Over the years, the Food and Drug Administration of the United States (FDA) has
ID: 3295818 • Letter: O
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Over the years, the Food and Drug Administration of the United States (FDA) has worked very hard to avoid making type II errors. A type II error occurs when the FDA approves a drug that is not both safe and effective. Despite the agency's efforts, however, bad drugs do on occasion get through to the public. For example Omniflox, an antibiotic, had to be recalled less than six months after its approval due to reports of severe adverse reactions, which included a number of deaths. Similarly, Fenoterol, an inhaled drug intended to relieve asthma attacks, was found to increase the risk of death rather than decrease it [17]. Is there any way for the FDA to completely eliminate the occurrence of type II errors? Explain. Data from the Framingham Study allow us to compare the distributions of initial serum cholesterol levels for two populations of males: those who go on to develop coronary heart disease and those who do not. The mean serum cholesterol level of the population of men who do not develop heart disease is mu = 219 mg/100 ml and the standard deviation is mu = 41 mg/100 ml [18]. Suppose, however, that you do not know the true population mean: instead, you hypothesize that mu is equal to 244 mg/100 ml. This is the mean initial serum cholesterol level of men who eventually develop the disease. Since it is believed that the mean serum cholesterol level for the men who do not develop heart disease cannot be higher than the mean level for men who do, a one-sided test conducted at the alpha = 0.05 level of significance is appropriate. (a) What is the probability of making a type I error? (b) If a sample of size 25 is selected from the population of men who do not go on to develop coronary heart disease, what is the probability of making a type II error? (c) What is the power of the test? (d) How could you increase the power? (e) You wish to test the null hypothesis H_0: mu greaterthanorequalto 244 mg/100 ml against the alternative H_a: muExplanation / Answer
14. a. The Type I error refer to rejecting the null hypothesis, when it is true. The probability of Type I error is the significance level, alpha=0.05.
b. From information given, sigma=41, n=25, sigma Xbar=sigma/sqrt n=41/sqrt 25=8.2
It is also known that H0:mu>=244 versus H1:mu<244
Thus, this is a left-tailed test. One would fail to reject the null (that is commit a Type II error) if one get a Z statistic greater than -1.64. Substitute the values in following Z score formula to compute X.
-1.64=(X-mu)/sigma xbar=(X-244)/8.2
X=244-1.64*8.2
=230.552
This implies that one will fail to reject null as long as one draws a sample mean greater than 230.552. Now, find probbaility of drawing sample mean greater than 230.552, given mu=219 and sigmaxbar=8.2, which gives the probbaility of making Type II error.
P[Z>(230.552-219)/8.2]=P(Z>1.41)
=1-0.9207 [z table gives area to the left of z, thus subtract the area corresponding to z=1.41 from 1]
=0.0793
c. Power of a test=1-P(Type II error)=1-0.0793=0.9207.
d. In order to increase the power, increase the sample size.
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