Assume that the heights of men are normally distributed with a mean of 69.0 inches and a standard deviation of 2.8 inches. if 64 men are randomly selected, find the probability that they have a mean height between 68 and 70 inches.

Answer: 0.9958Step-by-step explanation:We assume that the heights of men are normally distributed with Mean : [tex]\mu=69.0\text{ inches}[/tex]Standard deviation : [tex]\sigma=2.8\text{ inches}[/tex]Sample size : [tex]n=64[/tex]The value of z-score is given by:[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]Let x be the height of the randomly selected men.For x= 68[tex]z=\dfrac{68-69}{\dfrac{2.8}{\sqrt{64}}}\approx-2.86[/tex]For x= 70[tex]z=\dfrac{70-69}{\dfrac{2.8}{\sqrt{64}}}\approx2.86[/tex]Now, the probability that they have a mean height between 68 and 70 inches is given by :-[tex]P(68<x<70)=P(-2.86<z<2.86)\\=1-2P(z<-2.86)=1-2(0.0021182)=0.9957636\approx0.9958[/tex]Hence, the probability that they have a mean height between 68 and 70 inches = 0.9958