Question

the proportion of americans who have frequent migraines is 15.2% according to the cdc. an acupuncturist claims that her treatment can reduce this figure significantly. a random sample of 698 americans is administered the acupuncturists treatment and 80 report experiencing migraines. i highly recommend you explore the possible errors with a contingency table, making note of the meaning of each possibility. a. select the null and alternative hypotheses to the scenario using the correct symbols. $h_{0}: mu = 0.152$ $h_{a}: mu lt 0.152$ $h_{0}: p = 0.152$ $h_{a}: p lt 0.152$ $h_{0}: p lt 0.152$ $h_{a}: p = 0.152$ b. explain what a type 1 error is in this scenario. a type 1 error would be concluding that the migraine treatment can reduce occurence of frequent migraines when in fact, it can. a type 1 error would be concluding that the migraine treatment can reduce occurence of frequent migraines when in fact, it cannot. a type 1 error would be concluding that the migraine treatment cannot reduce occurence of frequent migraines when in fact it cannot. a type 1 error would be concluding that the migraine treatment cannot reduce occurence of frequent migraines when in fact it can

Explanation:

• Part a:
• In hypothesis testing for proportions, we use \(p\) (not \(\mu\) which is for means). The null hypothesis \(H_0\) is a statement of no - effect or no - change. Here, the claim is about reducing the proportion of people with frequent migraines. The null hypothesis should be the status - quo (the proportion is \(0.152\)), and the alternative hypothesis \(H_a\) is the claim (the proportion is less than \(0.152\)).
• Part b:
• A Type - 1 error is rejecting the null hypothesis \(H_0\) when \(H_0\) is actually true. In this context, \(H_0:p = 0.152\) (the treatment has no effect on reducing the proportion of migraines). Rejecting \(H_0\) means concluding that the treatment reduces the proportion of migraines. If \(H_0\) is true (treatment has no effect), then a Type - 1 error is concluding that the treatment reduces the occurrence of migraines when it actually does not.

Answer:

• a: \(H_0:p = 0.152\), \(H_a:p\lt0.152\) (the second option in part a)
• b: A type 1 error would be concluding that the migraine treatment can reduce occurence of frequent migraines when in fact, it cannot. (the second option in part b)