Question

A scientist believes that the frequency of cricket chirping is a good predictor of the ambient temperature. A random sample produced the following data where x is the number of cricket chirps in one minute and y is the ambient temperature in Fahrenheit.

x	y
128	70
227	93
188	84
150	76
210	89
185	83
137	72
171	80

1. (a) Find an equation of the least squares regression line. Round the slope and y-intercept value to two decimal places.
2. (b) Based on the equation from part (a), what is the predicted temperature when a cricket chirps 170 times in one minute?
3. (c) Based on the equation from part (a), what is the predicted temperature when a cricket chirps 320 times in one minute?
4. (d) Which predicted temperature that you calculated for (b) and (c) do you think is closer to the true temperature and why?

Answer 1

A scientist believes that the frequency of cricket chirping is a good predictor of the ambient temperature. A random sample produced the following data where x is the number of cricket chirps in one minute and y is the ambient temperature in Fahrenheit.

x	y
128	70
227	93
188	84
150	76
210	89
185	83
137	72
171	80

1. (a) Find an equation of the least squares regression line. Round the slope and y-intercept value to two decimal places

Now, the least squares regression line equation has the form Y=a+bX, where Y is the dependent variable (on the Y axis), X is the independent variable (on the X axis), b is the slope of the line and a is the y-intercept.

Using the above formula a and b can be calculated to be,

Sum of X = 1396
Sum of Y = 647
Mean X = 174.5
Mean Y = 80.875
Sum of squares = 8490
Sum of products = 1950.5

n = No. of observations = 8

ŷ = 0.22974X + 40.78522

Slope = 0.23, Y-intercept = 40.79 ( to two decimal places.)

ŷ = 0.23X + 40.79

b) Based on the equation from part (a), what is the predicted temperature when a cricket chirps 170 times in one minute?

Here, X=170

ŷ = 0.23*170 + 40.79=79.89

1. (c) Based on the equation from part (a), what is the predicted temperature when a cricket chirps 320 times in one minute?

X=320

ŷ = 0.23*320 + 40.79=114.39

(d) Which predicted temperature that you calculated for (b) and (c) do you think is closer to the true temperature and why?

The difference between the observed value of the dependent variable (y) and the predicted value (ŷ) is called the residual

1. Residual = Observed value - Predicted value

   e = y - ŷ

The closer the residual value is to 0, the closer is the predicted value to the observed value

Calculate the residual values and compare.