The heights of a certain population of corn plants follow a normal distribution with mean 145 cm and stan- dard deviation 22 cm (as in Exercise 4.S.4). (a) What percentage of the plants are between 135 and 155 cm tall? (b) Suppose we were to choose at random from the population a large number of samples of 16 plants each. In what percentage of the samples would the sample mean height be between 135 and 155 cm?

Answer with explanation:Given : The heights of a certain population of corn plants follow a normal distribution with mean [tex]\mu=145\ cm[/tex] and standard deviation [tex]\sigma=22\ cm[/tex]a) Using formula [tex]z=\dfrac{x-\mu}{\sigma}[/tex], the z-value corresponds to x= 135 will be [tex]z=\dfrac{135-145}{22}\approx-0.45[/tex]At x= 155, [tex]z=\dfrac{155-145}{22}\approx0.45[/tex]The probability that plants are between 135 and 155 cm tall :-[tex]P(-0.45<z<0.45)=P(z<0.45)-P(z<-0.45)\\\\=0.6736447- 0.3263552\\\\=0.3472895\approx0.3473=34.73\%[/tex]Hence, 34.73% of the plants are between 135 and 155 cm tall.b) Sample size : n= 16Using formula [tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex], the z-value corresponds to x= 135 will be [tex]z=\dfrac{135-145}{22}{\sqrt{16}}\approx-1.82[/tex]At x= 155, [tex]z=\dfrac{155-145}{22}{\sqrt{16}}\approx1.82[/tex]The probability that plants are between 135 and 155 cm tall :-[tex]P(-1.82<z<1.82)=P(z<1.82)-P(z<-1.82)\\\\= 0.9656205- 0.0343795\\\\=0.931241\approx0.9312=93.12\%[/tex]Hence,The percentage of the samples would the sample mean height be between 135 and 155 cm.= 93.12%