Question

20. 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

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

(b) Based on the equation from part (a), what is the predicted temperature when a cricket chirps 300 times in one minute? Show all work and justify your answer.

(c) Based on the equation from part (a), what is the predicted temperature when a cricket chirps 140 times in one minute? Show all work and justify your answer.

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

Answer 1

here to find the equation of the least square regression line we need the value of Σx, Σy, Σxy and Σx²

	x	Y	x^2	xy
	128	70	16384	8960
	227	93	51529	21111
	188	84	35344	15792
	150	76	22500	11400
	210	89	44100	18690
	185	83	34225	15355
	137	72	18769	9864
	171	80	29241	13680
Sum	1396	647	252092	114852

so lets say line equation is y^ = a +bx

Here n = 8

and a and b = line constants.

so by formulas

a = [(Σy) (Σx² ) - (Σx) (Σxy)]/ [ n (Σx² ) - (Σx)² ]

a = [647 * 252092 - 1396 * 114852]/ [8 * 252092 - 1396 * 1396]

a = 40.785

b = [ n(Σxy) - (Σx)((Σy)]/ [ n (Σx² ) - (Σx)² ]

b = [8 * 114852 - 1396 * 647]/[8 * 252092 - 1396 * 1396]

b = 0.2297

y^ = 40.785 + 0.2297x

(b) here for x = 300 times in one minute, the predicted temperature is

y^ = 40.785 + 0.2297 * 300 = 109.71

(c) here for x = 140 times in one minute, the predicted temperature is

y^ = 40.785 + 0.2297 * 140 = 72.95

(d) Here x = 300 times is not in the sample space of the independent variable (x). Here x = 140 times is in the independent variable range, so it should be closer to the true temeprature. X = 300 times is interpolation data which is out of the range.