# True population regression line passes through origin?

I'm having trouble interpreting a question: Test the null hypothesis that the true (population) regression line passes through the origin, for the simple linear regression of Rigidity on Density. Be sure to state your null and alternative hypotheses clearly in terms of the appropriate model parameter.

I'm not very good at statistics but I'm wondering why would it pass through the origin? There is an intercept so I thought the line would start at the intercept?

y = Bo + B1X

> summary(TimberMod2)
Call:
lm(formula = Rigidity ~ Density)
Residuals:
Min      1Q  Median      3Q     Max
-520.72  -96.04    5.50  100.14  599.32
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 206.723 129.866 1.592 0.118
Density 30.050 2.588 11.611 1.53e-15 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 217.3 on 48 degrees of freedom Multiple R-squared: 0.7374, Adjusted R-squared: 0.732 F-statistic: 134.8 on 1 and 48 DF, p-value: 1.525e-15

> mean(Rigidity)
[1] 1671.8
> var(Rigidity)
[1] 176210.2
> mean(Elasticity)
[1] 208.33
> var(Elasticity)
[1] 2502.445
> mean(Density)
[1] 48.754
> var(Density)
[1] 143.9005


This is all the information that is provided. I'm honestly not sure what to do with all the information. I thought that the null hypothesis would be: Ho: Bo = 0 vs Ha: Bo does not equal 0 but I'm sure that that is too simple? Also from the question I can't figure out if they're asking to test for the slope or Bo?

• The null hypothesis you state seems reasonable to me. As to whether it is reasonable to test if timber of Density zero would also have Rigidity zero I leave that to you. It seems nonsense to me but I am not a forestry scientist. Oct 13, 2017 at 15:06
• What makes physical sense as density tends to zero? It's this limiting behaviour that is being asked about, or so I guess, not whether wood can exist with zero density. Oct 13, 2017 at 15:09

I'm not very good at statistics but I'm wondering why would it pass through the origin? There is an intercept so I thought the line would start at the intercept?

There was a non-zero intercept calculated from the data. But that always happens. Or, more precisely, as long as there is an error term with a continuous distribution, probability(calculated intercept = 0) = 0; the calculated intercept is a statistic, just like the sample mean and sample std, and it will have some probability distribution. So what you need to do is figure out what that distribution is given the null hypothesis, find the probability mass for it being as extreme or more as your observed value (I'm far from certain that I'm reading the output correctly, but this appears to be given as 0.118), and compare that to your alpha.

I'm honestly not sure what to do with all the information. I thought that the null hypothesis would be: Ho: Bo = 0 vs Ha: Bo does not equal 0

You seem to be confusing statistics and parameters. The intercept of the true regression line is a parameter. The null hypothesis is that that parameter is equal to zero. The calculated intercept is a statistic. It is a random variable whose distribution depends on the intercept of the true regression line.

It's like how when you're testing the hypothesis that two distributions have different means, you have a null hypothesis that the population means are the same, then you take samples from each distribution and calculate the sample means. The population means are parameters. The sample means are statistics. Just because the sample means are different, that doesn't mean the population means are different. E.g. if a sample of people who received a drug did .01% better than a sample of people who didn't, then the sample means are different, but you still have to decide whether that reflects an actual difference of the drug's effectiveness compared to the control, or is just due to random chance.

• that was really helpful! So for the null hypothesis, would the parameter be timber? because some additional information stated "These data are for 50 varieties of timber, for modulus of rigidity, modulus of elasticity, and air dried density.” So they took a sample of timber, which is the population parameter? Am I understanding that correctly? Again thank you for helping me that cleared up a lot of the confusion. Oct 13, 2017 at 15:29
• The parameter would be the intercept of the true regression line. Oct 13, 2017 at 17:49