Question

Anyone who has been outdoors on a summer evening has probably heard crickets. Did you know that it is possible to use the cricket as a thermometer? Crickets tend to chirp more frequently as temperatures increase. This phenomenon was studied in detail by George W. Pierce, a physics professor at Harvard. In the following data, x is a random variable representing chirps per second and y is a random variable representing temperature (°F). x 20.5 16.0 20.0 17.4 16.5 15.5 14.7 17.1 y 89.0 71.2 94.3 82.7 79.2 75.2 69.7 82.0 x 15.4 16.2 15.0 17.2 16.0 17.0 14.4 y 69.4 83.3 79.6 82.6 80.6 83.5 76.3 Complete parts (a) through (e), given Σx = 248.9, Σy = 1198.6, Σx2 = 4172.81, Σy2 = 96,436.46, Σxy = 20,028.63, and r ≈ 0.833. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums Σx, Σy, Σx2, Σy2, Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.) Σx = Σy = Σx2 = Σy2 = Σxy = r = (c) Find x, and y. Then find the equation of the least-squares line y hat = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.) x = y = y hat = + x (d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot (e) Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r2 to three decimal places. Round your answers for the percentages to one decimal place.) r2 = explained % unexplained % (f) What is the predicted temperature when x = 17.0 chirps per second? (Round your answer to two decimal places.) °F

Answer 1

(a) We are now going to plot a scatter diagram for the data given above:

  

Scatter Plot of X and Y

  

(b)   

(c) By using R, we can derive the intercept(a) which is equals to 25.594 and the slope(b) which is equals to 3.273.

Hence, the simple linear regression model is given by, y = 25.594 + 3.273 * x.

(f) Given that the value of chirps per second (x) =17.

then the predicted temperature will be y = 25.594 + 3.273*17 = 81.235 ⁰F.