Settlement of two species of mussel larvae was studied. In a wave exposed location, a
collector pad yielded 108 larvae of Mytilus trossulus and 72 of M. californianus. In a wave
sheltered location, a collector pad yielded 54 larvae of M. trossulus and 16 of M.
californianus. Test for an association between wave energy and species. Use a 5% level of
significance.
To test for an association between wave energy and species in this study, we can use a chi-square test of independence. This test will help us determine whether there is a significant relationship between the two variables (wave energy and species) or if any observed differences could be due to chance.
First, let's summarize the data in a contingency table:
Wave Exposed Location | Wave Sheltered Location | Total
M. trossulus | 108 | 54 | 162
M. californianus | 72 | 16 | 88
Total | 180 | 70 | 250
Now, we can state our null and alternative hypotheses:
Null hypothesis (H0): There is no association between wave energy and species.
Alternative hypothesis (Ha): There is an association between wave energy and species.
To perform the chi-square test, we calculate the expected frequencies for each cell under the assumption that there is no association between the variables.
Expected frequency (E) = (row total * column total) / grand total
Wave Exposed Location | Wave Sheltered Location | Total
M. trossulus | (162 * 180) / 250 | (162 * 70) / 250 | 162
M. californianus | (88 * 180) / 250 | (88 * 70) / 250 | 88
Total | 180 | 70 | 250
Now, we can calculate the chi-square statistic:
chi-square = Σ[(O - E)² / E]
where Σ represents the sum of the values, O is the observed frequency, and E is the expected frequency.
chi-square = [(108-116.64)² / 116.64] + [(72-63.36)² / 63.36] + [(54-43.92)² / 43.92] + [(16-24.08)² / 24.08]
Finally, we compare the chi-square value to the critical chi-square value at a specified significance level (5% in this case) and degrees of freedom (df = (number of rows - 1) * (number of columns - 1)). If the calculated chi-square value is greater than the critical value, we reject the null hypothesis and conclude that there is an association between wave energy and species.
For df = (2-1) * (2-1) = 1, the critical chi-square value at a 5% level of significance is 3.841.
If the calculated chi-square value is less than or equal to the critical value, we fail to reject the null hypothesis and conclude that there is insufficient evidence of an association between wave energy and species.
Please note that the actual calculations for the observed and expected frequencies, as well as the chi-square statistic, may differ slightly depending on the specific software or calculator being used.