Hypothesis Testing Single Population

Hypothesis-Testing-Single-Population

  1. State the null hypothesis, $H_0$ and the alternative hypothesis, $H_1$
  2. Choose the level of significance, $\alpha$, and the sample size, n
  3. Determine the appropriate test statistic and sampling distribution
  4. Determine the critical values that divide the rejection and non-rejection regions
  5. Collect data and compute the value of the test statistic
  6. Make the statistical decision and state the managerial conclusion.
    • If the test statistic falls into the non-rejection region, do not reject the null hypothesis $H_0$.
    • If the test statistic falls into the rejection region, reject the null hypothesis.
    • Express the managerial conclusion in the context of the problem

p-Value Approach to Testing

p-value: Probability of obtaining a test statistic more extreme ($\geq$ or$\leq$ ) than the observed sample value given $H_0$ is true

  • Also called observed level of significance
  • Smallest value of $\alpha$ for which $H_0$ can be rejected

就是把求出的值代回z/t table求概率

If p-value $< \alpha$ , reject $H_0$ If p-value $\geq \alpha$, do not reject $H_0$

If $p < 0.10$, then there is some evidence to reject $H_0$ If $p < 0.05$, then there is strong evidence to reject $H_0$ If $p < 0.01$, then there is very strong evidence to reject $H_0$ If $p < 0.001$, then there is extremely strong evidence to reject $H_0$

Errors in Making Decisions

Type I Error

Reject a true null hypothesis 原假设是对的却拒绝 The probability of Type I Error is $\alpha$

Type II Error

Fail to reject a false null hypothesis 原假设是错的却接受 Fail to reject a false null hypothesis $\beta$

Type I and Type II errors cannot happen at the same time

  • Type I error can only occur if $H_0$ is true
  • Type II error can only occur if $H_0$ is false

If Type I error probability $\alpha \uparrow$ then Type II error probability $\beta \downarrow$ If Type I error probability $\alpha \downarrow$ then Type II error probability $\beta \uparrow$

Summary

Performed Z Test for the mean ($\sigma$ known) Performed t Test for the mean ($\sigma$ unknown) Performed Z Tests about a Population Proportion Discussed critical value and p–value approaches to hypothesis testing Type I & II errors

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