Example of how to make a conclusion in a test about a proportion.
- [Instructor] 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 used 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 of Z is approximately equal to 1.84 and a corresponding P value of approximately 0.033. Assuming the conditions for inference were met, which of these is an appropriate conclusion? And we have our four conclusions here. At any point in encourage you to pause this video and see if you can answer it for yourself, but now we will do it together. And just to make sure we understand what's going on, before 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 calculate a sample proportion of adults that support the tax increase. We see 113 out of 200 support it, 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 to figure out the P value. What is the probability of getting a result this much above the assumed proportion or greater, at least this 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 figure out is 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 1.84 standard deviations or at least 1.84 or more standard deviations above the mean? And they did 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 1/100, the P value right over here is over 3/100. It's roughly 3.3%. 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 supported the tax increase. 'cause remember our null hypothesis is that 50% do and we're failing to reject this. So that's not gonna be true. At that same significance level, they should conclude that less than 50% of adults 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%. So at the alpha equals to 5/100 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 0.033. 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 the situation at all. If we're rejecting our null hypothesis right over here, then it should suggest this alternative.