Experienced observers use aerial survey methods to estimate the number of snow geese in their summer range area west of Hudson Bay, Canada. A small aircraft flies over the range, and when a flock of geese is spotted, the observer estimates the number of geese in the flock.
To investigate the reliability of the estimates, an airplane carrying two goose observers flies over 45 flocks. Each observer makes an independent estimate of the number of geese in each flock. A photograph is taken of each flock and a count made of the number of geese in the photograph. The sample data for the 45 flocks appear in the DataView tool. [Data source: These data were obtained from Lunneborg, C. E. (1994). Modeling experimental and observational data. Pacific Grove, CA: Duxbury Press.]
Variables = 3
Observations = 45
Count from a Photo and Observers’ Estimates of the Size of Flocks of Geese
C. E. Lunneborg, Modeling Experimental and Observational Data, Duxbury Press, 1994
You will work with goose observer A’s estimates in this problem to examine how well observer A’s estimates predict counts from the associated photographs for the same flock. The photographs provide a highly accurate count of geese; optimally, the observer’s estimate would predict the photo-based count for a specific flock.
First, use the regression equation to predict Y values based on observer A’s estimates. The regression equation, in the format Ŷ = bX + a, is:
Ŷ = 0.88X + 26.65
where X = goose observer A’s estimate,
Ŷ = an estimate of the goose count from the photograph
In this problem, Y is the actual count of geese in the photograph.
Note: The estimated regression equation can also be obtained by going to the Correlation section in the DataView tool, specifying the proper dependent (Y) and independent (X) variables, and clicking on the Linear Regression button.
Find the predicted Y values for the flocks 2, 17, and 35. Ŷ for flock 2 is . Ŷ for flock 17 is . Ŷ for flock 35 is . You will need to use the Observations list in the DataView tool to identify goose observer A’s estimate for the appropriate flock. Click on the Observations button in the tool, and scroll to the appropriate flock number.
Calculate the residuals for the flocks identified. The residual for flock 2 is . The residual for flock 17 is . The residual for flock 35 is .
You want to test the significance of this regression equation. The null hypothesis can be phrased as:
The intercept of the regression equation is greater than zero.
The regression equation accounts for a significant portion of the variance in the y scores (counts from the photos).
The slope of the regression equation is greater than zero.
The regression equation does not account for a significant portion of the variance in the y scores (counts from the photos).
The Pearson correlation is r = 0.8662, SSXX = 327,086.00, and SSYY = 339,559.64. Calculate the predicted variability, the unpredicted variability, and the percentage of the variance explained by the regression equation mentioned previously. The predicted variability is . The unpredicted variability is . The percentage of the variance explained is .
Use the Distributions tool to help you test the null hypothesis.(Note: Do not use the F-table to calculate the critical value as the answer requires an of an exact df to calculate the value.)
Numerator Degrees of Freedom = 26
Denominator Degrees of Freedom = 26
To test the null hypothesis, you will first need to find the critical value of F at alpha = 0.05. F is . Next, calculate the F-ratio. The F-ratio is . Therefore, the null hypothesis is . On the basis of these results, you conclude that the regression equation accounts for a significant portion of the variance in the y scores (counts from the photos).