Question

a normally distributed population has mean 57.7 and standard deviation 12.1

a) find the probability that a single randomly selected element x of the population is less than 45

b) find the mean and the standard deviation of x-x for sample size 16

c) find the probability that the mean of sample of size 16 drawn from the population is less than 45

Answer 1

$mean\left(\mu \right)=57.7$

$Standard\:deviation\left(\sigma \right)=12.1000$

(a). Probability that a single randomly selected element X of the population is less than 45 :

$\:P\left(X<45\right)=\left(Z<\frac{\left(45-57.7\right)}{12.1}\right)=P\left(Z<-1.05\right)=0.1469$

(b).$n=16$

mean of sampling distribution $\mu x=57.7$

standard deviation of sampling distribution $\sigma \overline{x}=\frac{\sigma }{\sqrt{n}}=3.025$

(c). Probability that the mean of a sample of size 16 drawn from this population is less than 45

$P\left(X<45\right)=\left(Z<\frac{\left(45-57.7\right)}{3.025}\right)=P\left(Z<-4.2\right)=0.0000$