Clinical Trials of OxyContin OxyContin (oxycodone) is a drug u...

Question

Clinical Trials of OxyContin OxyContin (oxycodone) is a drug used to treat pain, but it is well known for its addictiveness and danger. In a clinical trial, among subjects treated with OxyContin, 52 developed nausea and 175 did not develop nausea. Among other subjects given placebos, 5 developed nausea and 40 did not develop nausea (based on data from Purdue Pharma L.P.). Use a 0.05 significance level to test for a difference between the rates of nausea for those treated with OxyContin and those given a placebo.

a. Use a hypothesis test.

a. Use a hypothesis test.

Accepted Answer

Welcome back, everyone. In this problem, a new medication, pain away, is being tested for its side effects. In a clinical study, 38 out of 190 patients treated with pain away reported headaches, while 7 out of 55 patients in the placebo group reported headaches. At the 0.05 significance level, use an appropriate confidence interval to identify if there is evidence of a difference in the rate of headaches between the pain away and the placebo groups. Now if we are going to. Use or find an appropriate confidence interval, OK, to see if there's evidence of a difference, then that means we'll first need to come up with our hypothesis based on the claim. Now so far we know that the sample proportions are from two simple random samples, so every individual has an equal chance of being selected. We also know they're independent because the study used separate groups with no overlapping patients and no matching or pairing. And for each of the two samples, there are at least 5 successes and at least 5 failures. OK? So. For another hypothesis we can say. That the first proportion is equal to the second proportion. In other words, we are assuming the status quo. There is no difference. The proportions are equal, which means our alternative hypothesis H1 is that the proportions are not equal. P1 is not equal to P2. Now, to solve, we'll need to calculate the sample proportions, find the margin of error, and then use that to help us find the confidence interval. So first, let's find the sample proportions, OK? Let's find the sample proportions. Now we know that the sample proportion is going to be equal to the number of successes divided by the total outcomes, so our first proportion P1 will be equal to X1 divided by N1, which is going to be 38 divided by 190 or 0.2. Likewise, Our second sample proportion P2 will be X2 divided by N2, 7 divided by 55, which equals or is approximately equal to 0.127. OK? So we have our sample proportions. Now we can use that to find the margin of error. Because remember that the margin of error E is going to be equal to our critical value, OK. In this case for our two-tail test, so it's a critical value for half of alpha multiplied by the square root, OK, of our sample proportion P1 multiplied by Q1, the probability of failure in our first sample divided by N1 plus P2Q2. Divided by N2. Now, we can, we already have P1 and P2. We know N1 and N2, and we can calculate Q1 and Q2, but we need our critical value. OK, Z half of alpha. At the 0.05 significance level, as stated in our problem. OK. For the two-tailed test, cuz remember we're testing if the proportions are equal or if they're not equal, we're testing for a difference, OK OK. Then we use a 95%. Confidence interval. So if we go to our table of common critical values for confidence levels, then this implies that our. Critical value, OK. Z half of alpha will be 1.96 for a confidence level of 95%, OK? So now that we have this critical value, we can plug it into our equation to find the margin of error. Because no, this would imply then that he. Is going to be equal to 1.96 multiplied by the square root of 0.2. Multiplied by 1 minus 0.2 divided by 190. Plus 0.127 multiplied by 1 minus 0.127. Divided by 55. I know when we go ahead and solve here, we get that to be approximately 0.105. Know that we have the margin of error, and we have our proportions, our sample proportions, then we can use that to figure out the confidence interval. So the confidence integral. OK. Confidence interval. For P1. Minus P2, OK, our population proportions is going to be. Expressed as our sample, the difference between our sample proportions minus E, OK. Less than P1 minus P2, less than the difference between our sample proportions, OK? Plus he OK. In other words, the difference between our population proportions will be equal to the difference between the sample proportions plus R minus E. Now, when we substitute the values that we know, we're really saying 0.2 minus, sorry, we're saying P1 minus P2 is going to be between 0.2 minus 0.127 minus 0.105, OK? And 0.2 minus 0.127 plus 0.5. 0.105. And when we simplify here, then we get P1 minus B2 to be to be between negative 0.0321. OK. And 0.178. So what does this mean? Well, since the 95% confidence interval for the difference in proportions includes zero, OK, because 0 is within this range, then there is insufficient evidence at the 0.05 significance level to conclude that there is a significant difference in the rate of headaches between the pain away and the placebo groups. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Hypothesis Testing

Significance Level

Contingency Table

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