Chi-Square test for Qualitative Data



WHEN SHOULD YOU USE THE CHI SQUARE TEST?

Qualitative data

Comparing PROPORTIONS of MORE THAN TWO groups OR

comparing NUMBERS of individuals in EACH CATEGORY

Sample data

Mary read that bees were attracted to the color blue as opposed to red, yellow, or white. She wondered if crickets would show a color preference. To test her hypothesis that crickets would be differentially attracted to colors, she placed 100 crickets in a container. The bottom of the container was divided into four equal sections covered by red, blue, yellow, or white paper. She observed the number of crickets on each color paper one hour after placing them in the container. The distribution of crickets was: 30 red, 40 blue, 12 yellow, and 18 white. By chance alone, an equal number of crickets on each color of paper would be expected.

Step 1: Make a two-way cross-classification table for the results.

Counts of Crickets Found Cross-Classified by Color of Background

| |Color of Background |

| |Red |Yellow |Blue |White |Total |

|Crickets |Yes |30 |40 |12 |18 |100 |

|Present | | | | | | |

| |No |70 |60 |88 |82 |300 |

|Total |100 |100 |100 |100 |400 |

Step 2: State the null hypothesis.

There is no significant difference between the proportions of crickets found in each color background.

Step 3: Determine the expected frequency distribution.

The expected frequency for each cell is obtained by multiplying the row total of that cell by the column total of that cell and dividing by the total sample size.

Expected Frequencies for Crickets in Each Color Background:

| |Color of Background |

| |Red |Yellow |Blue |White |Total |

|Crickets |Yes |25 |25 |25 |25 |100 |

|Present | | | | | | |

| |No |75 |75 |75 |75 |300 |

|Total |100 |100 |100 |100 |400 |

*Note: All expected frequencies should be greater than 1.0 or test will not be valid.

Step 4: Establish the level of significance (0.05).

Step 5: Calculate χ2 using the graphing calculator.

1. Press 2ND and x-1. Select “EDIT”, then ENTER to display the MATRIX[A] screen.

2. Enter the number of rows, then move to the next number and enter the number of columns. (In this example, we would have 2 X 4).

3. Use ( to move the cursor to the table and enter the observed frequencies into each cell. (Press ENTER to advance to the next cell).

4. Press 2ND, MODE after entering all the cell values.

5. Press 2ND and x-1. Select “EDIT”, then use ( to move the cursor to [B]. ENTER

6. Follow steps 2 and 3 above, this time entering the values for the expected frequencies.

7. Press 2ND, MODE after entering all cell values.

8. Press STAT and select “TESTS”.

9. Select “C: χ2-Test” and ENTER.

10. Verify that Observed is matrix A and Expected is matrix B.

11. Select “Calculate” and ENTER.

Results from TI-84:

χ2 – Test

χ2 = 24.96

p = 1.5740715E-5

df = 3

Step 6: Compare the calculated value for χ2 to the critical value of χ2.

Look at a table of critical values of χ2

Step 7: Decide to reject or not reject the null hypothesis.

Calculated χ2 < critical χ2 → null hypothesis not rejected

Calculated χ2 > critical χ2 → null hypothesis is rejected

Critical χ2 at 0.05 level.= 7.88; calculated χ2 of 24.96 > 7.88.

The null hypothesis is rejected.

Step 8: Determine whether the statistical findings support the research hypothesis.

IF Null hypothesis was rejected = research hypothesis was supported

IF Null hypothesis not rejected = research hypothesis was not supported

Because the null hypothesis was rejected, the research hypothesis that crickets would be differentially attracted to colors was supported.

Step 9: Construct a data table that communicates all statistics.

Table A: Frequencies of Crickets Found in Each Color Background

|Observed Frequencies |

| |Color of Background |

| |Red |Yellow |Blue |White |Total |

|Crickets |Yes |30 |40 |12 |18 |100 |

|Present | | | | | | |

| |No |70 |60 |88 |82 |300 |

|Total |100 |100 |100 |100 |400 |

| | | | | | |

|Expected Frequencies |

| |Color of Background |

| |Red |Yellow |Blue |White |Total |

|Crickets |Yes |25 |25 |25 |25 |100 |

|Present | | | | | | |

| |No |75 |75 |75 |75 |300 |

|Total |100 |100 |100 |100 |400 |

| |

|χ2 = 24.96 |Mode: Yellow |

|df = 3 | |

|Critical χ2 = 7.88 |χ2 of 24.96 > 7.88 p = 0.00 |

Step 10: Write a paragraph describing the results.

□ Write a topic sentence stating the independent and dependent variables, and a reference to

tables and graphs.

The distribution of crickets on various colors, as compared with the expected distribution is summarized in Table A and Graph A.

□ Write sentences comparing the mode or median and frequency distribution of the groups.

Since all crickets (100) had an equal opportunity to choose each of the colors, the expected frequencies would be 25 crickets on each of the four colors. The observed frequencies showed a greater proportion of crickets choosing the blue and red backgrounds and lower proportions choosing yellow and white. The mode for the observed frequencies was blue with 40% of the crickets.

□ Write sentences describing the statistical test, level of significance, and null hypothesis.

The Chi-square test was used to test the following null hypothesis at the 0.05 level of

significance: There is no significant difference between the proportions of crickets found in each color background.

□ Write sentences comparing the calculated χ2 value with the critical χ2 value and make a

statement about rejection of the null hypothesis.

The null hypothesis was rejected (χ2 = 24.96 > 7.88; p =0.00)

□ Write sentences stating support of the research hypothesis by the data.

The data did support the research hypothesis that crickets were differentially attracted to colors of backgrounds

Step 11: Construct a histogram to illustrate the frequency distribution of the observed results.

Step 12: Write an appropriate conclusion.

• What was the purpose of the experiment?

The relative attraction of crickets to the colors red, yellow, blue and white was investigated by comparing the distribution of 100 crickets on each of these colors to the expected distribution.

• What were the major findings? (Focus on results of the statistical test)

There was a significant difference among the proportions of crickets in each color group.

• Was the research hypothesis supported by the data?

The research hypothesis that crickets would be differentially attracted to various colors was supported.

• How did your findings compare (similarities and differences) with your preliminary research?

Although the research hypothesis was supported, the findings conflicted with the documented attraction of bees to blue (Bumble 118). The findings of this experiment indicate a greater proportion of crickets were attracted to the colors yellow and red, rather than blue.

(Keep going – there’s more!)

• What possible explanations can you offer for similarities and/or differences between your results and other researchers?

Differences in results for crickets and bees could be due to differences in the habitat preferences of the two species. Based on their visual abilities, bees are attracted to flowers with blue or purple petals and ultraviolet patches that highlight the center of the flower (Freeman 916). This ability to choose flowers for nectar is not necessary for a cricket.

• What recommendations do you have for further study and for improving the experiment?

Additional investigations should be conducted to determine if the frequency of the color was a factor or if true interspecies differences occur. Adaptive benefits of the color preferences could also be determined including food procurement or prey avoidance.

NOTE: You should be able to write much more than I did. After all, you did an extensive literature review before experimentation and you are the “expert” for your topic.

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