Tests about a population proportion
Constructing hypotheses for a significance test about a proportion
- [Instructor] We're told that Amanda read a report saying that 49% of teachers in the United States were members of a labor union. She wants to test whether this holds true for teachers in her state, so she is going to take a random sample of these teachers and see what percent of them are members of a union. Let P represent the proportion of teachers in her state that are members of a union. Write an appropriate set of hypotheses for her significance test. So, pause this video and see if you can do that. And let's do it together. So, what we want to do for this significance test is set up a null hypothesis and an alternative hypothesis. Now, your null hypothesis is the hypothesis that hey, there's no news here, it's what you would expect it to be and so, if you read a report saying that 49% of teachers in the United States were members of labor unions, well, then it would be reasonable to say that the null hypothesis, the no news here is that the same percentage of teachers in her state are members of a labor union, so that percentage, that proportion is P, so this would be the null hypothesis, that the proportion in her state is also 49% and now what would the alternative be? Well, the alternative is that the proportion in her state is not 49%. This is the thing that hey, there'd be news here, there'd be something interesting to report, there's something different about her state. And how would she use this? Well, she would take a sample of teachers in her state, figure out the sample proportion, figure out the probability of getting that sample proportion if we were to assume that the null hypothesis is true. If that probability is lower than a threshold which she should have said ahead of time, her significance level, then she would reject the null hypothesis which would suggest the alternative. But let's do another example here. According to a very large poll in 2015, about 90% of homes in California has access to the internet. Market researchers want to test if that proportion is now higher, so they take a random sample of 1,000 homes in California and find that 920 or 92% of homes sampled have access to the internet. Let P represent the proportion of homes in California that have access to the internet. Write an appropriate set of hypotheses for their significance test. So, once again, pause this video and see if you can figure it out. So, once again, we want to have a null hypothesis and we want to have an alternative hypotheses. The null hypothesis is that hey, there's no news here and so, that would say that it's kind of the status quo. That the proportion of people who have internet is still the same as the last study. It's still the same at 90% or I could write 90% or I could write 0.9 right over here. Now, some of you might have been tempted to put 92% there but it's very important to realize 92% is the sample proportion, that's the sample statistic. When we're writing these hypotheses, these are hypotheses about the true parameter. What is the true proportion of homes in California that now have the internet? And so, this is about the true proportion. And so, the alternative here is that it's now greater than 90% or I could say it's greater than 0.9, I could have written 90% or 0.9 here and so, they really in this question they wrote this to kind of distract you to make you think maybe I have to incorporate this 92% somehow and once again, how will they use these hypotheses? Well, they will take the sample in which they got 92% of homes sampled had access to the internet, so this right over here is my sample proportion and then they're gonna figure out well, what's the probability of getting this sample proportion for this sample size if we were to assume that the null hypothesis is true? If this probability of getting this is below thresholds, below alpha, below our significance level, then we'll reject the null hypothesis, which would suggest the alternative.