Medical groups have long warned about the dangers of indoor tanning. The America
ID: 3359072 • Letter: M
Question
Medical groups have long warned about the dangers of indoor tanning. The American Medical Association and the American Academy of Dermatology unsuccessfully petitioned the Food and Drug Administration in 1994 to ban cosmetic-tanning equipment. Three years later, the Federal Trade Commission warned the public to beware of advertised claims that “unlike the sun, indoor tanning will not cause skin cancer or skin aging” or that you can “tan indoors with absolutely no harmful side effects”.
In February 1999, still under pressure from the medical community, the FDA announced that current recommendations “may allow higher exposures” to UV radiation “than are necessary”. The agency proposed reducing recommended exposures and requiring simpler wording on consumer warnings. But it has not yet implemented either of these changes. An FDA spokeswoman told us that “the agency decided to postpone amendment of its standard pending the results of ongoing research and discussions with other organizations.
To make matters worse, only about half the states have any rules for tanning parlors. In some of these states, the regulation is minimal and may not require licensing, inspections, training, record keeping, or parental consent for minors. Despite this, nearly 30 million Americans, including a growing number of teenage girls, are expected to visit a tanning salon in 2005.
In a recent survey of 296 indoor-tanning facilities around the country, to our knowledge the first nationwide survey of its kind, we found evidence of widespread failures to inform customers about the possible risks, including premature wrinkling and skin cancer, and to follow recommended safety procedures, such as wearing eye goggles. Many facilities made questionable claims about indoor tanning; that it’s safer than sunlight, for example, and is well controlled.
a.) Given the fact that there are over 150,000 tanning facilities in the U.S., is the condition for independence of survey results satisfied? _______ Explain your answer.
b.) Assuming that the true proportion for tanning facilities that state "tanning in a salon is the same as tanning in the sun with respect to skin cancer" is 25%, describe the sampling distribution of p^ . State the 3 properties and explain.
c.) Calculate the probability that less than 22.6% of randomly selected tanning salon facilities would state that tanning in a salon is the same as tanning in the sun with respect to causing skin cancer.
d.) Assuming that the true proportion for tanning facilities that state that "tanning in a salon does not cause wrinkled skin" is 18%, describe the sampling distribution of p^. Show work.
e.) Calculate the probability that at least 14.2% will state that tanning in a salon does not cause wrinkled skin.
f.) Would it be unusual for 40 or fewer facilities to state that tanning in a salon does not cause wrinkled skin? Show the process to verify your answer.
Explanation / Answer
Solution :-
a)
Yes, the condition for independence of survey results is satisfied because the sample size is large.
b)
The sampling distribution follows normal distribution.
properties are : (1) Bell shape
(2) median is equal to the mean
(3) Can use standard normal table
c)
The probability is
P(phat <0.226) = P((phat-p) / sqrt(p*(1-p)/n) < (0.226-0.25) / sqrt(0.25*0.75/296))
= P(Z < -0.95) = 0.1711 (from standard normal table)
d)
The sampling distribution follows normal distribution
e)
The probability is
P(phat <0.142) = P(Z< (0.142-0.18) / sqrt(0.18*(1-0.18) / 296))
= P(Z<-1.7) = 0.0446
f)
It would be unusual for 40 or fewer facilities to state that tanning in a salon does not cause wrinkled skin because the probability is low.