a. Calculate the chi-square test statistic for the simulated table in Question 20. Since the table cell counts (at least for the concave malignant group) were assumed to be randomly generated, this test statistic can be thought of as one random sample drawn from a chi- square distribution with 1 degree of freedom.
b. Use software to generate 10,000 possible counts for the concave malignant group, using the cancer cell data. Calculate the chi-square test statistic for each of the 10,000 simulations. Create a histogram of the 10,000 chi-square test statistics. Describe the shape of this simulated chi-square distribution with 1 degree of freedom. Since 37 observations is still a fairly small sample size, the simulated distribution does not look completely like the theoretical chi- square distribution. Larger sample sizes will cause the simulated distribution to look much more like the true chi- square distribution.
c. Use statistical software to randomly generate 10,000 values from a chi-square distribution with 1 degree of freedom. Compare this distribution to the one generated in Part B. Do they look the same?
d. Repeat Part B for a larger sample size. Assume that there are 160 round and 210 concave nuclei. In addition, assume that there are 240 malignant and 130 benign cells. As in Part B, generate 10,000 possible counts for the concave malignant group for this new table. Calculate the chi-square test statistic for each of the 10,000 simulations. Create a histogram of the 10,000 chi-square test statistics. Describe the shape of this simulated chi-square distribution with 1 degree of freedom. The theoretical chi-square distribution is continuous. As shown in Part B and C, when the number of counts in each cell is small, the test statistic does not accurately follow a chi-square distribution. While the chi-square test works best with large sample sizes, most statisticians agree that the cell counts in Part B are sufficient to be modeled by the chi- square distribution.