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F-Test : Do we accept the null-hypothesis?

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,004
Hey!! :eek:

The below table shows the average monthly gross income of a sample of 44 developers. For each individual sample, it is indicated their country of employment and years of service in their field.

Calculate the regression line with the dependent variable the monthly gross income and independent the years of employee service and check the significance of the F criterion at $\alpha = 0.05$.


I have done the following:

Y_X.JPG

Therefore we get:
\begin{align*}\nu &=44 \\ \overline{X}&=\frac{\sum X}{\nu}=\frac{426.1}{44}=9.68 \\ \overline{Y}&=\frac{\sum Y}{\nu}=\frac{299767.60}{44}=6812.9 \\ \hat{\beta}&=\frac{\nu \sum \left (XY\right )-\left (\sum X\right )\left (\sum Y\right )}{\nu\sum X^2-\left (\sum X\right )^2}=\frac{44 \cdot 2911490.795-426.1\cdot 299767.60}{44\cdot 4417.39-426.1^2}=\frac{1.2810559498 \cdot 10^8-1.2773097436 \cdot 10^8}{194365.16-181561.21} \\ & =\frac{374620.62}{12803.95}=29.26 \\ \hat{\alpha}&=\overline{Y}-\hat{\beta}\cdot \overline{X}=6812.9-29.26\cdot 9.68=6812.9-283.2368=6529.66\end{align*}

So the regression line is \begin{equation*}\hat{Y}=29.26X+6529.66\end{equation*}


We consider a F-test whether the slope is $0$ or not.

We have the formula $\displaystyle{F=\frac{MSM}{MSE}=\frac{\text{explained variance}}{\text{unexplained variance}}}$.

We have that $\displaystyle{MSM = \frac{SSM}{DFM}}$ with $\displaystyle{SSM=\sum_{i=1}^n\left (\hat{Y}_i-\overline{Y}\right )^2}$ and $\displaystyle{DFM = p - 1}$.

We also have that $\displaystyle{MSE = \frac{SSE}{DFE}}$ with $\displaystyle{SSE=\sum_{i=1}^n\left (Y_i-\hat{Y}_i\right )^2}$ and $\displaystyle{DFE = \nu-p}$.


We have this table.

So we get
\begin{align*}&SSM=249138.5759 \\ &DFM=2-1=1 \\ &SSE=104926827.5 \\ &DFE=44-2=42 \\ &MSM=\frac{SSM}{DFM}=\frac{249138.5759}{1}=249138.5759 \\ &MSE=\frac{SSE}{DFE}=\frac{104926827.5}{42}=2498257.7976 \\ &F=\frac{MSM}{MSE}=\frac{249138.5759}{2498257.7976}=0.0997\end{align*}

Using at the R-program the command pf(0.0997, 1, 42, lower.tail=F) we get the p-value $0.7537537$.

That means that $\text{p-value} > \alpha$, does this mean that we accept the null hypothesis, i.e. that the slope is not significally different from $0$.

Is that correct? (Wondering)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,684
That means that $\text{p-value} > \alpha$, does this mean that we accept the null hypothesis, i.e. that the slope is not significally different from $0$.

Is that correct?
Hey mathmari !!

Formally it means we do not have sufficient evidence to conclude the slope is different from 0.
We 'keep' the null hypothesis, but we cannot conclude that the slope is 0. (Nerd)
 

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,004
Formally it means we do not have sufficient evidence to conclude the slope is different from 0.
We 'keep' the null hypothesis, but we cannot conclude that the slope is 0. (Nerd)
Ah ok!! Is there a way to check if my results are correct? (Wondering)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,684
Ah ok!! Is there a way to check if my results are correct?
Excel has the [M]LINEST[/M] function.
One of its outputs is the F-value when used in a 5x2 array context for a simple linear regression. (Thinking)
Excel also has the [M]F.DIST[/M] function to convert an F-value into a probability.

We can also do a t-test for the slope as we did before. It should yield the same p-value. (Thinking)

Additionally we can draw a graph of the points and the line to see if it makes sense that the explained variance is so much lower than the unexplained variance. (Thinking)
 

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,004
Excel has the [M]LINEST[/M] function.
One of its outputs is the F-value when used in a 5x2 array context for a simple linear regression. (Thinking)
Using the command [M]LINEST(D2:D45;B2:B45)[/M] I get $29.25833947$. If I used the command correctly, I must have a mistake at the calculations of $F$. But what? (Wondering)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,684
Using the command [M]LINEST(D2:D45;B2:B45)[/M] I get $29.25833947$. If I used the command correctly, I must have a mistake at the calculations of $F$. But what? (Wondering)
To get the F-value from LINEST, we need to specify 4 parameters. And we must make it an array formula with Ctrl+Shift+Enter. (Thinking)

You should also get for instance the slope and the y intercept. Do they match? (Wondering)