These data come from a case-controlled study to examine the potential relationship between age at first pregnancy and cervical cancer. In a case-control study a random sample of cases (i.e. people with the disease in question) and controls (i.e. people similar to those in the case group, except they do not have the disease) and the proportion of people with some potential risk factor are compared across the two groups. In this study we will be comparing the proportion of women who had their first pregnancy at or before the ages of 25, because researchers suspected that an early age at first pregnancy leads to increased risk of developing cervical cancer. The data is presented in the table below: Age at First Pregnancy 25 Cervical Cancer 42 7 Control 203 114 Were the conditions for this test met?

Question

These data come from a case-controlled study to examine the potential relationship between age at first pregnancy and cervical cancer.  In a case-control study a random sample of cases (i.e. people with the disease in question) and controls (i.e. people similar to those in the case group, except they do not have the disease) and the proportion of people with some potential risk factor are compared across the two groups.  In this study we will be comparing the proportion of women who had their first pregnancy at or before the ages of 25, because researchers suspected that an early age at first pregnancy leads to increased risk of developing cervical cancer.  The data is presented in the table below:

Age at First Pregnancy <=25	Age at First Pregnancy>25
Cervical Cancer	42	7
Control	203	114

Were the conditions for this test met?

BlackTom AI · Accepted Answer

Question restatement: The scenario describes a case-control study comparing the proportion of women who had their first pregnancy at age ≤25 versus >25, among those with cervical cancer and controls without the disease. The question asks: Were the conditions for this test met? Answer option provided: Yes, random sampling is met and all expected counts are at least 5.

Option analysis:
- Regarding random sampling in a case-control study: In a typical case-control design, cases (people with the disease) are identified, and controls (similar individuals without the disease) are selected separately. This selection is often not a simple random sample from the underlying population; controls are chosen to be comparable to cases but without the disease. Therefore, the claim that random sampling is met is generally incorrect for standard case-control designs. The study’s validity relies more on proper selection of cases and controls and on comparability between groups, rather than random sampling from the population.

- Regarding expected counts being at least 5: For a 2x2 table, the chi-square test requires that the expected counts in each cell be at least 5 to justify the approximation. We can compute the expected counts from the given data:
  - Total in cervical cancer group: 42 + 7 = 49
  - Total in control group: 203 + 114 = 317
  - Total for age ≤25: 42 + 203 = 245
  - Total for age >25: 7 + 114 = 121
  - Grand total N = 49 + 317 = 366
  - Expected count for (Cancer, ≤25): (49 * 245) / 366 ≈ 32.8
  - Expected count for (Cancer, >25): (49 * 121) / 366 ≈ 16.2
  - Expected count for (Control, ≤25): (317 * 245) / 366 ≈ 212.2
  - Expected count for (Control, >25): (317 * 121) / 366 ≈ 104.8
  All four expected counts are well above 5, so this condition is satisfied.

Conclusion given the analysis: The statement asserts both that random sampling is met and that all expected counts are at least 5. While the second part about expected counts is true, the first part about random sampling being met is not correct for a typical case-control design. Because the statement contains an incorrect component, the overall claim that the conditions for the test are met is not accurate in standard practice. In other words, the assertion is partially true (expected counts ≥5) but not fully correct due to the incorrect assumption about random sampling.

STAT_3615_20055_202501 Mod 9 HW Chi Squared Tests

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