Statistics and probability
Sal walks through an example about a neurologist testing the effect of a drug to discuss hypothesis testing and p-values. Created by Sal Khan.
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- Starting at4:22, why do you need to estimate the sample standard deviation when you already have it(.5)? He goes on to say that you put a hat on it to show that you estimated the population standard deviation by using the sample but why does the sigma have a hat for population estimate and have an x bar for sample? Is the notation correct on that section?(69 votes)
- Don't forget, we don't really care about the st.dv. of the sampl, we care about it's relationship to the population. So we have to take measures that involve the actual population. You must first see the video "standard error of the mean" to get this one.(8 votes)
- Why are you not using a t-distribution to find the probability of getting the sample result? I know that when the sample size is large (n = 100), a t-distribution is essentially the same as a normal distribution, but I think this lesson can be misleading when we are taught to use a t-distribution in the common case when the population standard deviation is not known and we are estimating it from the sample.(61 votes)
- The t-test is more conservative, if the sample size is small. I think you would opt for the more conservative test, knowing that with a larger sample size, there is essentially no difference between t and z. In general, when comparing two means, the t-test is used. Note from the results given above by ericp, that the conclusion from either test is the same. The two groups differ significantly. In scientific reports, p-value is reported to 2 decimal places. So using either the z or t test, you would report a significant difference "with p < .01".(21 votes)
- SHouldn't it be the other way around when calculating the Z value?
(1.05-1.2)/0.05 instead of (1.2-1.05)/0.05?
My professor always told me to do it that way. The final conclusion doesn't change in this case though, but just wanted to make sure if that's the proper way.(60 votes)
- since normal probability distribution (bell curve) is symmetric around the mean, it doesnt matter. It gives same result in terms of area under curve, thats why prof. wanted to make it less complex in saying that. But if we were dealing with a non symmetric prob. distr. like F distr, then it would matter.
hope that helps.(16 votes)
- Is it valid to assume the sample SD is close to the population SD? Even if the sample size is high, the rats in the sample have been injected, how do we know that doesn't affect the sample SD?(15 votes)
- It is an assumption you are making, justified by the fact that your Ho is that the drug has no effect, and that the populations (drug vs. no drug) will actually be identical. If the drug has no effect, then the standard deviation of drug and no drug rats should be the same. It is an assumption, justified with some logic, but not proven.
In a research paper, this would be recognized as a weakness, but an unavoidable one, because it is impossible to know the true standard deviation of either population - you only know the samples.(11 votes)
- I have a very fundamental question:
Short formulation of the question: Why is the hypothesis test designed the way it is? I want to know exactly why we can't calculate the probability of the alternative hypothesis given the sample directly and why we have to assume the null hypothesis is true?
Long formulation of the question: When conducting an experiment and setting up hypotheses about its outcome, what we actually want to know is whether our alternative hypothesis is true or at least how likely it is (i.e. the probability of the alternative hypothesis to be true), right?
The here presented hypothesis test only gives us the probability of the sample mean to be extreme, but not the probability of the real underlying population mean to be extreme.
So why do we have to go through this process of calculating the probability of the sample given the null-hypothesis is true and then use this result to infer the likelihood of the alternative hypothesis?
Unfortunately I have not found one textbook on statistics yet which answers this fundamental question but I was reading so many enlightening answers here and hope to get an answer!
From my understanding of the hypothesis test I would answer my own question like that:
Since we don't know anything about the underlying population except the tested sample, we just are not able to do any calculations of it. This includes calculating the probabilities of the alternative hypothesis because it is a hypothesis about the population.
We have to work under the assumption that the null hypothesis is true because otherwise we cannot really do anything, we wouldn't know where to center the normally distributed curve which we use to calculate significance.. ...but somehow I am not entirely convinced by my own answer..
Even if my own answer to the question happens to be not far away from the truth, I would appreciate it very much if someone could elaborate a bit.
Thank you!(12 votes)
- H_0: pop_mean that someone (including you) insists it's true
H_A: pop_mean what another (including you, again) insists H_0 can't be right, cause this, their own mean, is true
sample_mean (and sample_std): the only evidence for both sides to check which is right
in short, what you're doing with significance test is attacking someone's mean with a different mean based on a gathered data
if it's good enough to support you, you can kill H_0 and insist your own H_A as the next H_0 (that's how scientifical theories have been developed and challenged and so forth)
if not, you can't kill it. that's it. no more, no less (what about your precious H_A? just forget about that, not enough evidence)(1 vote)
- I don't understand where Sal got 99.7%... can anyone explain? (8:50)(8 votes)
- He mentioned this a couple of videos ago, but he is using the empirical rule, which states that, for a normal distribution, 99,7% of all values lie within 3 standard deviations of the mean. Similarly, 68,27% lies within 1 standard deviation and 95,45% within 2. See: http://en.wikipedia.org/wiki/68-95-99.7_rule(9 votes)
- Shouldn't we say that the alternative hypothesis is just μ<1.2s and not in both directions?(8 votes)
- That's an important question. In the end, it gets down to the reason that you are conducting the experiment. In this case, the null hypothesis is that the drug doesn't have an effect on response time, so you want to measure both tails. If your null hypothesis was that the drug doesn't have a negative effect on response time, then you would only measure one tail.(8 votes)
- How do you calculate the critical value? I cant find an explaination for it in your video list. Thank you!(6 votes)
- short answer: Critical values are generally chosen or looked up in a table (based on a chosen alpha).
In this video there was no critical value set for this experiment. In the last seconds of the video, Sal briefly mentions a p-value of 5% (0.05), which would have a critical of value of z = (+/-) 1.96. Since the experiment produced a z-score of 3, which is more extreme than 1.96, we reject the null hypothesis.
Generally, one would chose an alpha (a percentage) which represents the "tolerance level for making a mistake.*" Then the corresponding critical value can be looked up from a table. [* the "mistake" being to incorrectly reject the null hypothesis. In other words, we made the error of claiming that the experiment had an effect when it did not.]
The critical value is the cut-off point that corresponds to that alpha; any value beyond the critical value is less than alpha(%) likely to occur by chance.
see the wikipedia page for a z-tables and how to read them
note that for an alpha of 5%, in a cumulative table, you would first divide your alpha in half for a two-tailed test, then subtract that from 1. That is the value you are looking for in the table. So we get 1 - (.05/2) = 1 - .025 = 0.9750
We find 0.9750 in our table, look at the row: 1.9; look at the column: 0.06; add the two together to get the corresponding z-score: 1.96.(6 votes)
- If we assume that the null hypothesis is true, then why do we assume that the sample mean is 1.2 sec? We already know that it's 1.05 sec.(5 votes)
- Because that is the null hypothesis (H0).
What we are testing is how likely we are to have seen the data, under the assumption that H0 is true. Null hypothesis testing follows a somewhat backward seeming logic, but this is apparently pretty standard in mathematics.
1) We calculate how probable it is that we would have seen the observed data if H0 is true.
2) We then either reject H0 (or fail to reject it) depending on how often we are willing to wrongly reject H0 (this is the Type I error rate).
3) If we reject H0 then we provisionally conclude that our alternate hypothesis could be true ...(7 votes)
- Sal said: "Assuming the Null Hyphothesis was true, if the probability of getting the result from the sample is very small, then we reject the Null!"
But how could this be? Because if this probability is very small, it means that the Null is indeed true as a common sense. This is somewhat counter intuitive and I really get confused!!
Someone please, speak the kind of language where newbie like me can understand ?? I dont see any relationship here in this?? It seems that he took for granted that we already understood sth...but i actually many of us dont! So plze point out the logic that you make here related to the null, the z score, and the rejection. thank you!(2 votes)
- I think that your comment shows you came to the proper realization. Maybe this will help clarify or solidify the ideas, or in case others don't fully see it as you did:
First: There is some population (of rats injected with this drug), and this population has a mean, µ. We don't know the value of µ, but we can use a hypothesis test to gives us some information about it (if we knew µ, we wouldn't have to do a hypothesis test at all).
Second: We form a null hypothesis, in this case it is Ho: µ = 1.2. The 1.2 is just some value of interest to which we want to compare. In this case, it is the "status quo", the value of µ for rats NOT injected with the drug.
Third: We calculate our test statistic (the z-score) and p-value, assuming that Ho is correct. That is, we are assuming that the rats injected with the drug have the same value of µ as the rats not injected with the drug. It's important to remember that this is an assumption, and it may not accurately reflect reality. The data values will follow the real value of µ, because that is reality. So if µ=1, then the values will tend to group around 1. If µ=1.2, then the values will tend to group around 1.2.
Then, if the null hypothesis is wrong, then the data will tend to group at a point that is not the value in the null hypothesis (1.2), and then our p-value will wind up being very small. If the null hypothesis is correct, or close to being correct, then the p-value will be larger, because the data values will group around the value we hypothesized.(6 votes)
A neurologist is testing the effect of a drug on response time by injecting 100 rats with a unit dose of the drug, subjecting each to neurological stimulus and recording its response time. The neurologist knows that the mean response time for rats not injected with the drug is 1.2 seconds. The mean of the 100 injected rats response times is 1.05 seconds with the sample standard deviation of 0.5 seconds. Do you think that the drug has an affect on response time? So to do this we're going to set up two hypotheses. We're going to say, one, the first hypothesis is we're going to call it the null hypothesis, and that is that the drug has no effect on response time. And your null hypothesis is always going to be-- you can view it as a status quo. You assume that whatever your researching has no effect. So drug has no effect. Or another way to think about it is that the mean of the rats taking the drug should be the mean with the drug-- let me write it this way-- with the mean is still going to be 1.2 seconds even with the drug. So that's essentially saying it has no effect, because we know that if you don't give the drug the mean response time is 1.2 seconds. Now, what you want is an alternative hypothesis. The hypothesis is no, I think the drug actually does do something. So the alternative hypothesis, right over here, that the drug has an effect. Or another way to think about it is that the mean does not equal 1.2 seconds when the drug is given. So how do we think about this? How do we know whether we should accept the alternative hypothesis or whether we should just default to the null hypothesis because the data isn't convincing? And the way we're going to do it in this video, and this is really the way it's done in pretty much all of science, is you say OK, let's assume that the null hypothesis is true. If the null hypothesis was true, what is the probability that we would have gotten these results with the sample? And if that probability is really, really small, then the null hypothesis probably isn't true. We could probably reject the null hypothesis and we'll say well, we kind of believe in the alternative hypothesis. So let's think about that. Let's assume that the null hypothesis is true. So if we assume the null hypothesis is true, let's try to figure out the probability that we would have actually gotten this result, that we would have actually gotten a sample mean of 1.05 seconds with a standard deviation of 0.5 seconds. So I want to see if we assumed the null hypothesis is true, I want to figure out the probability-- and actually what we're going to do is not just figure out the probability of this, the probability of getting something like this or even more extreme than this. So how likely of an event is that? To think about that let's just think about the sampling distribution if we assume the null hypothesis. So the sampling distribution is like this. It'll be a normal distribution. We have a good number of samples, we have 100 samples here. So this is the sampling distribution. It will have a mean. Now if we assume the null hypothesis, that the drug has no effect, the mean of our sampling distribution will be the same thing as the meaning of the population distribution, which would be equal to 1.2 seconds. Now, what is the standard deviation of our sampling distribution? The standard deviation of our sampling distribution should be equal to the standard deviation of the population distribution divided by the square root of our sample size, so divided by the square root of 100. We do not know what the standard deviation of the entire population is. So what we're going to do is estimate it with our sample standard deviation. And it's a reasonable thing to do, especially because we have a nice sample size. The sample size is greater than 100. So this is going to be a pretty good approximator. This is going to be a pretty good approximator for this over here. So we could say that this is going to be approximately equal to our sample standard deviation divided by the square root of 100, which is going to be equal to our sample standard deviation is 0.5, 0.5 seconds, and we want to divide that by square root of 100 is 10. So 0.5 divided by 10 is 0.05. So the standard deviation of our sampling distribution is going to be-- and we'll put a little hat over it to show that we approximated it with-- we approximated the population standard deviation with the sample standard deviation. So it is going to be equal to 0.5 divided by 10. So 0.05. So what is the probability-- so let's think about it this way. What is the probability of getting 1.05 seconds? Or another way to think about it is how many standard deviations away from this mean is 1.05 seconds, and what is the probability of getting a result at least that many standard deviations away from the mean. So let's figure out how many standard deviations away from the mean that is. Now essentially we're just figuring out a Z-score, a Z-score for this result right over there. So let me pick a nice color-- I haven't used orange yet. So our Z-score-- you could even do the Z-statistic. It's being derived from these other sample statistics. So our Z-statistic, how far are we away from the mean? Well the mean is 1.2. And we are at 1.05, so I'll put that less just so that it'll be a positive distance. So that's how far away we are. And if we wanted it in terms of standard deviations, we want to divide it by our best estimate of the sampling distribution's standard deviation, which is this 0.05. So this is 0.05, and what is this going to be equal to? This result right here, 1.05 seconds. 1.2 minus 1.05 is 0.15. So this is 0.15 in the numerator divided by 0.05 in the denominator, and so this is going to be 3. So this result right here is 3 standard deviations away from the mean. So let me draw this. This is the mean. If I did 1 standard deviation, 2 standard deviations, 3 standard deviations-- that's in the positive direction. Actually let me draw it a little bit different than that. This wasn't a nicely drawn bell curve, but I'll do 1 standard deviation, 2 standard deviation, and then 3 standard deviations in the positive direction. And then we have 1 standard deviation, 2 standard deviations, and 3 standard deviations in the negative direction. So this result right here, 1.05 seconds that we got for our 100 rat sample is right over here. 3 standard deviations below the mean. Now what is the probability of getting a result this extreme by chance? And when I talk about this extreme, it could be either a result less than this or a result of that extreme in the positive direction. More than 3 standard deviations. So this is essentially, if we think about the probability of getting a result more extreme than this result right over here, we're thinking about this area under the bell curve, both in the negative direction or in the positive direction. What is the probability of that? Well we go from the empirical rule that 99.7% of the probability is within 3 standard deviations. So this thing right here-- you can look it up on a Z-table as well, but 3 standard deviation is a nice clean number that doesn't hurt to remember. So we know that this area right here I'm doing and just reddish-orange, that area right over is 99.7%. So what is left for these two magenta or pink areas? Well if these are 99.7% and both of these combined are going to be 0.3%. So both of these combined are 0.3-- I should write it this way or exactly-- are 0.3%. 0.3%. Or is we wrote it as a decimal it would be 0.003 of the total area under the curve. So to answer our question, if we assume that the drug has no effect, the probability of getting a sample this extreme or actually more extreme than this is only 0.3% Less than 1 in 300. So if the null hypothesis was true, there's only a 1 in 300 chance that we would have gotten a result this extreme or more. So at least from my point of view this results seems to favor the alternative hypothesis. I'm going to reject the null hypothesis. I don't know 100% sure. But if the null hypothesis was true there's only 1 in 300 chance of getting this. So I'm going to go with the alternative hypothesis. And just to give you a little bit of some of the name or the labels you might see in some statistics or in some research papers, this value, the probability of getting a result more extreme than this given the null hypothesis is called a P-value. So the P-value here, and that really just stands for probability value, the P-value right over here is 0.003. So there's a very, very small probability that we could have gotten this result if the null hypothesis was true, so we will reject it. And in general, most people have some type of a threshold here. If you have a P-value less than 5%, which means less than 1 in 20 shot, let's say, you know what, I'm going to reject the null hypothesis. There's less than a 1 in 20 chance of getting that result. Here we got much less than 1 in 20. So this is a very strong indicator that the null hypothesis is incorrect, and the drug definitely has some effect.