Answer to Question #307693 in Statistics and Probability for gurpinder

Question

3. The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams. A sample of 25 cans is to be selected. What is the probability that the sample mean will be between 100 and 120 grams?

Answer 1

We have a normal distribution, $\mu=110, \sigma=25, n=25.$

Let's convert it to the standard normal distribution, $z=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}$ ;

$z_1=\cfrac{100-110}{25/\sqrt{25}}=-2, z_2=\cfrac{120-110}{25/\sqrt{25}}=2,$

$P(100<\bar{X}<120)=P(-2<Z<2)=P(Z<2)-P(Z<-2)=$

$=0.97725-0.02275=0.9545$ (from z-table).