hypothesis testing

Question

Identify the null and alternative hypothesis(in symbolic and sentence form), test statistic, P-value(or critical values), conclusion about the null hypothesis, and final conclusion that addresses the original claim(Don’t just say Reject the null hypothesis or fail to reject the null hypothesis). If you used the calculator, identify everything mentioned above in addition to which calculator functions and inputs for each field were used.

In a clinical study of an allergy drug, 108 of the 202 subjects reported experiencing significant relief from their symptoms. At the 0.01 significance level, test the claim that more than half of all those using the drug experience relief.

Answer 1

We are dealing with a binary variable (let’s call it r). Its 2 values are

1. ·       Relief
2. ·       No relief

This experiment is like the result of flipping a coin 202 times and analyzing the result to determine if it’s fair. The null hypothesis is that the population proportion for r is 0.5 (usually expressed as π=0.5), meaning that half of the patients experience relief. The alternate hypothesis is π>0.5, meaning that more than half of the patients experience relief.

The sample proportion is 108/202 ≈ 0.5347. Is this far enough above the null hypothesis to reject it? 

The standard deviation of the binomial is 0.25. Adjusting for this sample size, divide that by the square root of 202. This gives a standard deviation of .0176.

The z-score for our sample is (.5347 - .5)/.0176 = 1.973.

The “reject” zone is the right tail only. Using normcdf on the TI-84+ with a lower bound of 1.973 and an upper bound of 10^99, the probability that this or a higher z-score could happen if the true proportion is .5 is .0242. This is the p-value. Since this is higher than .01, we do not reject the null hypothesis. We conclude that the medication does not provide relief at the 1% significance level.

Answer 2

If we let X ~ Binomial(n,p) be the random variable denoting the number of successes out of n trials, where n = 202 (the total number of trials in our experiment), then our null hypothesis is that p ≥ 0.5, against the alternative hypothesis that p < 0.5.

H₀: p ≥ 0.5

H₁: p < 0.5

For inference on a Binomial proportion, the test statistic is:

test.stat = X.hat / n, where X.hat is the observed value of the random variable (108 in our case).

The p-value is found by finding the probability of obtaining the value we get for the test.stat from our experiment, under the assumption that our null hypothesis is true.

P₀(X ≥ X.hat) = 0.8544 (found by using a Bin(202, 0.5) probability distribution)

Conclusion: Fail to reject the null hypothesis. We cannot conclude that the allergy drug will provide relief to more than half of its users.