In this tutorial, you're going to learn about a Hypothesis Test for Population Proportions. You will specifically focus on:
Look at a situation that would require proportions.
A popular consumer report reported that 80% of all supermarket prices end in the digits 9 or 5. Suppose you check a random sample of 115 items to just check it against the consumer report, and you find that only 88 end in 9 or 5. That's less than 80%. The question is, is that significantly less than 80%?
Is this evidence that, in fact, less than 80% of all items at the supermarket have a price ending in 9 or 5?
When running a hypothesis test, same four parts every time-- null and alternative hypotheses; conditions; the test statistic and the p-value; and based on that p-value, state a decision about the null hypothesis and conclusion in the context of the problem. So in this problem, what are the null and alternative hypotheses?
Your null hypothesis is the "nothing's going on" hypothesis. In this case, it's the "there is no reason to disbelieve the consumer report" hypothesis, whereas the alternative hypothesis suspects that something's up.
Rewrite it as p equals 0.8. The true proportion of all prices ending in 9 or 5 is 80% at the supermarket. Whereas the alternative hypothesis is going to say that p, the true proportion of prices ending in 9 or 5, is below 80%.
In this problem, choose a significance level of 0.10. With the decision rule, it carries is if the p-value is less than 0.10, you’ll reject the null hypothesis in favor of the alternative.
Part 2 is checking the conditions. You should be familiar with the conditions-- randomness, independence, and normality. Look at them one at a time.
Randomness. How was the sample obtained? For independence, make sure that the population is at least 10 times the size of the sample because you're sampling without replacement. And for normality, this is where it's a little different.
Because you're using the sampling distribution of p-hat instead of x-bar, there're different conditions for normality. Use the conditions np is at least 10 and nq is at least 10. We can't use the central limit theorem here because this is not the sampling distribution of x-bar. It's the sampling distribution of p-hat, sample proportions.
Let's check these.
In the problem, it does say that the items were randomly selected, so the simple random sample condition is OK. Assume the independence piece. Assume that the population of all items at the grocery store is at least 1,150. That seems reasonable.
For normality, you know what n is, and you know what p is. p is the value from the null hypothesis. It's the 80% that you're believing is the center of the distribution. n was the sample size, 115.
Multiply all that out and get 92 for n times p. That's greater than 10. And 23 for n times q, that's also greater than 10.
So the sampling distribution of sample proportions is going to be of an approximately normal sampling distribution. All three conditions have been checked, and we're good to go.
Now you can calculate the test statistic.
It's going to be statistic minus hypothesized parameter over standard error. And so the hypothesized center here of your sampling distribution was the 0.8.Your sample statistic was the 88 out of 115. 88 out of 115, and then 0.80, one minus 0.80, and 115.. When you do the fraction, when you can evaluate it, you get negative 0.93, which you can then find on the normal distribution that is the sampling distribution for p-hat, and you can find the tail probability.
The probability that your sample proportion would be less than the one that you got-- this is the 88 out of 115 number-- is the same as the probability that the z-statistic would be less than negative 0.93. You can find that area using the normal table, and you get about 18% of the time. This means that if the null hypothesis was true and this distribution was really centered at 0.8, the true proportion of prices ending in 9 or 5 was 0.8, you would find something at least as low as we got about 18% of the time.
You can also find this probability using technology.
Based on how your p-value compares to your chosen significance level, which you may recall was 0.10, you're going to make a decision about the null hypothesis and state the conclusion. Again, that's three parts, and we need all of them.
So in your case, 0.1762 is greater than 0.10.Your decision then is you fail to reject the null hypothesis.
The conclusion is that there's not sufficient evidence to conclude that less than 80% of supermarket prices end in 9 or 5. You don't have strong enough evidence to reject the claim of the consumer report.
The steps in any hypothesis test are always the same. Null and alternative hypotheses, this is where you would also state your alpha level. Then state and verify the conditions.
Calculate the test statistic and the p-value. Finally, based on your p-value, compare it to your alpha level and make a decision about the null hypothesis and state it in the context of the problem.
In this case, we did a z-test for population proportions, and it's analogous to any other hypothesis tests that you do. The only thing that you switched up was how you verified the normality condition. You needed np to be at least 10 and nq to be at least 10.
Source: this work is adapted from sophia author jonathan osters.
A hypothesis test where we compare to see if the sample proportion of "successes" differs significantly from a hypothesized value that we believe is the population proportion of "successes."
A type of hypothesis test used to test an assumed population proportion.