Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to use a calculator display to perform hypothesis testing on proportions. Here's our problem statement: A certain drug is used to treat asthma. In a clinical trial of the drug, 20 of 287 treated subjects experienced headaches, based on data from the manufacturer. The accompanying calculator display shows results from a test of the claim that less than 10% of treated subjects experienced headaches. Use the normal distribution as an approximation to the binomial distribution, and assume a 5% significance level to complete Parts A through E below.
OK, Part A asks, “Is the test two-tailed, left-tailed, or right-tailed?” To determine this, we need to know what our alternative hypothesis is. And the alternative hypothesis comes from the claim that's being tested. Here the test is that were claiming that less than 10% of treated subjects experience headaches. You can see that right here in the calculator display as well that the proportion that we're looking for is less than 10%. This has no equality with it, and so we can adopt this claim as the alternative hypothesis.
Since this is the alternative hypothesis, we know that this is a one-tailed test to the left because “less than” is like an arrow pointing to the left. Excellent!
Part B asks for the test statistic. You can see that here in the calculator display. Notice they've asked for a z-score — so “z equals” then there's your answer field. In here you've got your z-value, so you just take that right off the calculator display. There's no need to put any data into StatCrunch. There's no need to do any number crunching in StatCunch. The display for the results of the test is right here, so you can just take that from the display that they have right there. Well done!
Part C asks for the P-value, which is right below the test statistic in the calculator display. I'm asked around to four decimal places. Well done!
Part D asks, “What is the null hypothesis, and what do you conclude about it?” Well, the null hypothesis is by definition a statement of equality, so there's only one correct answer. And it's the one with a statement of equality. Excellent!
Now we're asked to decide whether to reject the null hypothesis. As we go back to our P-value, which is a little more than 4%, we're asked to test our claim at the 5% significance level. Our P-value is less than our significance level. Therefore, we’re inside the region of rejection, and we're going to reject the null hypothesis, because the P-value is less than or equal to the significance level alpha. Well done!
And now Part E asks, “What is the final conclusion?” Well, we rejected the null hypothesis, and so therefore there is sufficient evidence. But do we want to support the claim or do we want rejection of the claim? Well, to determine that, we go back to our alternative hypothesis. Remember, our alternative hypothesis matches the claim that was being made. Because it matches, we want to select “support the claim.” If it didn't match, then we would select “warrant rejection.” But it does match, so we're going to select “support the claim.” And we rejected the null hypothesis, so there is sufficient evidence. Nice work!
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