An advertising campaign for a new product is tergeted to make 20% of the adult population in a metropolitan area aware of the product. After the campaign, a random sample of 400 adults in the metropolitan area is obtained. i. Find the approximate probability that 57 or fewer adults in the sample are aware of the product. ii. Should the approximation in part i. be accurate?

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An advertising campaign for a new product is tergeted to make 20% of the adult population in a metropolitan area aware of the product. After the campaign, a random sample of 400 adults in the metropolitan area is obtained.  i. Find the approximate probability that 57 or fewer adults in the sample are aware of the product.  ii. Should the approximation in part i. be accurate?

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Part i   The probability of success is 20%=0.2 and the population is 400.   Therefore use normal distribution with μ=np=0.2×400=80  Standard deviation is σ=npq=400×0.2×0.8=8  The probability that less than 57 people are aware of the product is given by:  P(x≤57)=P(z≤57−808)  =P(z≤−2.88)  =0.00199  Part ii   The distribution can be well explained by the normal distribution but if the distribution is computed by Poisson or binomial distribution, the accuracy of the normal distribution is less than the two mentioned distributions.   So the calculations in part i are accurate if the distribution is uniform, if the distribution is non-uniform, it may encounter error and comparison with Poisson or binomial model will determine the error probability.

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