Question

The amount of fill in a half-liter (500 ml) soft drink bottle is normally distributed. The process has a standard deviation of 5 ml. The mean is adjustable.
a. Where should the mean be set to ensure a 95 percent probability that a half-liter bottle will not be underfilled?
b. Where should the mean be set to ensure a 99 percent probability that a half-liter bottle will not be underfilled?
c. Where should the mean be set to ensure a 99.9 percent probability that a half-liter bottle will not be underfilled?

Answer 1

Answer:

A. 491.775

B. 488.37

C. 484.55

Step-by-step explanation:

A. 100-95 = 5

thus

$\frac{5}{100} = 0.05$

using the formula;

$Z_{0.05} = \frac{X- \mu}{\sigma}$

where X = 500

         μ = mean

         σ = 5 (standard deviation)

$\mu= X - Z_{0.05}\sigma$

   = 500 - 1.645(5)

   = 491.775

(please note that $Z_{}0.05$ = 1.645 from the standard normal distribution table )

B. similarly,  following the above method we have

100 - 99 = 1

1/100 = 0.01

from the standard normal distribution table $Z_{0.01} = 2.326$

 μ = X - $Z_{0.01}$σ

    = 500 - 2.326(5)

    = 488.37

C. 100 - 99.9 = 0.1

0.1/100 = 0.001

from the standard normal distribution $Z_{0.001} = 3.09$

μ = X -  $Z_{0.001}$σ

   = 500 - 3.09(5)

    = 484.55