The Centers for Disease Control and Prevention reported that 20.9% of American a
ID: 3268458 • Letter: T
Question
The Centers for Disease Control and Prevention reported that 20.9% of American adults smoked regularly in 2004. Treat this as the true parameter value (i.e. p = 0.209) for the current population of American adults. Suppose you plan to take a random sample of 100 American adults and count the number in the sample who smoke.
Carefully state what each of the following are in this example:
a) the random variable, X,
b) a success,
c) the probability of a success, p,
d) the number of trials, n.
e) In your random sample of 100 American adults, will the sample proportion who smoke equal exactly 0.209? Explain.
f) What does the Central Limit Theorem (CLT) say about how all possible sample proportions (from samples of size 100) of people who smoke would vary from sample to sample? (Hint: comment on the shape, center, and spread of the sampling distribution.)
g). Draw a sketch of this sampling distribution predicted by the CLT. Appropriately label the axes.
h) In your sketch, shade the area corresponding to a sample proportion exceeding 0.25. What is the probability of obtaining a sample in which the proportion who smoke is greater than 0.25?
i) Suppose you instead plan to take a random sample of 400 American adults. How do you expect the probability that the sample proportion who smoke will exceed 0.25 to differ from when the sample size is 100? Explain your reasoning, including a well-labeled and shaded sketch of the sampling distribution when n = 400.
j). Calculate the probability asked for in part i above.
Explanation / Answer
a. The number of adults who smoked regularly in 2004 is a random variable, because it takes different values by chance.
b. Success refer to ratio of number of events to number of trials, where, number of trials refer to number of American adults selected, and number of events refer to number of American adults who smoked regularly in 2004.
c. According to reports from Centr eof disease control and prevention, the probability of success, p=0.209.
d. Number of trials refer to number of American adults sampled, that is n=100.