Here are some additional review questions on ch3/13; you may also find the odd-numbered problems in the textbook helpful.

[ click for answers ]

1. An experiment measured the mass of a certain variety of caterpillars after 3 weeks, in different temperatures. The results are shown below:
   

   Temp (C)	15	18	21	21	23	28
   Mass (g)	1.8	2.2	3.2	2.8	3.7	3.1

   1. What is the average temperature and average mass in the study sample?
      
      xbar = 21, ybar = 2.8
      
      
   2. Calculate SS(x), SS(y), and SS(xy).
      
      SS(x) = 98, SS(y) = 2.42, SS(xy) = 11.7
      
      
   3. Find the sample correlation r.
      
      r = SS(xy)/sqrt(SS(x) SS(y)) ≅ 0.75974 ≅ 0.76
      
      
   4. Construct a 95% confidence interval on the correlation.
      
      n = 6: about -0.05 < ρ < 0.96.
      
      
   5. At a level of significance of α = 0.05, is caterpillar mass correlated with temperature?
      
      df = n-2 = 4: 0.05 < p < 0.10: fail to reject H₀; insufficient evidence to show caterpillar mass is correlated with temperature.
      
      
   6. Find the equation of the best-fit line.
      
      b1 = SS(xy)/SS(x) ≅ 0.1194, b0 = 0.2929.
      Best-fit line is y = 0.2929 + 0.1194 x.
      
      
   7. What does the linear model predict should be the mass of caterpillars grown at a temperature of 25 C?
      
      y = 0.2929 + 0.1194 * 25 ≅ 3.2776 grams.
      
      
   8. Find SSE, the sum of squared errors (residuals).
      
      SSE = 1.0232.
      
      
   9. What are the assumptions of the linear model?
      
      Assumes that the sample is a random sample representative of the population. Assumes that the mass of caterpillars is normally distributed with a mean that depends upon the temperature according to the best-fit line and a standard deviation that does not depend on temperature.
      
      
   10. What does the linear model predict is the standard deviation of mass of caterpillars grown at a temperature of 25 C?
       
       s_e = sqrt(SSE/(n-2)) ≅ 0.5058.
       
       
   11. Find SE_b1, the standard error of estimate of the slope (b1) of the best-fit line.
       
       SE_b1 = s_e / sqrt(SS(x)) ≅ 0.0511
       
       
   12. At a level of significance of α = 0.02, is the slope of the best-fit line positive?
       
       t = (b1 - 0) / SE_b1 ≅ 2.34.
       df = n-2 = 4, one-tailed: 0.025 < p < 0.05
       fail to reject H₀: insufficient evidence to show that the slope of the best-fit regression line for mass and temperature is positive.