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 the Chi-square test of independence. This test will help determine if there is a significant relationship between the two variables.
Step 1: Set up hypotheses
- Null hypothesis (H₀): There is no association between wave energy and species.
- Alternative hypothesis (H₁): There is an association between wave energy and species.
Step 2: Calculate the expected values
To perform the Chi-square test, we need to calculate the expected values for each cell in the contingency table. Expected values can be calculated using the formula:
Expected value = (row total * column total) / grand total
Let's calculate the expected values:
For the wave exposed location:
- Expected value for Mytilus trossulus: (180 * 162) / 342 = 85.7
- Expected value for M. californianus: (180 * 162) / 342 = 94.3
For the wave sheltered location:
- Expected value for Mytilus trossulus: (70 * 162) / 342 = 33.0
- Expected value for M. californianus: (70 * 162) / 342 = 37.0
Step 3: Calculate the Chi-square statistic
The chi-square statistic can be calculated using the formula:
χ² = ∑ [(Observed value - Expected value)² / Expected value]
Let's calculate the Chi-square statistic for this study:
χ² = [(108 - 85.7)² / 85.7] + [(72 - 94.3)² / 94.3] + [(54 - 33.0)² / 33.0] + [(16 - 37.0)² / 37.0]
Step 4: Determine the degrees of freedom
The degrees of freedom (df) for a Chi-square test of independence can be calculated using the formula:
df = (number of rows - 1) * (number of columns - 1)
In this case, df = (2 - 1) * (2 - 1) = 1
Step 5: Determine the critical value
To determine the critical value, we need to consult a Chi-square distribution table with the appropriate degrees of freedom and significance level (5%). From the table, we find that the critical value at a 5% level of significance for 1 degree of freedom is approximately 3.841.
Step 6: Compare the Chi-square statistic with the critical value
If the calculated Chi-square statistic is greater than the critical value, we reject the null hypothesis (H₀) and conclude that there is a significant association between wave energy and species. If the calculated Chi-square statistic is less than or equal to the critical value, we fail to reject the null hypothesis.
Step 7: Interpret the result
Based on the comparison between the calculated Chi-square statistic and the critical value, we can draw conclusions about the association between wave energy and species.
I do not have the observed values for each species in the wave exposed and wave sheltered locations, but you can use the steps described above to calculate the Chi-square statistic and determine if there is a significant association between wave energy and species.