Answer to Question #244349 in Statistics and Probability for boo

Question

The width of white board markers manufactured by a company is approximately normally
distributed with a mean of 12 mm (millimetres) and a standard deviation of 4 mm. If a random
sample of size 16 white board markers is selected, what is
a) the distribution of the mean?
b) the probability that the sample mean is less than 15.6 mm?
c) the probability that the sample mean is between 12.8 mm and 13.2 mm?

Answer 1

a)

The sample of 16 is normally distributed with the sample average 12 mm and

$\sigma=4/\sqrt{16}=1$ mm

b)

$z=\frac{x-\overline{x}}{\sigma/\sqrt{n}}=\frac{15.6-12}{4/\sqrt{16}}=3.6$

$P(z<3.6)=0.9998$

c)

$z_1=\frac{12.8-12}{4/\sqrt{16}}=0.8$

$z_2=\frac{13.2-12}{4/\sqrt{16}}=1.2$

$P(12.8<\overline{x}<13.2)=P(z<1.2)-P(z<0.8)=0.0968$