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Chi square hand out (1)

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    Chi square hand out (1) Chi square hand out (1) Document Transcript

    • Chi-Square Test<br />What is chi-square testing?<br />
      • Identifies significant differences among the observed frequencies and the expected frequencies of a particular group
      • Attempts to identify whether any differences between the expected and observed frequencies are due to chance, or some other factor that is affecting it.
      • There are actually many types of Chi-square tests, but the most common one is the Pearson Chi-square Test.
      Terms and Definitions<br />
      • Categorical Data- 2 types
      • Numerical data- in form of numbers. (ex. 1,2,3,4)
      • Categorical data- comes in form of divisions. (ex. Yes or no)
      • Expected Frequencies
      • -values for parameters that are hypothesized to occur
      • -can be determined through:
      • 1) Hypothesizing that the frequencies are equal for each category.
      • 2) Hypothesizing the values on the basis of some prior knowledge.
      • 3) A mathematical method (see Page 3)
      Two applications of Pearson Chi-Square Test<br />
      • Chi-square test for Independence
      • -This tests whether the “category” from which the data comes from affects the data.
      • -May also be thought of as testing whether the categories in the experiment “prefer” certain kinds of data.
      • Example: Is there a difference in the car choices of male and females?
      • Chi-square test for goodness-of-fit
      • -This tests whether the observed data “fit” the expected data.
      • Example: Do the car sales this year match the car sales last year? (ie. Did we still sell around 50 blue cars? 25 red cars?)
      Requirements of the Chi-squared Test<br />1. The values of the parameters to be compared are quantitative and nominal.<br />2. There should be one or more categories in the setup.<br />3. The observations should be independent of each other.<br />4. An adequate sample size. (At least 10)<br />5. Most of the time, it is the frequency of the observations that are used.<br />Example<br />A student wants to see whether the food preferences of males and females differed. He tried to see whether males or females had a general difference in the preference for cooked and raw foods. A survey was conducted with the following results:<br />Twelve males preferred Cooked foods.<br />Eight males preferred Raw foods.<br />Five females preferred Cooked foods.<br />Five females preferred Caw foods.<br />Step 1: State the null hypothesis and the alternative hypothesis.<br />Ho: There is no significant difference between the food preferences of males and females.<br />Or<br />Food preference is independent of gender.<br /> <br />Ha: There is a significant difference between the food preferences of males and females.<br />Or<br />Food preference is affected by gender.<br />Step 2: State the level of significance. (Fish Thingy)<br />α = 0.05<br />0.05 is the level of significance for most scientific experiments.<br />Step 3: Set up a contingency table:<br />The contingency table summarizes the data.<br />The categories on the columns are the “preferences” that you are checking. <br />The categories on the rows are the “populations” whose preferences are being checked. A row total and column total is always included as well.<br />Preference Male Female Total (Row) Cooked 12 5 17 Raw 8 5 13 Total (Column) 20 10 30 <br />Step 4: Compute for the expected frequencies.<br />The chi-square test for independence usually uses the third method of getting expected frequencies.<br />Expected Frequency = (Row Total)(Column Total)<br />Grand total<br />This expected frequency is computed for EACH cell.<br />Preference Male Female Total (Row) Cooked (20)(17)/30 = 11.33 (10)(17)/30 = 5.67 17 Raw (13)(20)/30 = 8.67 (13)(10)/30 = 4.33 13 Total (Column) 20 10 30 <br />The fundamental formula for the Chi-squared test is:<br />-114300118110 <br />Where O is the observed frequencies<br />E is the expected frequencies<br />And x2 is the chi-square value<br />Step 5: Rearrange the table to show the observed and expected frequencies on the columns, and the subcategories on the rows.<br />Preference Observed Expected Chi-square Cooked Males 12 11.33 0.0396 Cooked Females 5 5.67 0.0792 Raw Males 8 8.67 0.0518 Raw Females 5 4.33 0.1037 Total 0.2743 <br />Step 6: Determine the degrees of freedom<br />The degrees of freedom is: df = (Rows – 1)(Columns – 1)<br />df = (2 – 1)(2 – 1) = 1<br />Step 7: Check the tabular Chi-squared value with your df and level of significance.<br />Checking the table, we see that the tabular chi-squared value for df = 1, and α = 0.05 is 3.841.<br />Since our calculated chi-square is less than this, the conclusion is to accept the null hypothesis. Hence, food preference is independent of gender.<br />If it were greater, we would reject the null hypothesis.<br />