Question

NewPop produces their brand of soda drinks in a factory where they claim that the

mean volume of the bottles is 24 ounces with a standard deviation of .14 ounces.

Part A: If 22,000 bottles are filled each day, how many bottles are within 14 ounces of
the mean?

Part B: Bottles are rejected if they contain less than 23.72 ounces. How many bottles
would you expect to be rejected in a sample of 22,000 bottles?

Answer 1

a) 14960 bottles

b) 502 bottles

Explanation:

Given that:

Mean (μ) = 24 ounces, standard deviation (σ) = 0.14 ounces

The z score (z) is used in statistics to measure how much a raw score is above or below the mean in a normal distributed population. It is given as:

`z=(x-\mu)/(\sigma)`

a) From empirical rule, 68% of the population are within 1 standard deviation of the mean i.e within 24±0.14 ounces. Therefore, 22,000*68% = 14960 bottles are filled with 24±0.14 ounces

b) x = 23.72

`z=(x-\mu)/(\sigma)=(23.72-24)/(0.14) =-2`

From the probability distribution table: P(X < 23.72) = P (Z < -2) = 0.0228

The number of rejected bottles = 22000 × 0.0228 = 502 bottles

Answer 2

A. 14960 bottles

B. 550 bottles

Explanation:

A normal distribution of the volumes is assumed.

A) From 68–95–99.7, we know that 68% of the outcomes are within 1 standard deviation of the mean. Then, 22,000*68% = 14960 bottles are filled with 24±0.14 ounces

B) mean = 24

standard deviation, sd = 0.14

value of interest, x = 23.72

Z-score = (x - mean)/sd = (23.72 - 24)/0.14 = -2

From the picture attached, we can see that:

P(Z < -2) = 2.35% + 0.15% = 2.5%

Then, it is expected that 22000*2.5% = 550 bottles would be rejected