Questions: One spring, Felipe observed a relationship between the temperature outside and the number of migrating waxwings he saw flying north. The chart shows his observations. Outside temperature (degrees F) Number of waxwings seen 64 2 68 10 72 18 76 26 What is the rate of change in the number of migrating waxwings that Felipe observed based on the temperature? Select an answer Enter an exact integer or decimal answer.

Solution Steps

The means of the temperature (\(x\)) and the number of waxwings seen (\(y\)) are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{64 + 68 + 72 + 76}{4} = 70.0 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = \frac{2 + 10 + 18 + 26}{4} = 14.0 \]

The correlation coefficient (\(r\)) is found to be:

\[ r = 1.0 \]

This indicates a perfect positive linear relationship between temperature and the number of waxwings.

The slope (\(\beta\)) is calculated using the following formulas:

Numerator for \(\beta\):

\[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 4080 - 4 \cdot 70.0 \cdot 14.0 = 160.0 \]

Denominator for \(\beta\):

\[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 19680 - 4 \cdot 70.0^2 = 80.0 \]

Thus, the slope is:

\[ \beta = \frac{160.0}{80.0} = 2.0 \]

The intercept (\(\alpha\)) is calculated as follows:

\[ \alpha = \bar{y} - \beta \bar{x} = 14.0 - 2.0 \cdot 70.0 = -126.0 \]

The line of best fit can be expressed as:

\[ y = -126.0 + 2.0x \]

The rate of change in the number of migrating waxwings based on the temperature is given by the slope:

\[ \text{Rate of change} = \beta = 2.0 \]

Final Answer