Comparing P-values to different significance levels | AP Statistics | Khan Academy

What we're going to do in this video is talk about significance levels, which are denoted by the Greek letter alpha.

We're going to talk about two things: the different conclusions you might make based on the different significance levels that you might set, and also why it's important to set your significance levels ahead of time before you conduct an experiment and calculate the p-values for, frankly, ethical purposes.

So, to help us get this, let's look at a scenario right over here which tells us Raheem heard that spinning rather than flipping a penny raises the probability above 50 percent that the penny lands showing heads. That's actually quite fascinating if that's true. He tested this by spinning 10 different pennies 10 times each, so that would be a total of 100 spins.

His hypotheses were: His null hypothesis is that by spinning, your proportion doesn't change; rather, versus flipping, it's still 50 percent. His alternative hypothesis is that by spinning, your proportion of heads is greater than 50, where p is the true proportion of spins that a penny would land showing heads.

In his 100 spins, the penny landed showing heads in 59 spins. Raheem calculated that the statistic, so this is the sample proportion here, is 59 out of 100 were heads, so that's 0.59 or 59 hundredths. He calculated an associated p-value of approximately 0.036.

So, based on this scenario, if ahead of time Raheem had set his significance level at 0.05, what conclusions would he now make?

And while you're pausing it, think about how that may or may not have been different if he set his significance levels ahead of time at 0.01. Pause the video and try to figure that out.

So, let's first of all remind ourselves what a p-value even is. You could view it as the probability of getting a sample proportion at least this large if you assume that the null hypothesis is true. If that is low enough, if it's below some threshold, which is our significance level, then we will reject the null hypothesis.

In this scenario, we do see that 0.036, our p-value, is indeed less than alpha. It is indeed less than 0.05, and because of that, we would reject the null hypothesis.

In everyday language, rejecting the null hypothesis is rejecting the notion that the true proportion of spins that a penny would land showing heads is 50. If you reject your null hypothesis, you could also say that suggests our alternative hypothesis: that the true proportion of spins that a penny would land showing heads is greater than 50.

Now, what about the situation where our significance level was lower? Well, in this situation, our p-value, our probability of getting that sample statistic if we assumed our null hypothesis were true, in this situation, it's greater than or equal to, and it's greater than in this particular situation than our threshold, than our significance level.

So here we would say that we fail to reject our null hypothesis. So we're failing to reject this right over here and it will not help us suggest our alternative hypothesis.

Because of the difference between what you would conclude given this change in significance levels, that's why it's really important to set these levels ahead of time. You could imagine it's human nature; if you're a researcher of some kind, you want to have an interesting result.

You want to discover something. You want to be able to tell your friends, "Hey, my alternative hypothesis, it actually is suggested! We can reject the assumption, the status quo. I found something that actually makes a difference."

It's very tempting for a researcher to calculate your p-values and then say, "Oh, well maybe no one will notice if I then set my significance values so that it's just high enough so that I can reject my null hypothesis." If you did that, that would be very unethical.

In future videos, we'll start thinking about the question of, "Okay, if I'm doing it ahead of time, if I'm setting my significance level ahead of time, how do I decide to set the threshold? When should it be 100? When should it be 50s? When should it be 100s? Or when should it be something else?"