Chapter 12: Inference on Categorical Data

Section 12.1: Goodness-of-Fit Test

Knowledge Prerequisites

declarative knowledge (definitions)
1. Section 1.1:
  - statistic
  - parameter
  - categorical data
2. Section 2.2:
  - skewed right distributions
  - normal distributions
3. Section 9.2:
  - degrees of freedom
4. Chapter 10:
  - most of the chapter
procedural knowledge
1. Section 6.1:
  - mean of discrete random variable (a.k.a., expected value)
2. Section 7.2:
  - how to calculate critical value, z_α [Analogous to calculating χ_α²]
3. Chapter 10:
  - most of the chapter
4. Section 11.1:
  - six steps to conduct a hypothesis test
  - calculate the test-statistic using the TI83/84
conditional knowledge
1. Section 1.3:
  - explain why a random sample is preferred over a convenient sample
2. Section 5.1:
  - know how to interpret probabilities
3. Chapter 10:
  - most of the chapter
4. Section 11.1:
  - identify hypothesis test using one sample and a hypothesis test using multiple samples
  - explain the meaning of P-value and α-level
  - explain why you cannot accept H0

Learning Goals

declarative knowledge (definitions)
1. Chi-squared distribution
2. properties of the chi-squared distribution
3. χ²-statistic
4. goodness-of-fit test
5. observed counts, O_i
6. expected counts, E_i
procedural knowledge
1. calculate expected counts [using the formula given below]
2. six steps to conduct a goodness-of-fit test
3. calculate the χ₀²-statistic [using the formula given below]
4. calculate the critical value, χ_α², using the TI83/84
conditional knowledge
1. identify the conditions needed to perform goodness-of-fit test
2. interpret the results of a goodness-of-fit test
3. explain what it means to make Type I and Type II errors in a hypothesis test

Chapter 12: Required Formulas – Need to Know for Tests

Expected Counts: E_i = μ_i = np_i, for i = 1, 2, ..., k
Chi-Squared Test Statistic: , for i = 1, 2, ..., k
Reject H₀: χ₀² > χ_α²
Fail to Reject H₀: χ₀² ≤ χ_α²