# Chapter 11: Chi – Square Goodness – of – Fit Tests

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Chapter 11: Chi – Square Goodness – of – Fit Tests
Standard: Statistical inference guides the selection of appropriate models. A. Estimation (point estimators and confidence intervals) B. Tests of significance Essential Question: Describe the display of observed or expected amount of an object distributed. Learning Target: Students will be able to construct a chart to display the distribution of expected amount versus observed amount.

Section 1 One – way table versus a two – way table:
One way: displays only the distribution of one categorical variable in one sample. Two way: distribution of one categorical variable in two or more samples or groups or the relationship between two categorical variables in one sample.

Recap If we wanted to only focus on one color of M&Ms – the proportion of this color occurring, then we would use the one sample z test. However, in this case, we are focusing on all the colors. Therefore, we want to turn to a new significance test (the chi-square goodness-of-fit test).

Comparing Observed and Expected Counts: the Chi-Square Statistic
The null hypothesis in a Chi-square will state a claim based upon the distribution of a single categorical variable. Example: look at the H0 on pp. 678 The alternate hypothesis is that the categorical variable does not have the specified distribution. Example: look at the Ha on pp. 678 You can also write in another form: pp. 679

Comparing Observed and Expected Counts: the Chi-Square Statistic
Idea behind Chi-square test Compare the observed counts from our sample with the cunts that would be expected if H0 was true. The more the observed counts differ from the expected counts, the more evidence we have against the null. The expected amounts can be obtained by taking the sample size and multiplying by each proportion of its distribution Take a look at Example on pp. 679 Another way to see numbers rather than a chart is by a bar graph We see large differences between the expected and observed values in this example, therefore we want to take a closer look. We want to calculate a statistic that measures how far apart the observed and expected values are from one another. We will use the chi-square statistic: x2 = summation of (observed – expected)2 / expected (look for further detail on pp. 680): take a look at example on pp. 680

Comparing Observed and Expected Counts: the Chi-Square Statistic
X2 is a measure of the distance of the observed counts from the expected counts. ** Remember distance values are only zero or a positive value. Large values of x2 indicates a strong evidence against the null. Small values of x2 suggests that the data is consistent with the null hypothesis. To know if the x2 is considered a large or small value, we will have to evaluate the p-value (which comes later in the chapter – so be patient).

The Chi-square Distributions and P-values
When the expected counts are all at least 5, then the sampling distribution of the x2 statistic is close to a chi-square distribution with df equal to the number of categories minus 1. The chi-square distributions are a family of distributions that take only positive values and are skewed to the right. A particular chi-square distribution is specified by giving its df. The chi-square goodness-of-fit test uses the chi-square distribution with df = # of categories – 1. Briefly look over pp. 682 below the yellow box Now take a look at the example on pp. 683 (you can also find p-value through the calculator – look in the back of the book) Remember to conclude to reject or fail to reject the null: if the p-value is less than significance level then reject, if it is greater than the fail to reject.