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# Using a confidence interval to test slope

AP.STATS:
UNC‑4 (EU)
,
UNC‑4.AH (LO)
,
UNC‑4.AH.1 (LO)

## Video transcript

hashem obtained a random sample of students and noticed a positive linear relationship between their ages and their backpack weights a 95% confidence interval for the slope of the regression line was 0.3 9 plus or minus 0.23 Hashem wants to use this interval to test the null hypothesis that the true slope of the population regression line so this is a population parameter right over here for the slope of the population regression line is equal to 0 versus the alternative hypothesis is that the true slope of the population regression line is not equal to 0 at the alpha is equal to 0.05 level of significance assume that all conditions for inference have been met so given the information that we just have about what Hashem is doing what would be his conclusion would he reject the null hypothesis which would suggest the alternative or would he be unable to reject the null hypothesis well let's just think about this a little bit we have a 95% confidence interval let me write this down so our 95 percent confidence confidence interval could write it like this or you could say that it goes from zero point three nine minus zero point two three so that'd be 0.16 so it goes from zero point one six until zero point three nine plus 0.23 is going to be what 0.62 now what a 95% confidence interval tells us is that 95% of the time that we take a sample and we construct a 95% confidence interval that 95% of time we do this it should overlap with the true population parameter that we are trying to estimate but in this hypothesis test remember we are assuming that the true population parameter is equal to zero and that does not overlap with this conference interval so assuming let me write this down assuming null hypothesis is true we are in the less than or equal to 5% chance of situations situations where we're beta not overlap overlap with 95% intervals and the whole notion of a hypothesis testing is you assume the null hypothesis you take your sample and then if you get statistics and if the probability of getting those statistics for something even more extreme than those statistics is less than your significance level then you reject the null hypothesis and that's exactly what's happening here and this null hypothesis value is nowhere even close to overlapping it's over 1600s below the low end of this bound and so because of that we would reject the null hypothesis or reject the null hypothesis which suggests the alternative which suggests the alternative hypothesis and one way to interpret this alternative hypothesis that beta is not equal to zero is that there is there is a non zero linear relationship a relationship between between ages and backpack weights ages and backpack weights and we are done
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