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Math  /  Data & Statistics

QuestionA researcher studied the relationship between the number of times a certain species of cricket will chirp in one minute and the temperature outside. Her data is expressed in the scatter plot and line of best fit below. Calculate the residual for the data point that corresponds to 70 chirps in one minute and a temperature of 56.3∘F56.3^{\circ} \mathrm{F}.

Studdy Solution

STEP 1

1. The line of best fit is a linear equation derived from the points (60, 53) and (80, 59).
2. The residual is the difference between the observed value and the predicted value from the line of best fit.

STEP 2

1. Determine the equation of the line of best fit.
2. Calculate the predicted temperature for 70 chirps using the line of best fit.
3. Calculate the residual for the data point (70, 56.3).

STEP 3

Find the slope m m of the line of best fit using the points (60, 53) and (80, 59).
m=59−5380−60=620=0.3 m = \frac{59 - 53}{80 - 60} = \frac{6}{20} = 0.3

STEP 4

Use the point-slope form to find the equation of the line. Using point (60, 53):
y−53=0.3(x−60) y - 53 = 0.3(x - 60)
Simplify to get the equation in slope-intercept form:
y=0.3x+35 y = 0.3x + 35

STEP 5

Calculate the predicted temperature for 70 chirps using the line equation y=0.3x+35 y = 0.3x + 35 .
y=0.3(70)+35=21+35=56 y = 0.3(70) + 35 = 21 + 35 = 56

STEP 6

Calculate the residual for the data point (70, 56.3).
Residual=Observed value−Predicted value=56.3−56=0.3 \text{Residual} = \text{Observed value} - \text{Predicted value} = 56.3 - 56 = 0.3
The residual for the data point (70, 56.3) is:
0.3 \boxed{0.3}

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