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ale4655 [162]
3 years ago
15

A study of homeowners in the 5th congressional district in Maryland found that their annual household incomes are normally distr

ibuted with a mean of $41,182 and a standard deviation of $11,990 (based on data from Nielsen Media Research). If an advertising campaign is to be targeted at those whose household incomes are in the top 20%, find the minimum income level for this target group.
Mathematics
1 answer:
tino4ka555 [31]3 years ago
4 0

Answer:

The minimum income level for this target group is of $51,253.6.

Step-by-step explanation:

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the z-score of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Mean of $41,182 and a standard deviation of $11,990

This means that \mu = 41182, \sigma = 11990

Find the minimum income level for this target group.

The 100 - 20 = 80th percentile, which is X when Z has a p-value of 0.8, so X when Z = 0.84.

Z = \frac{X - \mu}{\sigma}

0.84 = \frac{X - 41182}{11990}

X - 41182 = 0.84*11990

X = 51253.6

The minimum income level for this target group is of $51,253.6.

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