e-Statistics
Introduction
  • Advanced User Guide
    Assignment & Quiz
  • User Account
  • Guide to Worksheet 1-2
  • Guide to Worksheet 3-4
  • Guide to Worksheet 5-6
  • Guide to Worksheet 7-8
    Data in Statistical Studies
  • Overview
  • Data from SMDA
  • Data and Story Library
    Exploratory Graphics
  • Overview
  • Histogram
  • Stem and Leaf Plot
  • Boxplot
  • Scatter Plot
  • QQ plot
  • Bar Graph
    Descriptive Statistics
  • Overview
  • Summary Statistics
  • Grouped Data
  • Frequency Table
    Discrete Distributions
  • Frequency Function
  • Binomial Distribution
  • Normal Approximation
    Normal Distribution
  • Normal Density Function
  • Critical Point
  • Z-score
  • Central Limit Theorem
    Inference on Mean
  • t-Distribution
  • Confidence Interval
  • Hypothesis Test
  • Power of Test
  • One-sided CI
    Comparison of Two Groups
  • Overview
  • t-Test
  • Nonparametric Test
  • Test for Paired Data
    Inference on Variances
  • Chi-square Distribution
  • Testing a Standard Deviation
  • F-Distribution
  • F-Test
    Explore Statistics in Drug Safety

    • t-Distribution

    The t distribution is symmetric but comparatively flatter (see the solid line in the graph below) than the standard normal distribution (the dashed line below). The shape of particular t-distribution is determined by the degrees of freedom (df) = .

    We can calculate the critical region corresponding to the level $ \alpha$ .

    Level (p-value) $ \alpha =$
    Right-tailed region $ T > t_{\alpha,df} =$
    Two-sided region $ \vert T\vert > t_{\alpha/2,df} =$
    Left-tailed region $ T < -t_{\alpha,df} =$

    When the sample mean $ \bar{X}$ and the sample standard deviation $ S$ are obtained from the data of $ n$ observations, it is often assumed that the statistic $ T = \displaystyle\frac{\bar{X}-\mu}{S/\sqrt{n}}$ has the t distribution with $ df = n-1$ degrees of freedom if $ \mu$ is the true population mean. The appropriateness of this assumption can be ensured if (a) the sample distribution is approximately normal (the use of QQ plot is recommended), or (b) the sample size $ n$ is adequately large (as a rule of thumb it is desirable to have $ n \ge 30$ ).

    Conversely when the statistic $ T =$ is given, we can find the corresponding $ \alpha$ so that the value $ T$ belongs to the critical region, and call it p-value.