Heights for men are distributed according to a Normal model, with an average of 69 inches and a standard deviation of 3inches. The tallest 30% of men will be at what height or taller? Round your answer to the nearest hundredth (two decimalplaces), if necessary.

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Andre BalkonE · Accepted Answer

We are given that heights for men are distributed according to a Normal model with mean μ=69 inches and standard deviation σ=3 inches. We want to find the height h such that the tallest 30% of men are at that height or taller.To solve this problem, we can use the standard Normal distribution and the properties of the Normal distribution. Let Z be a standard Normal random variable, i.e., a Normal random variable with mean 0 and standard deviation 1. Then, we know that the cumulative distribution function (CDF) of Z is given by:Φ(z)=P(Z≤z)=12π∫−∞ze−t2/2dtUsing this CDF, we can find the height h such that the tallest 30% of men are at that height or taller by solving the following equation:P(X≥h)=0.3where X is a Normal random variable with mean μ=69 inches and standard deviation σ=3 inches. We can standardize X to obtain a standard Normal random variable Z:Z=X−μσ=X−693Then, we have:P(X≥h)&amp;amp;=P(Z≥h−μσ)=P(Z≥h−693)=1−Φ(h−693)=0.3Using a table of the standard Normal distribution or a calculator, we can find that Φ−1(0.3)≈−0.5244. Therefore,h−693≈−0.5244Solving for h, we get:h≈66.4272Rounding to the nearest hundredth, we get that the tallest 30% of men will be at a height of 66.43 inches or taller.

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