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## Statistics and probability

### Course: Statistics and probability > Unit 12

Lesson 3: Tests about a population proportion- Constructing hypotheses for a significance test about a proportion
- Writing hypotheses for a test about a proportion
- Conditions for a z test about a proportion
- Reference: Conditions for inference on a proportion
- Conditions for a z test about a proportion
- Calculating a z statistic in a test about a proportion
- Calculating the test statistic in a z test for a proportion
- Calculating a P-value given a z statistic
- Calculating the P-value in a z test for a proportion
- Making conclusions in a test about a proportion
- Making conclusions in a z test for a proportion

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# Constructing hypotheses for a significance test about a proportion

AP.STATS:

VAR‑6 (EU)

, VAR‑6.D (LO)

, VAR‑6.D.1 (EK)

, VAR‑6.D.2 (EK)

, VAR‑6.D.3 (EK)

, VAR‑6.D.4 (EK)

, VAR‑6.D.5 (EK)

Constructing hypotheses for a significance test about a proportion.

## Want to join the conversation?

- why is Ha in the second example p >0.9 instead of p not equal to 0.9?(5 votes)
- Because the problem statement indicates the researchers want to test if the proportion is now higher, hence the ">" sign. If it stated "test if the proportion is different" then you'd have an alternative hypothesis of p =\= 0.9(11 votes)

- In the second example, H_a is the hypothesis that p > 0.9 (internet access has increased significantly), while H_0 is the hypothesis that p = 0.9 (internet access is the same as before). Why isn't the null hypothesis that p <= 0.9 (internet access has not increased significantly)?

In other words, the alternative hypothesis is only capturing the "internet access has increased" tail of the normal distribution. There's a nonzero probability that internet access has decreased in a statistically-significant manner, but that probability isn't captured in either H_0 or H_a. Shouldn't that be taken into account?(4 votes)- Our null hypothesis is based on a certain poll which states that "...about 90% of homes in California had access to the internet". 90%, not 90% or fewer. Thus, null hypothesis is p = 0.9, not p <= 0.9.

Our alternative hypothesis is based on a theory that internet access has become more available. Quote: "...want to test if that proportion is now higher". Meaning, alternative hypothesis is p > 0.9, not p ≠ 0.9.

We do not know why the researchers had that theory, our hypothesis testing is simply based on the data presented.(3 votes)

- Out of curiosity, in the second example, what would happen if the researchers found after testing that p was less than 0.9? Would they reject the null hypothesis or not?(3 votes)
- What happens when knowledge in an area evolves and which changes a null hypothesis from true to false or vice versa?(3 votes)
- The video is not showing up for me. Any recommendations?(1 vote)
- I wonder whether we're trying to prove what the report says is true or challenge the result of the report in the first question?(1 vote)

## Video transcript

- [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.