# Confidence Intervals, P value and Null Hypothesis

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Studying for my MSc and I’m doing a research methology module. I’ve tried reading this multiple times and I just can’t get my head around it.

In: Mathematics

So say you have a bunch of points.

The confidence interval is the range you are most likely to find the points the majority of the time. There might be a couple points outside that range but it’s not not likely. Like if you have a strict daily schedule. You can predict where you will be with high confidence but occasionally you might have to change your schedule a little (outside the interval).

Null hypothesis is whatever condition you want really for your test. This what you are assuming is true about your points. For example, the mean is 0 or there is no difference between 2 groups.

Now the alternate hypothesis is that the null is not true. For example, the mean is not 0 or there is a difference between groups.

When you do a statistical test you will either gain evidence against the null or not. If the evidence is strong enough, you can reject the null and accept the alternate hypothesis. This is sometimes determined by p-values.

The p-values you get from some tests is the probability of the outcome matching your points assuming the null is correct. Essentially what are the chances of getting the same thing assuming the null.

So if p is really high (near 1.0) you learn the probability of the outcomes matching your points is high so the null is NOT rejected (i.e. it’s likely true). The chances are pretty good you will get the same as the null.

If p is lower than the probability threshold you set (e.g. 0.05) the probability is low that the outcome matches your data, so the null IS rejected (the alternative hypothesis is true). The chances are really low you’d get the null results.

Suppose you have this idea, and want to see if it’s true. How do you know your idea is right? Let’s say I want to prove that microwaved water isn’t as good as rain water for plants.

Well, test it – that’s science right. Nope – first I have to state my hypothesis formally. “Rain water is better for plants than microwaved tap water”. Which means I also have to state the “Null hypothesis” – AKA “what if I’m wrong”. In this case, “Rain water is no better for plants than microwaved water”. Do note: this isn’t “microwaved water is better than rain water” – it could be that both kinds of water are the same too. The Null Hypothesis has to account for anything other than you being right.

Basically, the Null Hypothesis is the uninteresting hypothesis. It’s the idea that nothing special or interesting, or at least nothing new, is happening here.

Okay, hypothesis done. Now I can test. So I grow two plants, water one with rain water, one with microwaved water: the one I water with microwaved water grows less. I’m right.

Not so fast. If you stop and think about it, one of my two plants was going to grow more, even if I did exactly the same thing. Even if I was wrong, there was a 50% chance to get this result.

Enter P values.

The p-value is the likelihood that the result I saw in my experiment would have happened anyway. Technically, it’s how likely this result *or a more extreme one* would happen. My experiment had a P-value of .5 (1 is “all the time”, 0 is “never”), which isn’t particularly convincing.

So I do my experiment again. 10 plants total: 5 watered with microwaved water, 5 with rain water. I measure how tall they are. Now what?

I’m going to skip the math on this – mostly because it turns out there’s a few ways I could do it, depending on the details of how I ran the experiment: you can take multiple college classes on just “how to calculate P-values”. But let’s say I get a P-value of .13. That’s about 1 in 7, 1 in 8 – it’s lower, but still not telling. Lower would be better: I probably don’t publish.

What if I don’t know anything?

Suppose I got these new plant seeds, and I don’t know anything about them. How tall do they grow?

There really isn’t a hypothesis yet. I just want information. So here, I’m not going to bother with making a hypothesis, or testing for a P-value. I just grow them all, and measure how tall they grow.

But they’re not all the same height: some are taller, some are shorter. So I take the average height, and publish that. BUT, I’m not sure – maybe I got some unusually tall ones, or short ones. That average height isn’t perfect.

So, I also publish my “confidence interval”. For example, after measuring all the plants and doing the math (again, I’m skipping over the math here. If you want more detail on the math, ask), I say that this plant grows an average of 86.5 +- 8.3 cm, 90% confident. That means I am 90% sure (If I got an unusual set of these plants, I might be wrong) that, if you grew all the plants of this type (in the same way I did), the average height would be between 78.2 and 94.8cm.