Question

3. The heights of all adults in a large city have an approximately normal distribution with a mean of 68 inches and a standard deviation of 4 inches. a) Find the probability that a randomly chosen height is less than 66 inches. b) Write the sampling distribution of sample mean for any sample size. Find the probability that the mean height of a random sample of 100 adults would be between 67.5 inches and 69 inches.

Answer 1

Let , $X\to N(\mu=68,\sigma=4)$

a) Now ,

- P(X < 66) = PG 66 - 68 -) = P(Z < -0.5) = D(-0.5) 4

$=1-\Phi(0.5)=1-0.6915=0.3085$ ; From standard normal distribution table

b) Now , the sampling distribution of the sample mean is ,

X + Ν(μχ = μ, σX = n

Therefore ,

$P(67.5<\bar{X}<69)=P(\frac{67.5-68}{4/\sqrt{100}}<\frac{\bar{X}-\mu_{\bar{X}}}{\sigma_{\bar{X}}}<\frac{69-68}{4/\sqrt{100}})$

$=P(-1.25<Z<2.50)=P(Z<2.50)-P(Z<-1.25)$

$=\Phi(2.50)-\Phi(-1.25)=\Phi(2.50)-[1-\Phi(1.25)]$

$=0.9938-1+0.8944=0.8882$ ; From standard normal distribution table