Answer to Question #164892 in Statistics and Probability for Muhammad Fahad Fayyaz

Question

The weights of packages filled by machine are normally distributed about a mean of 25 ounces, with a standard deviation of one ounce. What is the probability that n packages from the machine will have an average weight of less than 24 ounces if n = 1, 4, 16, 64? (10)

Answer 1

Mean, $u=25$

Standard deviation, $\sigma=1$

Average weight, $\bar{X}=24$

Case-1 When n=1

$P(z<24)=1-P(24<z<25) =1-\dfrac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}=1-\dfrac{24-25}{\frac{1}{\sqrt{1}}}=1-(-1)=2$

The value of z is 0.5080.

Case-2 When n=4

$P(z<24)=1-P(24<z<25) =1-\dfrac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}=1-\dfrac{24-25}{\frac{1}{\sqrt{4}}}=1-(-2)=3$

The value of z is 0.6852

Case -3 When n=16

$P(z<24)=1-P(24<z<25) =1-\dfrac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}=1-\dfrac{24-25}{\frac{1}{\sqrt{16}}}=1-(-4)=5$

The value of z is 0.8453

Case-4 When n=64,

$P(z<24)=1-P(24<z<25) =1-\dfrac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}=1-\dfrac{24-25}{\frac{1}{\sqrt{64}}}=1-(-8)=9$

The value of z is-0.9568