Using a table to estimate P-value from t statistic (video) | Khan Academy

Video transcript

– [ Instructor ] Caterina was testing her null hypothesis is that the true population bastardly of some data set is peer to zero versus her alternative guess, is that it ‘s not equal to zero and then she takes a sample distribution of six observations and then using that sample distribution her screen statistic, I can never say that, test statistic was T is equal to 2.75. Assume that the conditions for inference were met. What is the approximate P-value for Caterina ‘s test ? Like constantly, pause this video and see if you can figure it out. I barely always like to remind ourselves what ‘s going on here, so there ‘s some population here. She has a null hypothesis that the entail is peer to zero, or the option is that it ‘s not equal to zero. She wants to test her null guess so she takes a sample of size six. From that, since the population argument we care about is a population mean, she would calculate the sample distribution beggarly in order to estimate that and the sample distribution standard deviation. From that, we can calculate this T value. The T respect is going to be equal to the remainder between her sample mean and the wear population mean from the null guess, that ‘s what this little substitute zero means, it means it ‘s the bear base from the nothing guess, divided by our estimate of the standard deviation of the sampling and distribution. I say estimate because unlike when we were dealing with proportions, with proportions we can actually calculate the wear, based on the nothing guess, sampling distribution standard deviation, but here we have to estimate it. It ‘s going to be our sample standard deviation divided by the squarely root of N. In this exemplar, they calculated all of this for us. They said hey, this is going to be equal to 2.75 and so we can just use that to figure out our P-value. Let ‘s just think about what that is asking us to do. The nothing hypothesis is that the mean is zero. The alternative is is that it is not equal to nothing. This is a situation where, if we ‘re looking at the T distribution right over here, my quick string of a T distribution. If this is the mean of our T distribution, what we care about is things that are at least 2.75 above the bastardly and at least 2.75 below the bastardly because we care about things that are different from the mean, not fair things that are greater than the average or less than the mean. We would look at, we would say what ‘s the probability of getting a T rate that is 2.75 or more above the bastardly, and similarly, what ‘s the probability of getting a T value that is 2.75 or more below the mean ? This is damaging 2.75 good over there. What we have here is a T table and a T board is a small sting different than a Z table because there ‘s respective things going on. First of wholly, you have your degrees of freedom. That ‘s fair going to be your sample size subtraction one. In this exemplar, our sample distribution size is six, so six minus one is five, and then we are going to be in this row right over here. then what you want to do is, you want to look up your T rate. This is T distribution critical values, so we want to look up 2.75 on this row. We see 2.75, it ‘s a little bit less than that but that ‘s the closest value. It ‘s a good act more than this good over here, so it ‘s a little act closer to this value than this value. Our fag end probability, and remember, this is only giving us this probability correct over here, our tail probability is going to be between 0.025 and 0.02 and it ‘s going to be closer to this one. It ‘s gon sodium be approximately this. It ‘ll actually be a little spot greater because we ‘re gon na go a small spot in that direction because we are less than 2.757. We can say this is approximately 0.02. That ‘s 0.02 approximately, the T distribution is symmetrical, this is going to be approximately 0.02. Our P-value, which is going to be the probability of getting a T value that is at least 2.75 above the mean or 2.75 below the mean, the P-value is going to be approximately the kernel of these areas, which is 0.04. then of course, Caterina would want to compare that to her significance degree that she set ahead of prison term, and if this is lower than that, then she would reject the null hypothesis and that would suggest the alternative. If this is not lower than her significance level, well then she would not be able to reject her null hypothesis.

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