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To determine the distribution pattern of the oyster larvae at the different sites, we need to analyze the provided data for patterns. The data given in the table is as follows:
[tex]\[ \begin{array}{|l|c|c|c|c|} \cline { 2 - 5 } & \text{Site W} & \text{Site X} & \text{Site Y} & \text{Site Z} \\ \hline \text{Day 1} & 40 & 22 & 0 & 7 \\ \hline \text{Day 2} & 3 & 1 & 14 & 26 \\ \hline \text{Day 3} & 2 & 6 & 3 & 1 \\ \hline \end{array} \][/tex]
Here are the steps to analyze the distribution pattern:
1. Calculate the overall mean density of oyster larvae:
The mean of the entire data set can be found by summing all the values and dividing by the total number of observations.
[tex]\[ \text{Overall Mean} = \frac{40 + 22 + 0 + 7 + 3 + 1 + 14 + 26 + 2 + 6 + 3 + 1}{12} = 10.42 \][/tex]
2. Calculate the overall variance of the oyster larvae density:
Variance gives us an idea of how spread out the data is around the mean.
[tex]\[ \text{Overall Variance} = 146.91 \][/tex]
3. Calculate the mean density of oyster larvae at each site:
This involves averaging the values for each site across the three days.
[tex]\[ \text{Mean at Site W} = \frac{40 + 3 + 2}{3} = 15.00 \][/tex]
[tex]\[ \text{Mean at Site X} = \frac{22 + 1 + 6}{3} = 9.67 \][/tex]
[tex]\[ \text{Mean at Site Y} = \frac{0 + 14 + 3}{3} = 5.67 \][/tex]
[tex]\[ \text{Mean at Site Z} = \frac{7 + 26 + 1}{3} = 11.33 \][/tex]
4. Calculate the variance of the oyster larvae density at each site:
[tex]\[ \text{Variance at Site W} = 312.67 \][/tex]
[tex]\[ \text{Variance at Site X} = 80.22 \][/tex]
[tex]\[ \text{Variance at Site Y} = 36.22 \][/tex]
[tex]\[ \text{Variance at Site Z} = 113.56 \][/tex]
### Interpretation
From the computed variances and means, we can determine that:
- The overall variance is relatively large compared to the mean density.
- The variances at the individual sites also vary significantly.
If the overall variance is much higher than the mean, it usually indicates a clumped distribution. This means that the larvae tend to cluster in specific areas rather than being evenly spread out.
### Conclusion
Based on the results:
- Overall Mean = 10.42
- Overall Variance = 146.91
- Means per site = [15.00, 9.67, 5.67, 11.33]
- Variances per site = [312.67, 80.22, 36.22, 113.56]
Given these values, the distribution pattern exhibited by the oyster larvae is:
C. clumped
[tex]\[ \begin{array}{|l|c|c|c|c|} \cline { 2 - 5 } & \text{Site W} & \text{Site X} & \text{Site Y} & \text{Site Z} \\ \hline \text{Day 1} & 40 & 22 & 0 & 7 \\ \hline \text{Day 2} & 3 & 1 & 14 & 26 \\ \hline \text{Day 3} & 2 & 6 & 3 & 1 \\ \hline \end{array} \][/tex]
Here are the steps to analyze the distribution pattern:
1. Calculate the overall mean density of oyster larvae:
The mean of the entire data set can be found by summing all the values and dividing by the total number of observations.
[tex]\[ \text{Overall Mean} = \frac{40 + 22 + 0 + 7 + 3 + 1 + 14 + 26 + 2 + 6 + 3 + 1}{12} = 10.42 \][/tex]
2. Calculate the overall variance of the oyster larvae density:
Variance gives us an idea of how spread out the data is around the mean.
[tex]\[ \text{Overall Variance} = 146.91 \][/tex]
3. Calculate the mean density of oyster larvae at each site:
This involves averaging the values for each site across the three days.
[tex]\[ \text{Mean at Site W} = \frac{40 + 3 + 2}{3} = 15.00 \][/tex]
[tex]\[ \text{Mean at Site X} = \frac{22 + 1 + 6}{3} = 9.67 \][/tex]
[tex]\[ \text{Mean at Site Y} = \frac{0 + 14 + 3}{3} = 5.67 \][/tex]
[tex]\[ \text{Mean at Site Z} = \frac{7 + 26 + 1}{3} = 11.33 \][/tex]
4. Calculate the variance of the oyster larvae density at each site:
[tex]\[ \text{Variance at Site W} = 312.67 \][/tex]
[tex]\[ \text{Variance at Site X} = 80.22 \][/tex]
[tex]\[ \text{Variance at Site Y} = 36.22 \][/tex]
[tex]\[ \text{Variance at Site Z} = 113.56 \][/tex]
### Interpretation
From the computed variances and means, we can determine that:
- The overall variance is relatively large compared to the mean density.
- The variances at the individual sites also vary significantly.
If the overall variance is much higher than the mean, it usually indicates a clumped distribution. This means that the larvae tend to cluster in specific areas rather than being evenly spread out.
### Conclusion
Based on the results:
- Overall Mean = 10.42
- Overall Variance = 146.91
- Means per site = [15.00, 9.67, 5.67, 11.33]
- Variances per site = [312.67, 80.22, 36.22, 113.56]
Given these values, the distribution pattern exhibited by the oyster larvae is:
C. clumped
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