One-Sample t-Test Calculator | Compare Sample Mean to Known Value

Key Takeaways: One-Sample t-Test

Tip

• Purpose: Compare a sample mean to a known or hypothesized population value
• When to use: When testing if a sample differs from a specified value
• Assumptions: Random sampling, normality (or large sample size)
• Null hypothesis: The population mean equals the specified value (\(H_0: \mu = \mu_0\))
• Interpretation: If p < 0.05, there is a significant difference between the sample mean and the hypothesized value
• Common applications: Quality control, testing against standards or benchmarks, validating claims

What is the One-Sample t-Test?

The one-sample t-test is a statistical method used to determine whether the mean of a sample differs significantly from a known or hypothesized population value. It’s one of the simplest yet most useful statistical tests for comparing observed data against a standard or theoretical expectation.

Tip

When to use the one-sample t-test:

• When comparing a sample mean to a known standard or reference value
• When testing if a sample comes from a population with a specific mean
• When validating claims about population means based on sample data
• When quality testing products against specifications

This online calculator allows you to quickly perform a one-sample t-test, visualize your data, and interpret the results with confidence.

Calculator

#| '!! shinylive warning !!': |
#|   shinylive does not work in self-contained HTML documents.
#|   Please set `embed-resources: false` in your metadata.
#| standalone: true
#| viewerHeight: 1300

library(shiny)
library(bslib)
library(ggplot2)
library(bsicons)
library(vroom)
library(shinyjs)

ui <- page_sidebar(
  title = "One-Sample t-Test",
  useShinyjs(),  # Enable shinyjs for resetting inputs
  sidebar = sidebar(
    width = 400,

    card(
      card_header("Data Input"),
      accordion(
        accordion_panel(
          "Manual Input",
          textAreaInput("sample_input", "Sample Data [One value per row]", rows = 8,
                     placeholder = "Paste values here..."),
          div(
            actionLink("use_example", "Use example data", style = "color:#0275d8;"),
            tags$span(bs_icon("file-earmark-text"), style = "margin-left: 5px; color: #0275d8;")
          )
        ),
        accordion_panel(
          "File Upload",
          fileInput("file_upload", "Upload CSV or TXT file:",
                   accept = c("text/csv", "text/plain", ".csv", ".txt")),
          checkboxInput("header", "File has header", TRUE),
          conditionalPanel(
            condition = "output.file_uploaded",
            div(
              selectInput("sample_var", "Sample variable:", choices = NULL),
              actionButton("clear_file", "Clear File", class = "btn-danger btn-sm")
            )
          )
        ),
        id = "input_method",
        open = 1
      ),
      
      # Advanced Options accordion
      accordion(
        accordion_panel(
          "Advanced Options",
          
          card(
            card_header("Null Hypothesis:"),
            card_body(
              tags$div(style = "margin-bottom: 10px;",
                numericInput("null_hypothesis", "mu =", 0.0, width = "100%")
              )
            )
          ),
          
          card(
            card_header("Alternative Hypothesis:"),
            card_body(
              radioButtons("alternative", NULL,
                         choices = c("Population mean ≠ mu₀" = "two.sided", 
                                    "Population mean < mu₀" = "less",
                                    "Population mean > mu₀" = "greater"),
                         selected = "two.sided")
            )
          ),
          
          card(
            card_header("Confidence Level:"),
            card_body(
              sliderInput("conf_level", NULL, min = 0.8, max = 0.99, value = 0.95, step = 0.01)
            )
          )
        ),
        open = FALSE
      ),
      
      actionButton("run_test", "Run Test", class = "btn btn-primary")
    ),

    hr(),

    card(
      card_header("Interpretation"),
      card_body(
        div(class = "alert alert-info",
          tags$ul(
            tags$li("The one-sample t-test compares the mean of a sample to a hypothesized value."),
            tags$li(tags$b("Null hypothesis:"), " The population mean equals the specified value."),
            tags$li(tags$b("Alternative:"), " The population mean differs from (or is less/greater than) the specified value."),
            tags$li("If p-value < 0.05, there is a significant difference between the sample mean and the hypothesized value."),
            tags$li("This test assumes the data is approximately normally distributed.")
          )
        )
      )
    )
  ),

  layout_column_wrap(
    width = 1,

    card(
      card_header("Test Results"),
      card_body(
        navset_tab(
          nav_panel("Results", uiOutput("error_message"), verbatimTextOutput("test_results")),
          nav_panel("Explanation", div(style = "font-size: 0.9rem;",
            p("The One-Sample t-Test evaluates whether the population mean differs from a specified value:"),
            tags$ul(
              tags$li("It calculates the t-statistic by comparing the sample mean to the hypothesized value, considering the standard error."),
              tags$li("The test assumes that the data follows a normal distribution."),
              tags$li("Degrees of freedom = n-1, where n is the sample size."),
              tags$li("A small p-value indicates the population mean likely differs from the hypothesized value.")
            )
          ))
        )
      )
    ),

    card(
      card_header("Visual Assessment"),
      card_body(
        navset_tab(
          nav_panel("Histogram",
            navset_tab(
              nav_panel("Plot", plotOutput("histogram")),
              nav_panel("Explanation", div(style = "font-size: 0.9rem;",
                p("The histogram shows the distribution of your sample:"),
                tags$ul(
                  tags$li("The vertical dashed line shows the sample mean."),
                  tags$li("The vertical dotted line shows the hypothesized population mean value."),
                  tags$li("The curve represents the normal distribution with the same mean and standard deviation as your sample."),
                  tags$li("Comparing these lines helps you visualize the difference being tested.")
                )
              ))
            )
          ),
          nav_panel("QQ Plot",
            navset_tab(
              nav_panel("Plot", plotOutput("qqplot")),
              nav_panel("Explanation", div(style = "font-size: 0.9rem;",
                p("The Q-Q plot helps assess if your data follows a normal distribution:"),
                tags$ul(
                  tags$li("Points should follow the diagonal line if the data is normally distributed."),
                  tags$li("Deviations from the line suggest departures from normality."),
                  tags$li("The t-test assumes normality, so substantial deviations may affect the reliability of the test results.")
                )
              ))
            )
          ),
          nav_panel("Confidence Interval",
            navset_tab(
              nav_panel("Plot", plotOutput("ci_plot")),
              nav_panel("Explanation", div(style = "font-size: 0.9rem;",
                p("The confidence interval plot shows:"),
                tags$ul(
                  tags$li("The sample mean (point) and confidence interval (horizontal line)."),
                  tags$li("The vertical dotted line represents the hypothesized mean value."),
                  tags$li("If the confidence interval does not include the hypothesized value, the result is statistically significant."),
                  tags$li("The width of the interval reflects the precision of the estimate.")
                )
              ))
            )
          )
        )
      )
    )
  )
)

server <- function(input, output, session) {
  # Example data
  example_data <- "8.5\n7.2\n12.4\n10.8\n9.3\n6.7\n11.5\n8.9\n10.2\n7.8"

  # Track input method
  input_method <- reactiveVal("manual")
  
  # Function to clear file inputs
  clear_file_inputs <- function() {
    updateSelectInput(session, "sample_var", choices = NULL)
    reset("file_upload")
  }
  
  # Function to clear text inputs
  clear_text_inputs <- function() {
    updateTextAreaInput(session, "sample_input", value = "")
  }

  # When example data is used, clear file inputs and set text inputs
  observeEvent(input$use_example, {
    input_method("manual")
    clear_file_inputs()
    updateTextAreaInput(session, "sample_input", value = example_data)
  })

  # When file is uploaded, clear text inputs and set file method
  observeEvent(input$file_upload, {
    if (!is.null(input$file_upload)) {
      input_method("file")
      clear_text_inputs()
    }
  })

  # When clear file button is clicked, clear file and set manual method
  observeEvent(input$clear_file, {
    input_method("manual")
    clear_file_inputs()
  })
  
  # When text input changes, clear file inputs if it has content
  observeEvent(input$sample_input, {
    if (!is.null(input$sample_input) && nchar(input$sample_input) > 0) {
      input_method("manual")
      clear_file_inputs()
    }
  }, ignoreInit = TRUE)

  file_data <- reactive({
    req(input$file_upload)
    tryCatch({
      vroom::vroom(input$file_upload$datapath, delim = NULL, col_names = input$header, show_col_types = FALSE)
    }, error = function(e) {
      showNotification(paste("File read error:", e$message), type = "error")
      NULL
    })
  })

  observe({
    df <- file_data()
    if (!is.null(df)) {
      num_vars <- names(df)[sapply(df, is.numeric)]
      updateSelectInput(session, "sample_var", choices = num_vars)
    }
  })

  output$file_uploaded <- reactive({
    !is.null(input$file_upload)
  })
  outputOptions(output, "file_uploaded", suspendWhenHidden = FALSE)

  # Function to parse text input
  parse_text_input <- function(text) {
    if (is.null(text) || text == "") return(NULL)
    input_lines <- strsplit(text, "\\r?\\n")[[1]]
    input_lines <- input_lines[input_lines != ""]
    numeric_values <- suppressWarnings(as.numeric(input_lines))
    if (all(is.na(numeric_values))) return(NULL)
    return(na.omit(numeric_values))
  }

  # Get sample values
  sample_values <- reactive({
    if (input_method() == "file" && !is.null(file_data()) && !is.null(input$sample_var)) {
      df <- file_data()
      return(na.omit(df[[input$sample_var]]))
    } else {
      return(parse_text_input(input$sample_input))
    }
  })
  
  # Get mu value
  mu_value <- reactive({
    return(input$null_hypothesis)
  })

  # Validate input data
  validate_data <- reactive({
    sample <- sample_values()
    
    if (is.null(sample)) {
      return("Error: Please check your input. Make sure all values are numeric.")
    }
    
    if (length(sample) < 2) {
      return("Error: At least 2 observations are required for the t-test.")
    }
    
    # Check if all values are identical (no variance)
    if (var(sample) == 0) {
      return("Error: All values are identical. The t-test requires some variability in the data.")
    }
    
    # Check for normality using Shapiro-Wilk test (just a warning)
    if (length(sample) <= 50) {  # Shapiro-Wilk limited to 50 observations
      sw_test <- shapiro.test(sample)
      if (sw_test$p.value < 0.05) {
        return(paste0("Warning: Your data may not be normally distributed (Shapiro-Wilk p=", 
                      round(sw_test$p.value, 4), 
                      "). The t-test assumes normality."))
      }
    }
    
    return(NULL)
  })

  output$error_message <- renderUI({
    error <- validate_data()
    if (!is.null(error) && input$run_test > 0) {
      if (startsWith(error, "Warning")) {
        div(class = "alert alert-warning", error)
      } else {
        div(class = "alert alert-danger", error)
      }
    }
  })

  # Run the t-test
  test_result <- eventReactive(input$run_test, {
    error <- validate_data()
    if (!is.null(error) && startsWith(error, "Error")) return(NULL)
    
    # Run the t.test with the specified parameters
    t.test(
      sample_values(), 
      mu = mu_value(),
      alternative = input$alternative,
      conf.level = input$conf_level
    )
  })

  # Display test results
  output$test_results <- renderPrint({
    if (is.null(test_result())) return(NULL)
    
    result <- test_result()
    sample <- sample_values()
    
    # Use alternative input for interpretation
    alt_text <- switch(
      input$alternative,
      "two.sided" = "different from",
      "less" = "less than",
      "greater" = "greater than"
    )
    
    cat("One-Sample t-Test Results:\n")
    cat("==========================\n\n")
    cat("t-statistic:", round(result$statistic, 4), "\n")
    cat("Degrees of freedom:", round(result$parameter, 0), "\n")
    cat("p-value:", round(result$p.value, 6), "\n\n")
    
    conf_pct <- paste0(input$conf_level * 100, "%")
    cat(conf_pct, " Confidence Interval: [", 
        round(result$conf.int[1], 4), ", ", 
        round(result$conf.int[2], 4), "]\n\n", sep = "")
    
    cat("Sample Summary:\n")
    cat("---------------\n")
    cat("Sample size:", length(sample), "\n")
    cat("Sample mean:", round(result$estimate, 4), "\n")
    cat("Sample standard deviation:", round(sd(sample), 4), "\n")
    cat("Standard error of the mean:", round(sd(sample)/sqrt(length(sample)), 4), "\n\n")
    
    cat("Test Information:\n")
    cat("-----------------\n")
    cat("Hypothesized mean:", mu_value(), "\n")
    cat("Alternative hypothesis: true mean is", alt_text, mu_value(), "\n")
    cat("Confidence level:", conf_pct, "\n")
    
    
    # Effect size (Cohen's d)
    d <- (mean(sample) - mu_value()) / sd(sample)
    cat("\nEffect Size:\n")
    cat("--------------\n")
    cat("Cohen's d:", round(d, 4), "\n")
    
    effect_size_interpretation <- ""
    if (abs(d) < 0.2) {
      effect_size_interpretation <- "negligible"
    } else if (abs(d) < 0.5) {
      effect_size_interpretation <- "small"
    } else if (abs(d) < 0.8) {
      effect_size_interpretation <- "medium"
    } else {
      effect_size_interpretation <- "large"
    }
    
    cat("Interpretation: The effect size is", effect_size_interpretation, "\n\n")

    # Conclusion
    cat("Conclusion:\n")
    cat("-----------\n")
    if (result$p.value < 0.05) {
      cat("Interpretation: There is a significant difference (p < 0.05).\n")
      cat("The sample mean is significantly", alt_text, "the hypothesized value of", mu_value(), ".\n")
    } else {
      cat("Interpretation: No significant difference detected (p ≥ 0.05).\n")
      cat("We cannot conclude that the population mean is", alt_text, "the hypothesized value of", mu_value(), ".\n")
    }
    
  })

  # Generate histogram
  output$histogram <- renderPlot({
    req(input$run_test > 0, !is.null(sample_values()))
    
    sample <- sample_values()
    
    # Calculate mean and standard deviation for normal curve
    sample_mean <- mean(sample)
    sample_sd <- sd(sample)
    
    # Create data for the normal curve
    x_range <- seq(min(sample) - 2 * sample_sd, max(sample) + 2 * sample_sd, length.out = 100)
    normal_data <- data.frame(
      x = x_range,
      y = dnorm(x_range, mean = sample_mean, sd = sample_sd)
    )
    
    # Scale the normal curve to match histogram height
    hist_data <- hist(sample, plot = FALSE)
    max_height <- max(hist_data$density)
    max_norm <- max(normal_data$y)
    scale_factor <- max_height / max_norm
    normal_data$y <- normal_data$y * scale_factor
    
    ggplot() +
      geom_histogram(data = data.frame(value = sample), aes(x = value, y = after_stat(density)), 
                    bins = min(30, max(10, length(sample)/3)), 
                    fill = "#5dade2", color = "#2874a6", alpha = 0.7) +
      geom_line(data = normal_data, aes(x = x, y = y), 
               color = "#e74c3c", size = 1.2, linetype = "solid") +
      geom_vline(xintercept = sample_mean, linetype = "dashed", 
                color = "#c0392b", size = 1.5) +
      geom_vline(xintercept = mu_value(), linetype = "dotted", 
                color = "#8e44ad", size = 1.5) +
      annotate("text", x = sample_mean, y = 0, 
               label = paste("Mean =", round(sample_mean, 2)), 
               vjust = -1, hjust = ifelse(sample_mean > mu_value(), 1.1, -0.1),
               color = "#c0392b", fontface = "bold") +
      annotate("text", x = mu_value(), y = 0, 
               label = paste("H₀ =", mu_value()), 
               vjust = -3, hjust = ifelse(mu_value() > sample_mean, 1.1, -0.1),
               color = "#8e44ad", fontface = "bold") +
      labs(title = "Distribution of Sample Values with Normal Curve",
           subtitle = "With sample mean and hypothesized value",
           x = "Value",
           y = "Density") +
      theme_minimal(base_size = 14)
  })
  
  # Generate QQ plot
  output$qqplot <- renderPlot({
    req(input$run_test > 0, !is.null(sample_values()))
    
    sample <- sample_values()
    
    ggplot(data.frame(sample = sample), aes(sample = sample)) +
      stat_qq(color = "#3498db", size = 3) +
      stat_qq_line(color = "#e74c3c", size = 1.2) +
      labs(title = "Normal Q-Q Plot",
           subtitle = "Points should follow the line if normally distributed",
           x = "Theoretical Quantiles",
           y = "Sample Quantiles") +
      theme_minimal(base_size = 14)
  })
  
  # Generate confidence interval plot
  output$ci_plot <- renderPlot({
    req(input$run_test > 0, !is.null(test_result()))
    
    result <- test_result()
    sample <- sample_values()
    mu <- mu_value()
    
    # Extract confidence interval
    ci_lower <- result$conf.int[1]
    ci_upper <- result$conf.int[2]
    sample_mean <- mean(sample)
    
    # Create data frame for CI plot
    ci_data <- data.frame(
      parameter = "Mean",
      estimate = sample_mean,
      ci_lower = ci_lower,
      ci_upper = ci_upper
    )
    
    # Determine if CI includes hypothesized mean
    includes_mu <- ci_lower <= mu && mu <= ci_upper
    ci_color <- ifelse(includes_mu, "#3498db", "#e74c3c")
    
    ggplot(ci_data, aes(x = estimate, y = parameter)) +
      geom_vline(xintercept = mu, linetype = "dotted", 
                color = "#8e44ad", size = 1.2) +
      geom_pointrange(aes(xmin = ci_lower, xmax = ci_upper), 
                     color = ci_color, size = 1.5, linewidth = 1.2) +
      annotate("text", x = mu, y = 0.8, 
               label = paste("H₀ =", mu), 
               vjust = -1, color = "#8e44ad", fontface = "bold") +
      annotate("text", x = sample_mean, y = 1.2, 
               label = paste("Mean =", round(sample_mean, 2)), 
               vjust = 1.5, color = ci_color, fontface = "bold") +
      labs(title = paste0(input$conf_level * 100, "% Confidence Interval for the Mean"),
           subtitle = ifelse(includes_mu, 
                           "CI includes the hypothesized value (not significant)", 
                           "CI excludes the hypothesized value (significant)"),
           x = "Value",
           y = "") +
      theme_minimal(base_size = 14) +
      theme(axis.text.y = element_blank(),
            axis.ticks.y = element_blank())
  })
}

shinyApp(ui = ui, server = server)

Theory

How the One-Sample t-Test Works

The one-sample t-test evaluates whether the mean of a sample is statistically different from a specified value by comparing the observed mean difference to the variation within the sample.

oneSampleTTestFlowchart = {
  // Canvas setup
  const width = 1300;
  const height = 500;
  const padding = 60;
  
  // Create SVG with explicit viewBox
  const svg = d3.create("svg")
    .attr("width", width)
    .attr("height", height)
    .attr("viewBox", [0, 0, width, height])
    .attr("style", "max-width: 100%; height: auto; font: 16px sans-serif;");
  
  // Add title at the bottom
  svg.append("text")
    .attr("x", width / 2)
    .attr("y", height - 20)
    .attr("text-anchor", "middle")
    .attr("font-size", "20px")
    .attr("font-weight", "bold")
    .text("One-Sample T-Test Procedure");
  
  // Define nodes with horizontal layout
  const nodes = [
    // Input data and hypothesized value
    {id: "A", label: "Sample Data", x: padding + 80, y: 150},
    {id: "B", label: "Hypothesized\nPopulation Mean (μ₀)", x: padding + 80, y: 250},
    
    // Initial calculations
    {id: "C", label: "Calculate Sample\nMean (x̄)", x: padding + 280, y: 150},
    {id: "D", label: "Calculate Sample\nStandard Deviation (s)", x: padding + 280, y: 250},
    
    // Further calculations
    {id: "E", label: "Calculate Standard\nError (s/√n)", x: padding + 480, y: 250},
    
    // T-statistic
    {id: "F", label: "Calculate t-statistic\n(x̄ - μ₀)/(s/√n)", x: padding + 680, y: 200},
    
    // Final path
    {id: "G", label: "Determine\np-value", x: padding + 880, y: 200},
    {id: "H", label: "p-value < α?", x: padding + 1080, y: 200, isDecision: true},
    {id: "I", label: "Reject null\nhypothesis", x: padding + 990, y: 330},
    {id: "J", label: "Retain null\nhypothesis", x: padding + 1170, y: 330}
  ];
  
  // Define edges with draw order
  const edges = [
    // Sample data path
    {source: "A", target: "C", label: "", order: 1},
    {source: "A", target: "D", label: "", order: 1},
    
    // Hypothesized mean to t-statistic
    {source: "B", target: "F", label: "", order: 1},
    
    // Calculation path
    {source: "C", target: "F", label: "", order: 1},
    {source: "D", target: "E", label: "", order: 1},
    {source: "E", target: "F", label: "", order: 1},
    
    // Final path
    {source: "F", target: "G", label: "", order: 1},
    {source: "G", target: "H", label: "", order: 1},
    {source: "H", target: "I", label: "Yes", order: 1},
    {source: "H", target: "J", label: "No", order: 1}
  ];
  
  // Sort edges by order to control draw sequence
  edges.sort((a, b) => a.order - b.order);
  
  // Define arrow marker
  svg.append("defs").append("marker")
    .attr("id", "arrowhead")
    .attr("viewBox", "0 0 10 10")
    .attr("refX", 8)
    .attr("refY", 5)
    .attr("markerWidth", 8)
    .attr("markerHeight", 8)
    .attr("orient", "auto")
    .append("path")
    .attr("d", "M 0 0 L 10 5 L 0 10 z")
    .attr("fill", "#666");
  
  // Draw edges with improved path calculation
  const edgeLines = svg.selectAll("path.edge")
    .data(edges)
    .join("path")
    .attr("class", d => `edge order-${d.order}`)
    .attr("d", d => {
      const source = nodes.find(n => n.id === d.source);
      const target = nodes.find(n => n.id === d.target);
      
      // Calculate connector points
      let sourceX, sourceY, targetX, targetY;
      let path = "";
      
      // Special case for decision diamond
      if (source.isDecision) {
        if (d.label === "Yes") {
          // Going down-left to I
          sourceX = source.x - 25;
          sourceY = source.y + 20;
          targetX = target.x;
          targetY = target.y - 25;
          
          // Return path with offset to avoid overlapping
          path = `M${sourceX},${sourceY} L${sourceX},${(sourceY + targetY)/2} L${targetX},${(sourceY + targetY)/2} L${targetX},${targetY}`;
        } else if (d.label === "No") {
          // Going down-right to J
          sourceX = source.x + 25;
          sourceY = source.y + 20;
          targetX = target.x;
          targetY = target.y - 25;
          
          // Return path with offset to avoid overlapping
          path = `M${sourceX},${sourceY} L${sourceX},${(sourceY + targetY)/2} L${targetX},${(sourceY + targetY)/2} L${targetX},${targetY}`;
        }
      }
      // Special case: Sample Data to Standard Deviation
      else if (source.id === "A" && target.id === "D") {
        // A to D (diagonal down)
        sourceX = source.x + 30;
        sourceY = source.y + 25;
        targetX = target.x - 70;
        targetY = target.y;
        
        // Create angled path
        path = `M${sourceX},${sourceY} L${(sourceX + targetX)/2},${(sourceY + targetY)/2} L${targetX},${targetY}`;
      }
      // Hypothesized mean to t-statistic
      else if (source.id === "B" && target.id === "F") {
        // B to F (diagonal up)
        sourceX = source.x + 70;
        sourceY = source.y;
        targetX = target.x - 70;
        targetY = target.y;
        
        // Create a curved path that avoids other nodes
        const midX1 = sourceX + 120;
        const midY1 = sourceY;
        const midX2 = midX1 + 100;
        const midY2 = (sourceY + targetY) / 2;
        const midX3 = targetX - 100;
        const midY3 = targetY;
        
        path = `M${sourceX},${sourceY} C${midX1},${midY1} ${midX2},${midY2} ${targetX},${targetY}`;
      }
      // Sample mean to t-statistic
      else if (source.id === "C" && target.id === "F") {
        // C to F (diagonal down)
        sourceX = source.x + 70;
        sourceY = source.y;
        targetX = target.x - 70;
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Mathematical Procedure

1. State hypotheses:

   • Null hypothesis \(H_0: \mu = \mu_0\) (the population mean equals the hypothesized value)
   • Alternative hypothesis \(H_a: \mu \neq \mu_0\) (two-tailed), \(H_a: \mu > \mu_0\) (right-tailed), or \(H_a: \mu < \mu_0\) (left-tailed)

2. Calculate the t-statistic:

   \[t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}\]

   Where:

   • \(\bar{x}\) is the sample mean
   • \(\mu_0\) is the hypothesized population mean
   • \(s\) is the sample standard deviation
   • \(n\) is the sample size

3. Determine degrees of freedom:

   \[df = n - 1\]

4. Calculate p-value by comparing the t-statistic to the t-distribution with \(n - 1\) degrees of freedom

5. Make a decision:

   • If p < \(\alpha\) (typically 0.05): Reject the null hypothesis
   • If p ≥ \(\alpha\): Fail to reject the null hypothesis

Effect Size (Cohen’s d)

For a one-sample t-test, Cohen’s d is calculated as:

\[d = \frac{|\bar{x} - \mu_0|}{s}\]

Interpretation: - d ≈ 0.2: Small effect - d ≈ 0.5: Medium effect - d ≈ 0.8: Large effect

Assumptions of the One-Sample t-Test

1. Random sampling: The sample should be randomly selected from the population
2. Normality: The data should follow an approximately normal distribution
   • For large samples (n > 30), the t-test is robust to violations of normality due to the Central Limit Theorem
3. Measurement level: The dependent variable should be measured on a continuous scale

Statistical Power Considerations

Important

Statistical Power Note: The power of a one-sample t-test depends on: - Sample size - Effect size (difference between sample mean and hypothesized value) - Significance level (α) - Variability within the sample

To achieve 80% power (standard convention) for detecting: - Small effect (d = 0.2): Need approximately 199 observations - Medium effect (d = 0.5): Need approximately 34 observations - Large effect (d = 0.8): Need approximately 15 observations

These calculations assume α = 0.05 for a two-tailed test.

Examples

Example 1: Testing Battery Life Claims

A manufacturer claims their new batteries last 40 hours on average. A consumer group tests 20 batteries and wants to verify this claim.

Data (battery life in hours): 38.6, 42.4, 39.8, 43.1, 40.2, 41.6, 39.4, 40.9, 37.8, 42.3, 41.8, 38.5, 40.4, 39.2, 41.5, 40.7, 39.6, 42.8, 40.1, 41.3

Analysis Steps:

1. State hypotheses:
   • \(H_0: \mu = 40\) (the mean battery life equals 40 hours)
   • \(H_a: \mu \neq 40\) (the mean battery life differs from 40 hours)
2. Calculate descriptive statistics:
   • Sample mean: \(\bar{x} = 40.60\) hours
   • Sample standard deviation: \(s = 1.52\) hours
   • Sample size: \(n = 20\)
3. Calculate the t-statistic:
   • \(t = \frac{40.60 - 40}{1.52/\sqrt{20}} = \frac{0.60}{0.34} = 1.76\)
4. Determine degrees of freedom:
   • \(df = 20 - 1 = 19\)
5. Calculate p-value:
   • For a two-tailed test with df = 19, t = 1.76: p = 0.094
6. Calculate effect size:
   • \(d = \frac{|40.60 - 40|}{1.52} = 0.39\) (small to medium effect)

Results: - t(19) = 1.76, p = 0.094, d = 0.39 - 95% CI for mean: [39.96, 41.24] - Interpretation: There is no statistically significant evidence that the actual mean battery life differs from the claimed 40 hours (p > 0.05).

How to Report: “A one-sample t-test was conducted to determine whether the average battery life differed from the manufacturer’s claim of 40 hours. The mean battery life of the tested batteries (M = 40.60, SD = 1.52) was not significantly different from the claimed 40 hours, t(19) = 1.76, p = 0.094, d = 0.39, 95% CI [39.96, 41.24]. These results suggest that the manufacturer’s claim about battery life is plausible.”

Example 2: Testing Student Performance Against Standard

A teacher wants to determine if her class performs better than the national average score of 75 on a standardized test.

Data Summary: - National average (hypothesized value): 75 - Class average (sample mean): 78.6 - Standard deviation of class scores: 8.4 - Sample size: 25 students - Alternative hypothesis: Class mean > national average (one-tailed test)

Results: - t(24) = 2.14, p = 0.021, d = 0.43 - 95% CI for mean: [75.4, 81.8] - Interpretation: There is statistically significant evidence that the class performs better than the national average (p < 0.05).

How to Report: “The class’s mean test score (M = 78.6, SD = 8.4) was significantly higher than the national average of 75, t(24) = 2.14, p = 0.021, d = 0.43, 95% CI [75.4, 81.8]. This indicates that the class is performing above the national standard.”

Reporting

How to Report One-Sample t-Test Results

When reporting the results of a one-sample t-test in academic papers or research reports, include the following elements:

"The sample mean (M = [value], SD = [value]) was significantly [higher/lower/different from] 
the hypothesized value of [μ₀], t([df]) = [t-value], p = [p-value], d = [effect size], 
95% CI [lower bound, upper bound]."

For example:

"The sample mean (M = 78.6, SD = 8.4) was significantly higher than 
the hypothesized value of 75, t(24) = 2.14, p = 0.021, d = 0.43, 95% CI [75.4, 81.8]."

Additional information to consider including: - Direction of the effect (higher or lower than hypothesized value) - Whether assumptions were met (e.g., normality) - Whether the test was one-tailed or two-tailed

APA Style Reporting

For APA style papers (7th edition), report the one-sample t-test results as follows:

We conducted a one-sample t-test to examine whether [variable] differed from [hypothesized value]. 
Results indicated that the sample mean (M = [value], SD = [value]) was significantly 
[higher/lower than/different from] the hypothesized value of [μ₀], 
t([df]) = [t-value], p = [p-value], d = [effect size], 95% CI [lower, upper].

Reporting in Tables

When reporting multiple one-sample t-test results in a table, include these columns: - Variable tested - Hypothesized value (μ₀) - Sample mean and standard deviation - t-value - Degrees of freedom - p-value - Effect size (Cohen’s d) - 95% confidence interval

Quiz

Test Your Understanding

1. What does the one-sample t-test primarily compare?
   • 1. Two sample means to each other
   • 2. A sample mean to a hypothesized population value
   • 3. A sample median to a hypothesized value
   • 4. Two population variances
2. What is the formula for the degrees of freedom in a one-sample t-test?
   • 1. n, where n is the sample size
   • 2. n - 1, where n is the sample size
   • 3. n - 2, where n is the sample size
   • 4. n + 1, where n is the sample size
3. A researcher finds t(18) = 2.65, p = 0.016 when testing if a sample differs from a hypothesized value. What can they conclude?
   • 1. There is no significant difference from the hypothesized value
   • 2. There is a significant difference from the hypothesized value
   • 3. The test is invalid
   • 4. More data is needed
4. What sample size would you need to detect a medium effect size (d = 0.5) with 80% power in a one-sample t-test?
   • 1. Approximately 15
   • 2. Approximately 34
   • 3. Approximately 64
   • 4. Approximately 200
5. Which assumption becomes less critical for the one-sample t-test when the sample size is large (n > 30)?
   • 1. Random sampling
   • 2. Independent observations
   • 3. Normality of the data
   • 4. Continuous measurement scale

Answers: 1-B, 2-B, 3-B, 4-B, 5-C

Common Questions About the One-Sample t-Test

When should I use a one-sample t-test versus a two-sample t-test?

Use a one-sample t-test when comparing a single sample to a known or hypothesized value. Use a two-sample t-test (independent or paired) when comparing two separate samples to each other.

What if my data isn’t normally distributed?

If your sample size is large (n > 30), the t-test is generally robust to violations of normality due to the Central Limit Theorem. For smaller samples with non-normal data, consider using the non-parametric alternative, the one-sample Wilcoxon signed-rank test.

How do I report one-sample t-test results in research papers?

Include: t-value, degrees of freedom, p-value, sample mean and standard deviation, effect size (Cohen’s d), and the 95% confidence interval. For example: “The sample (M = 25.3, SD = 4.2) was significantly higher than the test value of 20, t(29) = 6.94, p < .001, d = 1.27, 95% CI [23.7, 26.9].”

What’s the difference between one-tailed and two-tailed tests?

A two-tailed test examines whether the sample mean differs from the hypothesized value in either direction (higher or lower). A one-tailed test examines only one direction (specifically higher than or lower than). Two-tailed tests are more conservative and generally recommended unless there’s a strong theoretical reason for testing only one direction.

How do I choose the hypothesized value to test against?

The hypothesized value (μ₀) should be based on: - Theoretical expectations or previous research findings - Industry standards or regulatory thresholds - Claimed values that need verification - Historical averages or baseline measurements - Meaningful reference points for your specific context

Can I use the one-sample t-test with very small samples?

Yes, the one-sample t-test can be used with small samples, but its reliability depends on how closely your data follows a normal distribution. With very small samples (n < 10), examine the data carefully for normality, or consider non-parametric alternatives if you have concerns about the distribution.

Examples of When to Use the One-Sample t-Test

1. Quality control: Testing if product measurements meet specifications
2. Educational assessment: Comparing class performance to national standards
3. Medical research: Testing if a treatment changes metrics from baseline values
4. Manufacturing: Validating if production output meets target levels
5. Environmental science: Comparing pollution levels to regulatory thresholds
6. Psychology: Testing if a group’s scores differ from normative values
7. Consumer research: Verifying if customer ratings meet expected satisfaction levels
8. Finance: Testing if investment returns differ from market benchmarks
9. Sports science: Comparing athletic performance to established standards
10. Agriculture: Testing if crop yields meet expected production levels

Step-by-Step Guide to the One-Sample t-Test

1. Check Assumptions

Before interpreting the results, verify these assumptions:

1. Random sampling: The data should represent a random sample from the population
2. Normality: The data should follow an approximately normal distribution
   • Check using the Q-Q plot in the ‘Assumptions’ tab
   • With larger samples (n > 30), the t-test is robust to normality violations
3. Independence: The observations should be independent of each other

2. Interpret the Results

1. Check the p-value:
   • If p < 0.05, there is a statistically significant difference between the sample mean and the hypothesized value
   • If p ≥ 0.05, there is not enough evidence to conclude the sample mean differs from the hypothesized value
2. Examine the confidence interval:
   • If it doesn’t include the hypothesized value, the difference is statistically significant
   • The width indicates precision of the estimated mean
3. Assess the effect size:
   • Cohen’s d indicates the practical significance of the difference
   • Consider whether the magnitude of the effect is meaningful in your context

Further Reading and Related Tools

If you found this calculator helpful, you may also be interested in these related tools:

• Independent Samples t-Test Calculator - For comparing two separate groups
• Paired Samples t-Test Calculator - For comparing matched or repeated measurements
• One-Sample Wilcoxon Test Calculator - Non-parametric alternative to the one-sample t-test
• One-Sample Z-Test Calculator - For large samples when population standard deviation is known
• Confidence Interval Calculator - For estimating population parameters from sample statistics
• Normality Test Calculator - For checking if your data meets t-test assumptions

References

• Student. (1908). The probable error of a mean. Biometrika, 6(1), 1-25.
• Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
• Cumming, G. (2014). The new statistics: Why and how. Psychological Science, 25(1), 7-29.
• Sawilowsky, S. S. (2009). New effect size rules of thumb. Journal of Modern Applied Statistical Methods, 8(2), 597-599.
• Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4, 863.
• Lumley, T., Diehr, P., Emerson, S., & Chen, L. (2002). The importance of the normality assumption in large public health data sets. Annual Review of Public Health, 23(1), 151-169.