Studies that compare treatments for chronic medical conditions such as headaches can use the same subjects for each treatment. This type of study is commonly referred to as a crossover design. With a crossover design, each person crosses over from using one treatment to another during the study. One such study considered a drug (a pill called Sumatriptan) for treating migraine headaches in a convenience sample of children. 8 The study observed each of 30 children at two times when he or she had a migraine headache. The child received the drug at one time and a placebo at the other time. The order of treatment was randomized and the study was double-blind. For each child, the response was whether the drug or the placebo provided better pain relief. Let p denote the proportion of children having better pain relief with the drug, in the population of children who suffer periodically from migraine headaches. Can you conclude that p 7 0.50, with more than half of the population getting better pain relief with the drug, or that p 6 0.50, with less than half getting better pain relief with the drug (i.e., the placebo being better)? Of the 30 children, 22 had more pain relief with the drug and 8 had more pain relief with the placebo.
a. For testing H0: p = 0.50 against Ha: p = 0.50, show that the test statistic z = 2.56.
b. Show that the P-value is 0.01. Interpret.
c. Check the assumptions needed for this test, and discuss the limitations due to using a convenience sample rather than a random sample.

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