Question

A simple random sample of size n=14 is obtained from a population with μ=64 and σ=19.

Answer 1

Answer:

a) A. The population must be normally distributed

b) P(X < 68.2) = 0.7967

c) P(X  ≥  65.6) = 0.3745

Step-by-step explanation:

a) The population is normally distributed having a mean ($\mu_x$) = 64  and a standard deviation ($\sigma_x$) = $\frac{19}{\sqrt{14} }$

b) P(X < 68.2)

First me need to calculate the z score (z). This is given by the equation:

$z=\frac{x-\mu_x}{\sigma_x}$ but μ=64 and σ=19 and n=14,  $\mu_x=\mu=64$ and $\sigma_x=\frac{ \sigma}{\sqrt{n} }=\frac{19}{\sqrt{14} }$

Therefore: $z=\frac{68.2-64}{\frac{19}{\sqrt{14} } }=0.83$

From z table, P(X < 68.2) = P(z < 0.83) = 0.7967

P(X < 68.2) = 0.7967

c) P(X  ≥  65.6)

First me need to calculate the z score (z). This is given by the equation:

$z=\frac{x-\mu_x}{\sigma_x}$

Therefore: $z=\frac{65.6-64}{\frac{19}{\sqrt{14} } }=0.32$

From z table,  P(X  ≥  65.6) =  P(z  ≥  0.32) = 1 -  P(z  <  0.32) = 1 - 0.6255 = 0.3745

P(X  ≥  65.6) = 0.3745

P(X < 68.2) = 0.7967