scoresvideos
Honors Statistics
Table of Contents
Unit 11 Overview: The Chi–Square Distribution
11.1 Facts About the Chi-Square Distribution
11.2 Goodness-of-Fit Test
11.3 Test of Independence
11.4 Test for Homogeneity
11.5 Comparison of the Chi-Square Tests
11.6 Test of a Single Variance
11.7 Lab 1: Chi-Square Goodness-of-Fit
11.8 Lab 2: Chi-Square Test of Independence

The goodness-of-fit test helps determine if observed data matches a specific theoretical distribution. It uses a chi-square test statistic to compare observed frequencies with expected frequencies, allowing researchers to assess the fit of data to a hypothesized distribution.

Interpreting results involves comparing the test statistic to a critical value or examining the p-value. Rejecting the null hypothesis suggests the data doesn't follow the specified distribution, while failing to reject indicates insufficient evidence against the distribution's fit.

Goodness-of-Fit Test

Goodness-of-fit test for distributions

  • Determines if observed data matches a specific theoretical distribution (uniform, normal, binomial, Poisson, multinomial distribution)
  • Null hypothesis ($H_0$): Data follows the specified distribution
  • Alternative hypothesis ($H_a$): Data does not follow the specified distribution
  • Test procedure:
    1. Calculate expected frequencies for each category based on the theoretical distribution
    2. Compute the test statistic using observed and expected frequencies
    3. Determine the critical value using the significance level and degrees of freedom
    4. Compare the test statistic to the critical value and decide whether to reject or fail to reject the null hypothesis

Test statistic in chi-square distribution

  • Goodness-of-fit test statistic follows a chi-square distribution
  • Chi-square test statistic formula: $\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}$
    • $\chi^2$: Chi-square test statistic
    • $O_i$: Observed frequency for category $i$
    • $E_i$: Expected frequency for category $i$
    • $k$: Number of categories
  • Degrees of freedom: $df = k - 1$
    • Additional degrees of freedom lost if distribution parameters are estimated from the data

Interpretation of goodness-of-fit results

  • Goodness-of-fit test is a right-tailed test
    • Large test statistic values provide evidence against the null hypothesis
  • Interpreting results:
    • Compare calculated test statistic to the critical value from the chi-square distribution
      • Reject $H_0$ if test statistic > critical value
      • Fail to reject $H_0$ if test statistic < critical value
    • Alternatively, calculate the p-value and compare it to the significance level
      • Reject $H_0$ if p-value < significance level
      • Fail to reject $H_0$ if p-value > significance level
  • Rejecting $H_0$: Sufficient evidence to suggest data does not follow the specified distribution
  • Failing to reject $H_0$: Insufficient evidence to suggest data does not follow the specified distribution

Additional Considerations

  • Contingency tables are often used to organize categorical data for goodness-of-fit tests
  • Sample size affects the power of the test and the reliability of results
  • Effect size measures the magnitude of the difference between observed and expected frequencies

Key Terms to Review (22)

Null Hypothesis: The null hypothesis, denoted as H0, is a statistical hypothesis that states there is no significant difference or relationship between the variables being studied. It represents the default or initial position that a researcher takes before conducting an analysis or experiment.
Sample Size: Sample size refers to the number of observations or data points collected in a study or experiment. It is a crucial aspect of research design and data analysis, as it directly impacts the reliability, precision, and statistical power of the conclusions drawn from the data.
Normal Distribution: The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetrical and bell-shaped. It is a fundamental concept in statistics and probability theory, with widespread applications across various fields, including the topics covered in this course.
Uniform Distribution: The uniform distribution is a continuous probability distribution where the probability of any outcome within a specified range is equally likely. It is characterized by a constant probability density function over a defined interval.
Categorical Data: Categorical data refers to variables that can be classified into distinct groups or categories. These variables do not have a numerical value, but rather represent qualitative characteristics or attributes.
Contingency Table: A contingency table, also known as a cross-tabulation or cross-tab, is a type of table that displays the frequency distribution of two or more categorical variables. It allows for the analysis of the relationship between these variables and is a fundamental tool in various statistical analyses.
Binomial Distribution: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes: success or failure. It is a fundamental concept in probability theory and statistics, with applications across various fields.
Expected Frequency: Expected frequency refers to the anticipated or predicted number of observations in each category or cell of a contingency table, assuming the null hypothesis is true. It is a crucial concept in various statistical tests, including the goodness-of-fit test, test of independence, and chi-square goodness-of-fit analysis.
Degrees of Freedom: Degrees of freedom (df) is a fundamental statistical concept that represents the number of independent values or observations that can vary in a given situation. It is an essential parameter that determines the appropriate statistical test or distribution to use in various data analysis techniques.
Chi-Square Distribution: The chi-square distribution is a probability distribution that arises when independent standard normal random variables are squared and summed. It is a continuous probability distribution that is widely used in statistical hypothesis testing, particularly in assessing the goodness of fit of observed data to a theoretical distribution, testing the independence of two attributes, and testing the homogeneity of multiple populations.
Poisson Distribution: The Poisson distribution is a discrete probability distribution that describes the number of events occurring in a fixed interval of time or space, given that these events happen with a known average rate and independently of the time since the last event. It is commonly used to model rare events that occur randomly and independently over time or space.
Alternative Hypothesis: The alternative hypothesis, denoted as H1 or Ha, is a statement that contradicts the null hypothesis and suggests that the observed difference or relationship in a study is statistically significant and not due to chance. It represents the researcher's belief about the population parameter or the relationship between variables.
Test Statistic: A test statistic is a numerical value calculated from a sample data that is used to determine whether to reject or fail to reject the null hypothesis in a hypothesis test. It is a crucial component in various statistical analyses, as it provides the basis for making inferences about population parameters.
Critical Value: The critical value is a threshold value in statistical analysis that determines whether to reject or fail to reject a null hypothesis. It is a key concept in hypothesis testing and is used to establish the boundaries for statistical significance in various statistical tests.
P-value: The p-value is a statistical measure that represents the probability of obtaining a test statistic that is at least as extreme as the observed value, given that the null hypothesis is true. It is a crucial component in hypothesis testing, as it helps determine the strength of evidence against the null hypothesis and guides the decision-making process in statistical analysis across a wide range of topics in statistics.
Effect Size: Effect size is a quantitative measure that indicates the magnitude or strength of the relationship between two variables or the difference between two groups. It provides information about the practical significance of a statistical finding, beyond just the statistical significance.
Significance Level: The significance level, denoted as α, is the probability of rejecting the null hypothesis when it is true. It represents the maximum acceptable probability of making a Type I error, which is the error of concluding that an effect exists when it does not. The significance level is a critical component in hypothesis testing, as it sets the threshold for determining the statistical significance of the observed results.
Right-Tailed Test: A right-tailed test is a statistical hypothesis test where the alternative hypothesis specifies that the parameter of interest is greater than a certain value. It is used when the researcher is interested in determining if a sample statistic is significantly larger than a hypothesized population parameter.
Goodness-of-Fit Test: The goodness-of-fit test is a statistical hypothesis test used to determine whether a sample of data fits a particular probability distribution. It evaluates how well the observed data matches the expected data under a specified distribution model.
Chi-Square Test Statistic: The chi-square test statistic is a statistical measure used to determine the goodness-of-fit between an observed set of data and an expected set of data. It is a fundamental concept in hypothesis testing that helps assess whether the differences between observed and expected frequencies are statistically significant.
Observed Frequency: Observed frequency refers to the actual or empirical count of the number of occurrences of a particular event or outcome in a dataset or experiment. It is a fundamental concept in the analysis of categorical data and is central to various statistical tests, such as the goodness-of-fit test and the test of independence.
Multinomial Distribution: The multinomial distribution is a generalization of the binomial distribution, where the random variable can take on more than two possible outcomes. It is used to model the probabilities of obtaining different categories or outcomes from a single experiment with multiple possible results.