Question
In: Statistics and Probability
A researcher wants to know how the number of days a vampire goes without blood (X) is related to the number of violent behaviors that a vampire engages in (Y)
A researcher wants to know how the number of days a vampire goes without blood (X) is related to the number of violent behaviors that a vampire engages in (Y). To investigate this, he measures both variables in a sample of 7 (n = 7) vampires. The goal of the study was to find the least-squares regression equation for the data, and to then test whether the slope of this equation is any different from 0. Assume all assumptions for the test are met. Do all hypothesis testing for this question using α =0.01. (See table below for sample data.)
|
Days without blood |
Violent behaviors |
|
X |
Y |
|
12 |
16 |
|
15 |
18 |
|
6 |
12 |
|
6 |
13 |
|
11 |
13 |
|
9 |
12 |
|
10 |
19 |
|
Σx =69 |
Σy =103 |
|
MX =9.86 |
MY =14.71 |
A) State the hypotheses for testing the slope?
B) Find Fcrit .
Find the equation for the regression line for these data. You may find it helpful to use the table below. (There are points for each answer labeled “i” through “vi” below the table.)
- the table.)
|
X |
Y |
||||||
|
Sums |
|||||||
|
Means |
Solutions
Expert Solution
Solution:
Calculation Summary
Sum of X = 69
Sum of Y = 103
Mean X = 9.8571
Mean Y = 14.7143
Sum of squares (SSX) = 62.8571
Sum of squares (SSY)=51.42857
Sum of products (SP) = 37.7143
Regression Equation = ŷ = bX + a
b = SP/SSX = 37.71/62.86 = 0.6
a = MY - bMX = 14.71 - (0.6*9.86) = 8.8
ŷ = 8.8 + 0.6X
| x | y | X - Mx | Y - My | (X - Mx)2 | (Y - My)2 | (X - Mx)(Y - My) |
| 12 | 16 | 2.1429 | 1.2857 | 4.5918 | 1.653024 | 2.7551 |
| 15 | 18 | 5.1429 | 3.2857 | 26.449 | 10.79582 | 16.898 |
| 6 | 12 | -3.8571 | -2.7143 | 14.8776 | 7.367424 | 10.4694 |
| 6 | 13 | -3.8571 | -1.7143 | 14.8776 | 2.938824 | 6.6122 |
| 11 | 13 | 1.1429 | -1.7143 | 1.3061 | 2.938824 | -1.9592 |
| 9 | 12 | -0.8571 | -2.7143 | 0.7347 | 7.367424 | 2.3265 |
| 10 | 19 | 0.1429 | 4.2857 | 0.0204 | 18.36722 | 0.6122 |
| M: 9.8571 | 14.7143 | SS: 62.8571 | 51.42857 | SP: 37.7143 |
| SUMMARY OUTPUT | ||||||||
| Regression Statistics | ||||||||
| Multiple R | 0.663325 | |||||||
| R Square | 0.44 | |||||||
| Adjusted R Square | 0.328 | |||||||
| Standard Error | 2.4 | |||||||
| Observations | 7 | |||||||
| ANOVA | ||||||||
| df | SS | MS | F | Significance F | ||||
| Regression | 1 | 22.62857 | 22.62857 | 3.928571 | 0.104303 | |||
| Residual | 5 | 28.8 | 5.76 | |||||
| Total | 6 | 51.42857 | ||||||
| Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Lower 95.0% | Upper 95.0% | |
| Intercept | 8.8 | 3.118741 | 2.821651 | 0.037037 | 0.783021 | 16.81698 | 0.783021 | 16.81698 |
| X Variable 1 | 0.6 | 0.302715 | 1.982062 | 0.104303 | -0.17815 | 1.378154 | -0.17815 | 1.378154 |
Correlation coefficient
The value of R is 0.6633.
This is a moderate positive correlation, which means there is a tendency for high X variable scores go with high Y variable scores (and vice versa).
The value of R2, the coefficient of determination, is 0.44.
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