10 Here is one way in which nature regulates the size of animal populations: high population density attracts predators, which remove a higher proportion of the population than when the density of the prey is low. One study looked at kelp perch and their common predator, the kelp bass. On each of four occasions, the researcher set up four large circular pens on sandy ocean bottoms off the coast of southern California. He randomly assigned young perch to 1 of 4 pens so that one pen had 10 perch, one pen had 20 perch, one pen had 40 perch, and the final pen had 60 perch. Then he dropped the nets protecting the pens, allowing bass to swarm in, and counted the number of perch killed after two hours. A regression analysis was performed on the 16 data points using = number of perch in pen and = proportion of perch killed. Given is computer output from the least-squares regression analysis of the perch data.
Predictor
Coef
Stdev.
t-ratio
p
Constant
0.12049
0.09269
1.30
0.215
Perch
0.008569
0.002456
3.49
0.004

S = 0.1886
R-Sq = 46.5%
R-Sq(adj) = 42.7%

What is the estimate for ? Interpret this value.
The estimate for is =0.12049=0.12049. For each increase of 1 perch, the proportion killed will increase by 0.12049.
The estimate for is = 0.008569. For each increase of 1 perch, the average proportion killed increases by 0.008569.
The estimate for is =0.008569. For each increase of 1 perch, the proportion killed will increase by 0.008569.
The estimate for is = 0.12049. For each increase of 1 perch, the average proportion killed increases by 0.12049.
The estimate for is =0.002456. For each increase of 1 perch, the average proportion killed increases by 0.002456.

The correct interpretation is: The estimate for β₁ (slope coefficient) is 0.008569. For each increase of 1 perch, the proportion killed will increase by 0.008569.

Step-by-step explanation:

Certainly! Let's break down the interpretation step by step based on the provided regression output:

1. **Constant (Intercept):**

- Coefficient (Coef) for the constant term is 0.12049.

- Interpretation: This is the estimated intercept when the number of perch (the predictor variable) is zero.

2. **Perch (Predictor Variable):**

- Coefficient (Coef) for the Perch variable is 0.008569.

- Interpretation: This is the estimated slope of the regression line, indicating the change in the response variable (proportion of perch killed) for a one-unit increase in the predictor variable (number of perch).

3. **Interpretation of Coefficients:**

- The estimate for the intercept is 0.12049. This means that when there are zero perches (hypothetically), the estimated proportion killed is 0.12049.

- The estimate for the Perch variable (slope) is 0.008569. This means that for each additional perch added, the estimated proportion killed increases by 0.008569.

4. **Standard Error (Stdev.) and t-ratio:**

- The standard error for the Perch coefficient is 0.002456.

- The t-ratio is 3.49.

- Interpretation: The t-ratio is a measure of how many standard errors the coefficient is away from zero. A higher t-ratio suggests more evidence against the null hypothesis that the true coefficient is zero.

5. **p-value:**

- The p-value for the Perch coefficient is 0.004.

- Interpretation: A p-value less than 0.05 indicates that the Perch variable is statistically significant, suggesting that there is evidence to reject the null hypothesis that the true coefficient is zero.

6. **Model Fit:**

- Standard Error (S) is 0.1886.

- R-Squared (R-Sq) is 46.5%.

- Adjusted R-Squared (R-Sq(adj)) is 42.7%.

- Interpretation: R-Squared indicates the proportion of the variability in the response variable explained by the model. In this case, about 46.5% of the variability in the proportion of perch killed is explained by the number of perch in the pens.

In summary, the estimate for the Perch variable (0.008569) indicates that for each additional perch, there is an estimated increase of 0.008569 in the proportion killed, and this estimate is statistically significant based on the p-value.