One of nature's patterns connects the percent of adult birds in a colony that return from the previous year and the number of new adults that join the colony. Following are data for 13 colonies of sparrowhawks:
Percent return 73 61 79 54 75 62 53 46 57 50 62 45 35
New adults 4 7 7 10 13 16 16 15 16 18 18 20 20
Use your calculator or software to find the correlation (± 0.001) between the percent of returning birds and the number of new birds:
Below is the scatterplot of the original data (green) and two new points added. Point A (blue): 9% return, 25 new birds. Point B (red): 40% return, 9 new birds.
Find the new correlation (± 0.001) for the original data plus Point A:
Find the new correlation (± 0.001) for the original data plus Point B:
Adding point A makes correlation
Weaker
Stronger
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To calculate the correlation between the percent of returning birds and the number of new birds, you can use the Pearson correlation coefficient formula. Here are the steps to get the answer:
Step 1: Calculate the mean for both the percent return and new adults:
- Add up all the percent return values and divide by the total number of colonies (13). This will give you the mean for percent return.
- Add up all the new adult values and divide by the total number of colonies (13). This will give you the mean for new adults.
Step 2: Calculate the deviations for each data point:
- For each percent return value, subtract the mean percent return calculated in step 1.
- For each new adult value, subtract the mean new adults calculated in step 1.
Step 3: Calculate the product of the deviations:
- Multiply each deviation for percent return with the corresponding deviation for new adults.
Step 4: Calculate the sum of the products from step 3.
Step 5: Calculate the standard deviation for both percent return and new adults:
- Calculate the square root of the sum of the squared deviations for percent return.
- Calculate the square root of the sum of the squared deviations for new adults.
Step 6: Calculate the correlation coefficient:
- Divide the sum of the products from step 4 by the product of the standard deviations calculated in step 5.
Using these steps, you can calculate the correlation coefficient for the original data. Remember to round the result to ±0.001.
To find the new correlation after adding a point, repeat the steps above, including the added point in the calculation. For each scenario (with Point A and with Point B), calculate the correlation coefficient using the updated data.
After finding the new correlations, compare them to the correlation of the original data. If the new correlation is closer to 1 or -1, it means the relationship has become stronger. If the new correlation is closer to 0, it means the relationship has become weaker.