In: Statistics and Probability
The authors of an article found that the speed of a prey (twips/s) and the length of a prey (twips ✕ 100) are good predictors of the time (s) required to catch the prey. (A twip is a measure of distance used by programmers.) Data were collected in an experiment in which subjects were asked to "catch" an animal of prey moving across his or her computer screen by clicking on it with the mouse. The investigators varied the length of the prey and the speed with which the prey moved across the screen.
The following data are consistent with summary values and a graph given in the article. Each value represents the average catch time over all subjects. The order of the various speed-length combinations was randomized for each subject.
Prey Length | Prey Speed | Catch Time |
---|---|---|
7 | 20 | 1.11 |
6 | 20 | 1.21 |
5 | 20 | 1.24 |
4 | 20 | 1.41 |
3 | 20 | 1.49 |
3 | 40 | 1.40 |
4 | 40 | 1.35 |
6 | 40 | 1.30 |
7 | 40 | 1.28 |
7 | 80 | 1.41 |
6 | 60 | 1.37 |
5 | 80 | 1.39 |
7 | 100 | 1.43 |
6 | 100 | 1.42 |
7 | 120 | 1.71 |
5 | 80 | 1.49 |
3 | 80 | 1.40 |
6 | 100 | 1.51 |
3 | 120 | 1.91 |
(a) Fit a multiple regression model for predicting catch time using prey length and speed as predictors. (Use x1 for prey length and x2 for speed. Round your answers to three decimal places.)
=( ____) +(____ x1) +( _____ x2)
(b) Predict the catch time for an animal of prey whose length is 6 and whose speed is 50. (Round your answer to three decimal places.)
(_____ s)
(C) Calculate the test statistic. (Round your answer to two decimal places.)
F = (_____)
(d) The authors of the article suggest that a simple linear regression model with the single predictor
x = length / speed
might be a better model for predicting catch time. Calculate the x values and use them to fit this linear regression model. (Round your answers to three decimal places.)
= (____) + (_____ x)
we will use excel to perform regression analysis
the output of regression analysis
SUMMARY OUTPUT | ||||||
Regression Statistics | ||||||
Multiple R | 0.8582 | |||||
R Square | 0.7365 | |||||
Adjusted R Square | 0.7036 | |||||
Standard Error | 0.0958 | |||||
Observations | 19.0000 | |||||
ANOVA | ||||||
df | SS | MS | F | Significance F | ||
Regression | 2.0000 | 0.4102 | 0.2051 | 22.3648 | 0.0000 | |
Residual | 16.0000 | 0.1467 | 0.0092 | |||
Total | 18.0000 | 0.5569 | ||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
Intercept | 1.4342 | 0.0856 | 16.7447 | 0.0000 | 1.2526 | 1.6157 |
Prey Length | -0.0509 | 0.0150 | -3.3927 | 0.0037 | -0.0827 | -0.0191 |
Prey Speed | 0.0040 | 0.0006 | 6.2130 | 0.0000 | 0.0026 | 0.0053 |
a)
Catch Time=1.434-0.051*Prey Length+0.004*Prey Speed
b)
Predict the catch time for an animal of prey whose length is 6 and whose speed is 50. (Round your answer to three decimal places.)
Catch Time=1.434-0.051*Prey Length+0.004*Prey Speed
prey whose length is 6 and whose speed is 50
Catch Time=1.434-0.051*6+0.004*50
Catch Time = 1.328
(C) Calculate the test statistic. (Round your answer to two decimal places.)
F = 22.36
d)
single predictor
x = length / speed
create new variable
x |
0.3500 |
0.3000 |
0.2500 |
0.2000 |
0.1500 |
0.0750 |
0.1000 |
0.1500 |
0.1750 |
0.0875 |
0.1000 |
0.0625 |
0.0700 |
0.0600 |
0.0583 |
0.0625 |
0.0375 |
0.0600 |
0.0250 |
output
SUMMARY OUTPUT | ||||||
Regression Statistics | ||||||
Multiple R | 0.7169 | |||||
R Square | 0.5139 | |||||
Adjusted R Square | 0.4853 | |||||
Standard Error | 0.1262 | |||||
Observations | 19.0000 | |||||
ANOVA | ||||||
df | SS | MS | F | Significance F | ||
Regression | 1.0000 | 0.2862 | 0.2862 | 17.9741 | 0.0006 | |
Residual | 17.0000 | 0.2707 | 0.0159 | |||
Total | 18.0000 | 0.5569 | ||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
Intercept | 1.5831 | 0.0496 | 31.8906 | 0.0000 | 1.4783 | 1.6878 |
x | -1.3686 | 0.3228 | -4.2396 | 0.0006 | -2.0497 | -0.6875 |
Catch Time = 1.583-1.369*x