Question

A popular soft drink is sold in 2​-liter ​(​2000-milliliter) bottles. Because of variation in the filling​ process, bottles have a mean of and a standard deviation of 16 ​milliliters, normally distributed. Complete parts a and b below.

Answer 1

Answer:

The answer is below.

Explanation:

The z score is a used in statistics to determine by how many standard deviations the raw score is above or below the mean. The z score is given by:

$z=\frac{x-\mu}{\sigma}\\\\where\ x=raw\ score, \mu=mean,\sigma=standard\ deviation\\\\For\ a\ sample\ size(n):\\\\z=\frac{x-\mu}{\sigma/\sqrt{n} }$

a) Given that n = 100, μ = 2000, σ = 18

For x < 1995 millimeters:

$z=\frac{x-\mu}{\sigma/\sqrt{n} }=\frac{1995-2000}{18/\sqrt{100} } =-2.78$

From the normal distribution table, P(x < 1995) = P(z < -2.78) = 0.0027

b) P(z > z*)  = 10% = 0.1

P(z < z*) = 1 - 0.1 = 0.9

z* = 1.28

$z*=\frac{x-\mu}{\sigma/\sqrt{n} }\\\\1.28=\frac{x-2000}{18/\sqrt{100} }\\\\x-2000 =-2.304\\\\x=2002.3\ ml$

From the normal distribution table, P(z < z