If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Making conclusions in a test about a proportion

AP.STATS:
DAT‑3 (EU)
,
DAT‑3.B (LO)
,
DAT‑3.B.2 (EK)
,
DAT‑3.B.3 (EK)
,
DAT‑3.B.4 (EK)
,
DAT‑3.B.5 (EK)
,
DAT‑3.B.8 (EK)
,
DAT‑3.B.9 (EK)

## Video transcript

a public opinion survey investigated whether a majority more than 50% of adults supported a tax increase to help fund the local school system a random sample of 200 adults showed that 113 of those sampled supported the tax increase researchers use these results to test the null hypothesis is that the proportion is 0.5 the alternative hypothesis is that it's greater than 0.5 where P is the true proportion of adults that support the tax increase they calculated a test statistic statistic of Z is approximately equal to one point eight four and a corresponding p-value of approximately zero point zero three three assuming the conditions for inference were met which of these is an appropriate conclusion and we have our four conclusions here at any point I encourage you to pause this video and see if you can answer it for yourself but now we will do it together and just make sure we understand what's going on before even we even cut to the chase and get to the answer so what we do is we have this population and we are going to sample it so n is equal to 200 from that sample we can't calculate a sample proportion of adults that support the tax increase we see 113 out of 200 supported which is going to be equal to let's see that is the same thing as 56.5% so 56.5% and so the key is is to figure out the p-value what is the probability of getting a result this much above the assumed proportion or greater at least as much above the assumed proportion if we assume that the null hypothesis is true and if that probability if that p-value is below a preset threshold if it's below our significance level they haven't told it to us yet it looks like they're gonna give some in the choices well then we would reject the null hypothesis which would suggest the alternative if the p-value is not lower than this then we will fail to reject the null hypothesis now to calculate that p-value to calculate that probability what we what we figure out is well how many in our sampling distribution how many standard deviations above the mean of the sampling distribution and the mean of the sampling distribution would be our assumed population proportion how many standard deviations above that mean is this right over here and that is what this test statistic is and then we can use this to look at a Z table and say all right well in a normal distribution what percentage or what is the area under the normal curve that is further than one point eight four standard deviations or at least one point eight four or more standard deviations above the mean and they give that for us as well so really what we just need to do is compare this p-value right over here to the significance level if the p-value is less than our significance level then we reject reject our null hypothesis and that would suggest the alternative if this is not true then we would fail to reject the null hypothesis so let's look at these choices and if you didn't answer it the first time I encourage you to pause the video again so at the alpha is equal to 0.01 significance level they should conclude that more than 50% of adults support the tax increase so if the Alpha is 100 the p-value right over here is over 300s it's roughly 3.3 percent so this is a situation where our p value our p value is greater than or equal to alpha in fact it's definitely greater than alpha here and so here we would fail to reject we would fail to reject our null hypothesis and so we wouldn't conclude that more than 50% of adults support the tax increase because remember our null hypothesis is that 50% do and we're failing to reject this so that's not going to be true at that same significance level they should conclude that less than 50% of adult support the tax increase no we can't say that either we just failed to reject this null hypothesis that the true proportion is 50% at the alpha is equal to so at the alpha equals to 500th significance level they should conclude that more than 50% of adults support the tax increase well yeah in this situation we have our p-value which is zero point zero three three it is indeed less than our significance level in which case we reject reject the null hypothesis and if we reject the null hypothesis that would suggest the alternative that the true proportion is greater than 50% and so I would pick this choice right over here and then choice D at that same significance level they should conclude that less than 50% of adults support the tax increase no not not the situation at all if we're rejecting our null hypothesis right over here then we should it should suggest this alternative