You are the director of product development for the Frobozz magic thermometer company. You need to determine the markings for the scale printed on the thermometer tube. You take a tube and fill it with mercury, then measure the height of the mercury above a mark near the bottom of the tube at seven different temperatures. You perform the preliminary calculations for a linear regression as follows:
degrees F y
Height, cm x2 y2 xy
60 -1.0 3600 1.00 -60
65 2.9 4225 8.41 188.5
70 4.3 4900 18.49 301
75 7.6 5625 57.76 570
80 10.2 6400 104.04 816
85 13.0 7225 169.00 1105
90 14.1 8100 198.81 1269
a. Find the equation for the least-squares line.
For parts (b) - (e), assume that the residuals of the least-squares fit are well behaved.
b. Do you have sufficient evidence at the 10% level to support the claim that the height of the mercury in the tube increases with increasing temperature? Explain your answer (don't just say yes or no).
c. Construct a 90% confidence interval for the slope.
d. How high would you expect the mercury to rise at a temperature of 72 degrees?
e. Find a 95% prediction interval for the height of the mercury at 72 degrees.
This is a case study on the product development for the Frobozz magic thermometer company.