Answer to Question #244829 in Statistics and Probability for Amore

Question

2. 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

$\mu=12 \\ \sigma=4 \\ n=16$

a) Distribution will be normal with mean = 12 mm and standard deviation $= \frac{4}{\sqrt{16}}=1$

b)

$P(\bar{X} <15.6) = P(Z < \frac{15.6-12}{4 / \sqrt{16}}) \\ = P(Z<3.6) \\ = 0.9998$

c)

$P(12.8< \bar{X}<13.2) = P(\bar{X}<13.2) -P(\bar{X}<12.8) \\ = P(Z< \frac{13.2-12}{4/ \sqrt{16}}) -P(Z< \frac{12.8-12}{4 /\sqrt{16}}) \\ = P(Z< 1.2) -P(Z<0.8) \\ = 0.8849 -0.7881 \\ = 0.0968$