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MANE 6313

Week 3, Module B

Student Learning Outcome

Analyze simple comparative experiments and experiments with a single factor.

Module Learning Outcome

Review of (statistical) hypothesis testing.

Hypothesis Testing

  • A Statistical Hypothesis is a statement about the values of the parameters of a probability distribution. For example:
\[ \begin{aligned} H_0 : \mu &=& 1.500\\ H_A : \mu & \neq & 1.500\end{aligned} \]

  • The Null Hypothesis is given by \(H_0\) and is assumed to be true

  • The Alternative Hypothesis is given by \(H_A\). We are trying to gather evidence to support the claim of the alternative hypothesis

  • Hypotheses may be either two-sided or one-sided

Hypothesis Testing

  • You should be familiar with hypothesis testing.

  • You are responsible for the following material:

    • Tests on means with variance known (Table 2-4)

    • Tests on means of normal distribution with variance unknown(Table 2-4)

    • Paired comparison test (section 2-5)

    • Tests on variances of normal distribution (Table 2-8)

Overview: Conducting a Test of Hypothesis

  1. Take a random sample from the population under study

  2. Compute the appropriate statistic

  3. Decide to either reject or fail to reject \(H_0\)

The set of values of the test statistic leading to the reject of \(H_0\) is called the rejection region or critical region.

Errors in Hypothesis Testing

  • A Type I error occurs when the null hypothesis is true but the decision is made to reject \(H_0\)
\[ \alpha=P\left\{\mbox{type I error}\right\} = P\left\{\mbox{reject }H_0|H_0\mbox{ is true}\right\} \]
  • A Type II error occurs when the null hypothesis is false but the decision is made not to reject \(H_0\)
\[ \beta=P\left\{\mbox{type II error}\right\} = P\left\{\mbox{ fail to reject }H_0|H_0\mbox{ is false}\right\} \]

Decision Errors

Source[^3]

Questions

  1. Which type of error is the producer's risk?

  2. Which type of error is the consumer's risk?

Classical Approach

Steps taken from [^1]

  1. Find parameter of interest
  2. State null hypothesis, \(H_0\)
  3. State alternative hypothesis, \(H_1\)
  4. Calculate test statistic
  5. Construct rejection region
  6. State conclusion(s)

P-value Approach

  • Similar in structure to classical approach

  • Find parameter of interest

  • State null hypothesis, \(H_0\)
  • State alternative hypothesis, \(H_1\)
  • Calculate \(p\)-value

  • State conclusion(s)

  • Decision rule

  • If \(p\)-value < \(\alpha\) reject \(H_0\)

  • else (\(p\)-value > \(\alpha\)) fail to reject \(H_0\)

[^1]: Montgomery and Runger (2014). Applied Statistics and Probability for Engineers, 6th edition. John Wiley & Sons.

[^3]: DeVor, Chang, Sutherland (2007). Statistical Quality Design and Control: Contemporary Concepts and Methods, 2nd edition. Pearson: Prentice-Hall.