A phycologist is interested in determining the proportion of algae samples from a local rivulet that belong to a particular phyla, and he believes the phyla should be uniformly distributed. A simple random sample of 60 alga was obtained, and each was categorized as being a member of Rhodophyta, Chlorophyta, or Heterokontophyta. The observed counts were 25, 25, and 10, respectively. What is the p-value of the chi-square statistic in this case?

Question

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The null hypothesis is that the proportions of algae samples from the local rivulet belonging to each phyla are all equal, i.e. Rhodophyta = Chlorophyta = Heterokontophyta = 1/3. The alternative hypothesis is that at least one proportion is not equal to the others.
We can use the chi-square test to compare the observed counts with the expected counts under the null hypothesis. The expected counts can be calculated as follows:
Expected count for Rhodophyta = (60 x 1/3) = 20
Expected count for Chlorophyta = (60 x 1/3) = 20
Expected count for Heterokontophyta = (60 x 1/3) = 20
The chi-square test statistic is calculated as:
X^2 = Σ (O-E)^2 / E
where O is the observed count and E is the expected count for each category.
Using the given data, we get:
X^2 = [(25-20)^2 / 20] + [(25-20)^2 / 20] + [(10-20)^2 / 20]
X^2 = 5 + 5 + 10
X^2 = 20
The degrees of freedom for the test are (k-1), where k is the number of categories. Here, k = 3, so df = 2.
We can find the p-value for the test from the chi-square distribution table. From the table, at df = 2 and significance level α = 0.05, the critical value of X^2 is 5.99. Since the calculated value of X^2 (20) is greater than the critical value (5.99), we reject the null hypothesis.
The p-value is the probability of getting a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. Since the calculated value of X^2 is greater than the critical value, the p-value is less than 0.05 but greater than 0.01. Therefore, the best choice is C.

A Phycologist Is Interested in Determining the Proportion of Algae