# Why Is 0.05 Commonly Used as a Significance Level? Suppose someone tosses a coin you can’t…

Why Is 0.05 Commonly Used as a Significance Level?

Suppose someone tosses a coin you can’t confirm ahead of time is fair. Based on a couple of tosses, you need to decide whether you can reject H0: p = probability of head = 0.50 (fair coin) in favor of Ha: p  0.50. Consider the following:

On the first toss, the coin lands heads. Would you reject H0 and conclude that the coin is not fair?

On the second toss, the coin lands heads. Would you now reject H0 and conclude that the coin is not fair?

On the third toss, the coin also lands heads. Would you now reject H0 and conclude that the coin is not fair?

On the fourth toss, the coin lands heads again. Are you now starting to doubt that H0 is true and willing to believe that Ha may be true?

On the fifth toss, the coin lands heads again. Are you now ready to reject H0 and conclude that the coin cannot be fair?

If you are like many people, by the time you see the fifth straight head, you are willing to predict that the c Activity 1 coin is not fair. (By then, you may even visually inspect the coin to make sure it has different sides.) If the null hypothesis that p = 0.50 is actually true, then by the binomial distribution, the probability of five heads in a row is (0.50)5 = 1/32 = 0.03. For a two-sided test, this result gives a P-value of 210.032 = 0.06, close to 0.05. So, for many people, it takes a P-value near 0.05 before they feel there is enough evidence to reject a null hypothesis. (For comparison, many people would not get too suspicious of the null hypothesis after four straight heads, for which the P-value is 0.125.) This may be one reason the significance level of 0.05 has become common over the years in a wide variety of disciplines that use significance tests.

Exercise 9.129 presents this activity in the context of comparing a drug (for instance, against migraine) to a placebo to test which one is better. You start by assuming they are equally effective (the null hypothesis). One patient after another tries both the drug and the placebo (in random order) and reports which one is better. After the first five patients report better results with the placebo, are you willing to accept that the placebo works better?

Exercise 9.129

A study compares a new drug for pain relief in patients with chronic migraine against a placebo. Each patient is given both the drug and the placebo (in random order) and reports which one works better. Let p denote the probability that the pain relief is better with the drug. You have to decide whether you can reject H0: p = 0.50 in favor of Ha: p   0.50 based on seeing the results of one patient after another. Ahead of time, you have no idea whether the drug or placebo works better, and you assume they are equally effective (i.e., you assume the null hypothesis is true.) Each student should indicate

a. How many consecutive patients reporting better relief with the drug would be necessary before he or she would feel comfortable rejecting H0: p = 0.50 in favor of Ha: p   0.50 and concluding that the drug works better.

b. How many consecutive patients reporting better relief with the placebo would be necessary before he or she would feel comfortable rejecting H0: p = 0.50 in favor of Ha: p  0.50 and concluding that the placebo works better.

The instructor will compile a “distribution of significance levels” for the two cases. Are they the same? In principle, should they be?

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