesis that \(\mu_0 = 0.2\). We know what our conclusion is going to be beased on the CI, but let’s get some practice anyway.

Write the hypotheses: \(H_0: \mu = 0.2\) versus \(H_0: \mu \ne 0.2\).

Calculate our test statistic:

Now let’s calculate our p-value:

The p-value for a two-sided \(t\)-test 0.016. This means that there’s a 1.6% chance of getting an average keystroke at least as extreme as the one we did if the null hypothesis were true.

Conclude: Our p-value is less than the 5% significance level, so we reject the null hypothesis that the mean of these keystrokes could be 0.2 in favour of the two-sided alternative.

10 Built-in R function: t.test

R is for stats, and this is basic stats. This is how many, many, many scientific papers actually calculate their statistics, and is a great way for you to check your answer to textbook problems that provide the data.

Our values match exactly!

• mu is the value from the null hypothesis.
• alternative can be either "two.sided", "less", or "greater", as appropriate.
• level is the confidence level, \(1 - \alpha\).

Important Note: The WeBWorK assignments expect you to use the \(t\)-table! Use R to check your answers, but make sure you can use the tables for WeBWorK and the exam.

11 Addendum: \(t_{n-1}\rightarrow Z\) as \(n\rightarrow\infty\)

#| standalone: true
#| viewerHeight: 700

library(shiny)

ui <- fluidPage(
    sidebarLayout(
        sidebarPanel(
            sliderInput("n",
                "Sample size (n)",
                min = 2,
                max = 100,
                value = 5,
                step = 1,
                animate = TRUE),
            sliderInput("obs",
                "Test statistic",
                min = -4,
                max = 4,
                value = 1.96,
                step = 0.01,
                animate = TRUE)
        ),
        
        mainPanel(
            plotOutput("distPlot")
        )
    )
)

server <- function(input, output) {
    
    output$distPlot <- renderPlot({
      x <- seq(-4, 4, length.out = 200)
      z <- dnorm(x)
      t <- dt(x, df = input$n - 1)

      pz <- round(2 * pnorm(-abs(input$obs)), 3)
      pt <- round(2 * pt(-abs(input$obs), df = input$n - 1), 3)
      dz <- dnorm(input$obs)
      dt <- dt(input$obs, df = input$n - 1)
      
      plot(z ~ x,
        type = "l",
        main = "Z vs t",
        ylim = c(0, max(z)),
        xlab = "Test statistic", ylab = "Density"
      )
      lines(c(input$obs, input$obs), c(0, dz))
      lines(t ~ x, col = 2, lwd = 2)
      lines(c(input$obs, input$obs), c(0, dt), col = 2)
      # dt and dz are swapped so the higher label is the larger p-value
      text(input$obs, dt, label = paste0("sigma known\np=", pz), pos = 2)
      text(input$obs, dz, label = paste0("sigma unknown\np=", pt), pos = 4)
    })
}

# Run the application 
shinyApp(ui = ui, server = server)

Intuition check: Is the p-value shown a one-sided or two-sided p-value?