According to the American Academy of Cosmetic Dentistry, 75% of adults believe t

Question

According to the American Academy of Cosmetic Dentistry, 75% of adults believe that an unattractive smile hurts career success. Suppose that 25 adults are randomly selected. What is the probability that 15 or more of them would agree with the claim? Please give the answer to four decimal places.

Explanation / Answer

This problem can be solved using the Tables for the cumulative binomial probability distribution, or by using the normal approximation. From Table , for n = 25, p(success) = 0.75: p(k14) = 0.97 since these probabilities must sum to one. answer -> 0.97 Alternatively, using the normal approximation, with mean = n*p = 25*0.75 = 18.75. and variance = n*p*(1-p) = 25* 0.75 *0.25 = 4.6875, so the standard deviation is the square root of this number, i.e. 2.165. So the standardized normal variate, z, for k=15 is: z = (15 – 18.75) /2.165 = - 1.73. From Table , for the normal distribution, z = 1.73 has an area (probability) above the mean of 0.4582 so 0.5 minus 0.4582 = 0.04 is in the upper tail implying that p(z-1.73) = 0.96 = p(k>=15), close to the previous answer

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