Using a P-value to make conclusions in a test about slope | AP Statistics | Khan Academy

Alicia took a random sample of mobile phones and found a positive linear relationship between their processor speeds and their prices. Here is computer output from a least squares regression analysis on her sample.

So just to be clear what's going on: she took a sample of phones; they're not telling us exactly how many, but she took a number of phones, and she found a linear relationship between processor speed and prices. So this is price right over here, and this is processor speed right over here. Then she plotted her sample; for every phone would be a data point, and so you see that.

Then she put those data points into her computer, and it was able to come up with a line, a regression line for her sample. Her regression line for her sample, if we say that's going to be y or y hat, is going to be a plus bx. For her sample, a is going to be 127.092, so that's that over there.

For her sample, the slope of the regression line is going to be the coefficient on speed. Another way to think about it, this x variable right over here, speed, so the coefficient on that is the slope. But we have to remind ourselves that these are estimates of maybe some true truth in the universe.

If you were able to sample every phone in the market, then she would get the true population parameters. But since this is a sample, it's just an estimate. Just because she sees this positive linear relationship in her sample, it doesn't necessarily mean that this is the case for the entire population. She might have just happened to sample things that had this positive linear relationship.

That's why she's doing this hypothesis test. In a hypothesis test, you actually assume that there isn't a relationship between processor speed and price. So beta right over here, this would be the true population parameter for regression on the population.

If this is the population right over here, and if somehow—where is price on the vertical axis and processor speed on the horizontal axis—and if you were able to look at the entire population; I don't know how many phones there are, but it might be billions of phones. Then do a regression line; this is our null hypothesis: the slope of the regression line is going to be zero.

The regression line might look something like that, where the equation of the regression line for the population y hat would be alpha plus beta times x. So our null hypothesis is that beta is equal to zero, and the alternative hypothesis, which is her suspicion, is that the true slope of the regression line is actually greater than zero.

Assume that all conditions for inference have been met. At the alpha equals 0.01 level of significance, is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones?

So pause this video and see if you can have a go at it. Well, in order to do this hypothesis test, we have to say, well, assuming the null hypothesis is true—assuming this is the actual slope of the population regression line—what is the probability of us getting this result right over here?

What we can do is use this information and our estimate of the sampling distribution of the sample regression line slope, and we can come up with a t statistic. For this situation, where our alternative hypothesis is that our true population regression slope is greater than zero, our p-value can be viewed as the probability of getting a t-statistic greater than or equal to this.

So getting a t-statistic greater than or equal to 2.999. Now, you could be tempted to say, hey, look, there's this column that gives us a p-value; maybe they just figured out for us that this probability is 0.004.

We have to be very, very careful here because here they're actually giving us— I guess you call it a two-sided p-value. If you think of a t-distribution and they would do it for the appropriate degrees of freedom, this is saying what's the probability of getting a result where the absolute value is 2.999 or greater.

So if this is t equals 0 right here in the middle, and this is 2.999, we care about this region; we care about this right tail. This p-value right over here is giving us not just the right tail, but it's also saying, well, what about getting something less than negative 2.999 or including negative 2.999? So it's giving us both of these areas.

So if you want the p-value for this scenario, we would just look at this. As you can see, because this distribution is symmetric—the t-distribution is going to be symmetric—you take half of this. So this is going to be equal to 0.002.

What you do in any significance test is then compare your p-value to your level of significance. If you look at 0.002 and compare it to 0.01, which of these is greater? Well, at first, your eyes might say, hey, 2 is greater than 1, but this is 2 thousandths versus 100; this is 10 thousandths right over here.

So in this situation, our p-value is less than our level of significance. We will say, hey, the probability of getting a result this extreme or more extreme is so low if we assume our null hypothesis that, in this situation, we will reject—we will decide to reject our null hypothesis, which would suggest the alternative.

So is there sufficient evidence to conclude a positive linear relationship between these variables for all mobile phones? Yes. Why? Because p-value is less than our significance level, and so we reject our null hypothesis, which suggests our alternative hypothesis.