Question 1: The Association between Gene Mutation and Lung Cancer (2x2 table)

A biomedical study was conducted to examine whether there is an association between a specific gene mutation (Gene X) and lung cancer occurrence. The researchers collected the following data:

Cancer Present Cancer Absent
Mutation Present 45 30
Mutation Absent 20 105

Is there a significant association between the mutation in Gene X and the occurrence of lung cancer? Please report your test statistic, the degrees of freedom, the p-value, and your interpretation of the result.


Answer to Question 1:


  • Test statistic: 41.37
  • Degree of freedom: 1
  • p-value: P<0.0001
  • Conclusion: The results of Chi-squared test indicate a significant association between the variables (cancer presence and mutation presence/absence). The calculated chi-squared value was 39.386 with 1 degree of freedom, resulting in a very low p-value (P<0.0001). These findings provide strong evidence to reject the null hypothesis and suggest a significant association between the mutation and the occurrence of cancer.


(Manual calculation)

The first step in solving a Chi-squared problem is to calculate the expected counts. For a 2x2 table, this can be done by taking (row total * column total) / grand total for each cell.

For the “Mutation Present and Cancer Present” cell, the expected count would be (75 * 65) / 200 = 24.375 For the “Mutation Present and Cancer Absent” cell, the expected count would be (75 * 135) / 200 = 50.625 For the “Mutation Absent and Cancer Present” cell, the expected count would be (125 * 65) / 200 = 40.625 For the “Mutation Absent and Cancer Absent” cell, the expected count would be (125 * 135) / 200 = 84.375

The test statistic is calculated as the sum of (observed-expected)^2/expected over all cells.

Chi-squared = (45-24.375)^2/24.375 + (30-50.625)^2/50.625 + (20-40.625)^2/40.625 + (105-84.375)^2/84.375 = 20.589

In a 2x2 table, the degrees of freedom are (2-1)(2-1) = 1

Using a Chi-squared distribution table, we can compare our test statistic with the critical value for our degrees of freedom (which is 3.841 for α = 0.05). Since our test statistic is greater than the critical value, we reject the null hypothesis that gene mutation and cancer occurrence are independent.



Question 2: Exploring the Relationship between Cancer Type and Genetic Variants (RxC table)

A comprehensive genomics study was performed to understand if there is a significant association between four different types of cancer (Lung, Breast, Colorectal, and Prostate) and the presence of five distinct genetic variants (Variant A, B, C, D, E). The following data was collected:

Variant A Variant B Variant C Variant D Variant E
Lung Cancer 32 40 45 38 35
Breast Cancer 35 38 50 33 34
Colorectal Cancer 40 42 55 50 60
Prostate Cancer 50 48 45 55 52

Is there a significant association between the mutation in Gene X and the occurrence of lung cancer? Please report your test statistic, the degrees of freedom, the p-value, and your interpretation of the result.


Answer to Question 2:


  • Test statistic: 9.714
  • Degree of freedom: 12
  • p-value: 0.6410
  • Conclusion: The Chi-squared test result yielded a p-value of 0.6410, which is greater than the significance level of 0.05. This indicates no significant association between the type of cancer and the genetic variants in our sample.



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