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

    • Hypothesis Test

    The process of determining ``yes'' or ``no'' from the outcome of experiment is called a hypothesis test. A widely used formalization of this procedure is due to Neyman and Pearson. Suppose that a researcher is interested in whether a new drug works. Then null hypothesis may be that the drug has no effect --it is often the reverse of what he or she actually believe, why? Because the researcher hopes to reject the hypothesis and announce that the new drug leads to significant improvements. If the null hypothesis is not rejected, the researcher announces nothing and goes on to a new experiment.

    The test procedure, known as t-test, is based upon the sample mean $ \bar{X} =$ and the sample standard deviation $ S =$ from a data set of size $ n =$ . Then the discrepancy between the data set and the ``assumed'' population mean $ \mu_0$ is measured by the test statistic

    $ T = \displaystyle\frac{\bar{X} - \mu_0}{S/\sqrt{n}} =$

    Here we are interested in the plausibility of the hypothesis

    $ H_A:\hspace{0.05in}\mu$ $ \mu_0 =$

    regarding the ``true'' population mean $ \mu$ . $ H_A$ is called an alternative hypothesis, and together with null hypothesis $ H_0$ it forms the basis of hypothesis testing. $ H_0$ becomes the opposite of $ H_A$ , and is used in the context of determining whether we can reject ``$ H_0$ in favor of $ H_A$ .''

    The significance level $ \alpha =$ has to be chosen from $ \alpha = 0.01$ or $ 0.05$ ( $ \alpha = 0.1$ is not common in this particular test). Under the null hypothesis $ H_0$ , it is ``unlikely'' that the t-statistic $ T$ lies in the critical region specified in the table below. If so, it suggests significant evidence against the null hypothesis $ H_0$ .

    Hypotheses Critical region to reject $ H_0$
    $ H_0: \mu = \mu_0$ versus $ H_A: \mu \neq \mu_0$ . $ \vert T\vert > t_{\alpha/2,n-1} =$
    $ H_0: \mu \ge \mu_0$ versus $ H_A: \mu < \mu_0$ . $ T < -t_{\alpha,n-1} =$
    $ H_0: \mu \le \mu_0$ versus $ H_A: \mu > \mu_0$ . $ T > t_{\alpha,n-1} =$

    Alternatively, the p-value can be calculated so that ``p-value $ < \alpha$ '' is equivalent to the t-statistic $ T$ being observed in the critical region. When the null hypothesis $ H_0$ is rejected (i.e., p-value $ < \alpha$ ), it is reasonable to calculate the confidence interval estimating the population mean $ \mu$ .