Question

6. The heights of men selected from a particular population have a normal distribution with mean 68 inches and standard deviation 4 inches. a. Find the percentage of men in the population who are taller than 72 inches. b. Find the percentage of men who are between 64 and 74 inches tall. c. Suppose 25% of men in this population are taller than Ralph. How tall is Ralph

Answer 1

Answer:

a) the proportion is 0.159 (15.9%)

b) the proportion is 0.524 (52.4%)

c) Ralph is 70.958 inches tall

Step-by-step explanation:

defining our random variable X= heights of men , then we can define the standard random variable Z as

Z= (X- μ)/σ , where  μ is the population's mean and σ is the corresponding standard deviation of X

then for X=72

Z= (X- μ)/σ

Z= (72-68)/4 = 1

a) P(X> 72) = P(Z> 1) = 1-0.841 = 0.159 (using standard normal distribution tables)

b) for X=64 and X=74

Z₁= (64 -68)/4 = -1

Z₂= (74-68)/4 = 1.5

then

P(68<X< 74) = P(-1<Z <1.5) 0 = P(Z<1.5) - P(Z<-1) =  0.933- 0.159 =0.774

c) Z₃ = 0.933 - 0.774*0.25 = 0.7395

thus

Z₃= (X₃- μ)/σ →X₃ =μ+ Z₃*σ = 68 + 0.7395*4  = 70.958 inches

therefore Ralph is 70.958 inches tall