Answer to Question #224231 in Statistics and Probability for Arham

Question

A company has designed a coffee vending machine which fills 210 ml coffee in each cup, on average. You are supposed to perform find some statistics as a part of quality control department. If the amount of drink dispensed regularly from this machine is normally distributed with a standard deviation equal to 13.5 ml,
a) What percentage of the cups will contain more than 231 ml?

b) What would be the chance that a cup will contain between 180 and 220 ml?

c) Calculate the number of cups that will have a probability to overflow if 235 ml cups are used for 900 drinks sold at any random day?

d) If only less than 15% cups are overflowed, what would the cutoff filling volume be?

Answer 1

Let $X=$ the amount of coffee in a cup: $X\sim N(\mu, \sigma^2).$

Given $\mu=210\ ml, \sigma=13.5\ ml.$

a)

$$P(X>231)=1-P(Z\leq\dfrac{231-210}{13.5})$$

$$=1-P(Z\leq \dfrac{14}{9})\approx0.0599$$

6% of the cups will contain more than 231 ml.

b)

$$P(180<X<220)=P(X<220)-P(X\leq180)$$

$$=P(Z<\dfrac{220-210}{13.5})-P(Z\leq\dfrac{180-210}{13.5})$$

$$\approx0.7705747-0.0131341\approx0.757441$$

$0.757441$

c)

$$P(X>235)=1-P(Z\leq\dfrac{235-210}{13.5})$$

$$=1-P(Z\leq\dfrac{50}{27})\approx0.032024$$

$$0.032024(900)=29$$

$29$ cups

d)

$$P(X>x)=0.15$$ $$P(X>x)=1-P(Z\leq\dfrac{x-210}{13.5})=0.15$$

$$P(Z\leq\dfrac{x-210}{13.5})=0.85$$

$$\dfrac{x-210}{13.5}\approx1.036433$$

$$x=224$$

$224$ ml.