Chi-Square Chapter 14

Chi Square Introduction A population can be divided according to gender, age group, type of personality, marital status, religion, annual income, political affiliation, education, and more. These division are known as classifications of the population. In this chapter, we will explore a technique that determines if two classifications of a population are independent (unrelated) ore dependent (related). Here are some of the questions that we can ask:

Chi Square Introduction Are annual income and education dependent or independent? If they are dependent, we would expect people with low levels of education to have low incomes and people with high levels of education to have high incomes. If they are independent, we expect people with high levels of education to have low incomes and people with low levels of education to have high incomes.

Chi Square Introduction Are movie preference and gender dependent or independent? If they are dependent, then the relationship between classifications might be that females generally prefer romance type movies and males generally prefer adventure type movies. If they are independent, then knowing one’s gender would not give you a clue to their movie preference.

Chi Square Introduction Are male personality types and incidence of heart attack dependent or independent? If they are dependent, then we might expect to find males with type A personalities more susceptible to heart attacks than others types. If they are independent, the one’s personality has no relation to the incidence of heart attack.

Chi Square Introduction The chi-square test of independence is a statistical technique used to investigate whether the classifications of a population are related. A random sample is selected from a population. The sample data is separated into different categories for each classification of the population.

Chi Square Introduction The number of responses falling into the categories for each classification are recorded into a table. These are called observed frequencies. The table is referred to as a contingency table.

A contingency table always has a row classification, and a column classification. Here, the column classification is movie preference, separated into 5 categories: romance, comedy, drama, adventure, and mystery. The row classification is gender, which is separated into two categories, male and female. Each individual table entry is called a cell.

Some observed frequencies we notice from this contingency table: The number of males who prefer adventure movies over all other types of movies is 40. The number of females who prefer romance movies over all other type of movies is 50. There are a total of 90 males. There are 45 people who prefer comedy movies over all other types of movies.

Observed vs. Expected Frequencies Consider that there are a total of 90 men and 110 women. If there are 60 people who like romance movies best… and gender and movie preference are independent, then how many males and females would you expect to like romance movies? We would expect 27 men and 33 women to like romance movies. These are called expected values.

Observed vs. Expected Frequencies The values in the contingency table are observed values, taken from a particular sample. To test whether the two classifications, gender and movie preference, are related, we’ll compare the observed frequencies to expected frequencies. The expected frequencies will be determined assuming that there is no dependence between the two classifications. Expected frequencies can be found with the formula:

Observed vs. Expected Frequencies The comparison of the observed frequencies and expected frequencies (assuming independence) is calculated using a formula: Χ 2 is called the Pearson’s chi-squared test statistic. O is the observed frequency of a cell in the table. E is the expected frequency of a cell in the table.

Properties of the Chi-Square Distribution The value of χ 2 is never negative. It is always either 0 or greater. The graph of the chi-square distribution is not- symmetric. It is skewed right and extends infinitely to the right of 0. Smaller values of χ 2 Larger values of χ 2

Chi-Square Hypothesis Test for Independence – Example 13.1 A study was conducted at the 1% level of significance to determine if there was a relationship between gender and one’s view on capital punishment. A random sample of 100 adults was selected and asked their opinion on the question: “Do you believe that the death penalty should be given to those convicted of first degree murder?” Based on these results, can we conclude that there is a relationship between gender and one’s view on capital punishment at α=1%?

Chi-Square Hypothesis Test for Independence – Example 13.1

Step 1: Formulate the hypotheses H 0 will always have the form: the population classifications are independent. i.e. the classifications have no relationship. H a will always have the form: the population classifications are dependent. i.e. the classifications have a relationship. Chi-Square Hypothesis Test for Independence – Example 13.1

Step 1: Formulate the hypotheses H 0 : The population classifications gender and one’s view on capital punishment are independent. H a : The population classifications gender and one’s view on capital punishment are dependent.

All chi-square test curves will look like similar to this, with the specific α identified, and the value for χ 2 α identified. Chi-Square Hypothesis Test for Independence – Example 13.1

Step 4: Analyze the sample data We will calculate Pearson’s χ 2 statistic using the TI- 83/84. Before running the test, we will input the contingency table into a matrix on the calculator. In math, a matrix is a collection of quantities arranged by rows and columns. We can use the views on capital punishment table to build a matrix. The matrix will not contain row or column totals. Chi-Square Hypothesis Test for Independence – Example 13.1

Practice with using the Matrix feature on the TI 83/84 You can input a matrix into your calculator by pressing: 2 nd -> Matrix-> Edit -> 1:[A] -> enter This matrix will have 2 rows and 3 columns: 2 x 3 Enter the numbers and use the arrow keys to move through the matrix

Step 4: Analyze the sample data - continued To run the χ 2 –Test on the calculator: STAT->TESTS->C: χ 2 –Test To put a new matrix name into the χ 2 –Test entry screen: 2 nd -> Matrix ->1:[A] Scroll down to “Calculate” and hit enter. The test statistic is the χ 2 value at the top, and the p- value is just under it. Chi-Square Hypothesis Test for Independence – Example 13.1

Chi-Square Hypothesis Test for Independence (p-Values) – Example 13.2 A study was conducted to determine if there is a significant relationship between the frequency of times meat is served as a main meal per month for individuals living in the eastern, central, and western United States. A questionnaire was administered to a random sample of 300 families and the results are summarized in the table.

Do these sample results indicated that there is a significant relationship between the frequency of meat served as a main meal per month and geographic living area at α=5%? Step 1: Formulate the hypotheses H 0 : The population classifications frequency of meat served as a main meal per month and geographic living area are independent. H a : The population classifications frequency of meat served as a main meal per month and geographic living area are dependent. Chi-Square Hypothesis Test for Independence (p-Values) – Example 13.2

Step 2: Determine the model to test the null hypothesis, H 0. The hypothesis testing model is a chi-square distribution. Chi-Square Hypothesis Test for Independence (p-Values) – Example 13.2

Step 3: Formulate the decision rule The level of significance is α=5%. Decision Rule: Reject H 0 if the p-value of the χ 2 test statistic less than is α=5%. Chi-Square Hypothesis Test for Independence (p-Values) – Example 13.2

Step 4. Analyze the sample data and find the p-value. STAT->Tests->C: χ 2 –Test Observed: [A] where [A] is the matrix with the observed frequencies. Expected: [B] This is where the calculator will put the expected frequencies. Calculate -> Enter χ 2 = P-value = 3.32E-4 =

Chi-Square Hypothesis Test for Independence (p-Values) – Example 13.2 Step 5. State the conclusion. Since the p-value of the χ 2 test statistic, , is less than α=5%, we reject H 0 and accept H a at α=5%. Therefore, there is a statistically significant relationship between the frequency of times meat is served as a main meal and one’s geographic living location at α=5%. The classifications are dependent.

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