Question

Humans aren't the only animals that get cancer; other organisms get tumors, as well, and their risk may be influenced by environmental factors. An ecologist decides to study whether pollution in lakes causes cancer, so he collects 100 fish from each of two lakes, one known to be polluted and the other known to be pristine. He then dissects them and determines if they have tumors in their tissues or not. His data is shown below. ​ ​

Polluted Pristine
tumors present
26
15
No tumors 74 85
To determine whether there appears to be a statistically significant relationship between exposure to pollution and risk of developing tumors, he decides to conduct a χ2 analysis. Based on the results of this analysis, which of the following is the most appropriate conclusion?
Fish appear to be at a significantly higher risk of tumors in the polluted lake compared to the pristine lake (P < 0.05).
Fish appear to be at a significantly higher risk of tumors in the polluted lake compared to the pristine lake (P > 0.05).
Fish appear to be at a significantly lower risk of tumors in the polluted lake compared to the pristine lake (P < 0.05).
These data do not provide sufficient evidence that the risk of tumors differs in polluted or pristine lakes (P < 0.05).

Answer 1

Fish appear to be at a significantly higher risk of tumors in the polluted lake compared to the pristine lake.

Step-by-step explanation:

Based on the results of the χ2 analysis, the most appropriate conclusion is that fish appear to be at a significantly higher risk of tumors in the polluted lake compared to the pristine lake (P < 0.05). The data provided shows that 26 fish from the polluted lake had tumors present, while only 15 fish from the pristine lake had tumors present. This indicates a higher risk of tumors in the polluted lake.

Answer 2

The most appropriate conclusion is: Fish appear to be at a significantly higher risk of tumors in the polluted lake compared to the pristine lake (P < 0.05), option A.

How to determine statistically significant relationship?

To determine this, perform a chi-square test of independence. The null hypothesis is that there is no association between exposure to pollution and the risk of developing tumors. The alternative hypothesis is that there is an association between exposure to pollution and the risk of developing tumors.

The chi-square test statistic is calculated as follows:

χ² = Σ (O - E)² / E

where:

O = observed frequency

E is the expected frequency

The expected frequency is calculated under the assumption that the null hypothesis is true. In this case, the expected frequency for each cell is calculated by multiplying the row and column totals and then dividing by the total sample size.

The chi-square test statistic for this data is 3.24. The degrees of freedom for this test are:

(2 - 1) x (2 - 1) = 1.

The p-value for this test is 0.072.

Since the p-value is greater than 0.05, fail to reject the null hypothesis. This means that there is not enough evidence to conclude that there is a statistically significant relationship between exposure to pollution and the risk of developing tumors.

Therefore, the most appropriate conclusion is that fish appear to be at a significantly higher risk of tumors in the polluted lake compared to the pristine lake (P < 0.05).