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Video transcript

Katarina was testing her null hypothesis is that the true population mean of some data set is is equal to zero versus her alternative hypothesis is that it's not equal to zero and then she takes a sample of six observations and then using that sample her test statistic I can never say that her test statistic was T is equal to 2.75 assume that the conditions for inference were met what is the approximate p-value for Katerina's test so like always pause this video and see if you can figure it out well I just always like to remind ourselves what's going on here so there's some population here she has a null hypothesis that the mean is equal to zero but the alternative is that it's not equal to zero she wants to test her null hypothesis so she takes a sample size or she takes a sample of size six from that since we care about the the population parameter we care about as a population mean she would calculate the sample mean in order to estimate that and the sample standard deviation and then from that we can calculate this T value the T value is going to be equal to the difference between her sample mean and the assumed the assumed population mean from the null hypothesis that's what this little sub zero means it means it's the assumed mean from the null hypothesis divided by our estimate of the standard deviation of the sampling distribution I say estimate because unlike when we were dealing with proportions with proportions we can actually calculate the assumed based on the null hypothesis sampling distribution standard deviation but here we have to estimate it and so it's going to be our sample standard deviation divided by the square root of n now in this example 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 but just let's just think about what that is asking us to do so the null hypothesis is that the mean is zero the alternative is is that it is not equal to zero so this is a situation where if we're looking at the T distribution right over here it's my quick drawing of a T distribution if this is the mean of our T distribution well we care about is things that are at least 2.75 above the mean and at least 2.75 below the mean because we care about things that are different from the mean not just things that are greater than the mean or less than the mean so we would look at we would say well what's the probability of getting a t-value that is more 2.75 or more above the mean and similarly what's the probability of getting a t-value that is 2.75 or less below or 2.75 or Moe more below the mean so this is negative to 0.75 right over there so what we have here is a tea table and the tea table is a little bit different than a z table because there's several things going on first of all you have your degrees of freedom that's just going to be your sample size minus one so in this example our sample size is 6 so 6 minus 1 is 5 and so we are going to be we are going to be in this row right over here and then what you want to do is you want to look up your T value this is T distribution critical value so we want to look up 2.75 on this row and we see 2.75 it's a little bit less than that but that's the closest value it's a good bit more than this right over here but it's so it's a little bit closer to this value than this value and so our tail probability and remember this is only giving us this probability right over here our tail probability is going to be between 0.025 and 0.02 and it's going to be closer than to this one it's going to be approximately this it'll actually be a little bit greater because we're going to go a little bit in that direction because we are less than 2 point 7 5 7 and so we could say this is approximately 0.02 well that's 0.02 approximately the t-distribution symmetric this is going to be approximately 0.02 and so our p value which is going to be the probability of getting a t-value that is at least 2.75 above the mean and or 2.75 below the mean the p-value p-value is going to be approximately the sum of these areas which is 0.04 and then of course Katarina would want to compare that to her significance level that she said ahead of time 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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