Question

A vending machine automatically pours soft drinks into cups. The amount of soft drink dispensed into a cup is normally distributed with a mean of 7.6 ounces and standard deviation of 0.4 ounce. Examine the figure below and answer the following questions.
(a) Estimate the probability that the machine will overflow an 8-ounce cup. (Round your answer to two decimal places.)
1
(b) Estimate the probability that the machine will not overflow an 8-ounce cup. (Round your answer to two decimal places.)
2
(c) The machine has just been loaded with 868 cups. How many of these do you expect will overflow when served?
3 cups

Answer 1

Answer:

a) $P(X>8)=P(\frac{X-\mu}{\sigma}>\frac{8-\mu}{\sigma})=P(Z>\frac{8-7.6}{0.4})=P(Z>1)$

And we can find this probability using the complement rule and the standard normal table or excel:

$P(Z>1)=1-P(Z$

b)$P(X$

And we can find this probability using the standard normal table or excel:

$P(Z$

c) For this case we have a total of 868 cups and we know that the fraction of cups that will be overflow is 0.159, so then the expected number of cups overflow would be:

$N = 868*0.159= 138.012 \approx 138$

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the amount of soft frink of a population, and for this case we know the distribution for X is given by:

$X \sim N(7.6,0.4)$  

Where $\mu=7.6$ and $\sigma=0.4$

We are interested on this probability

$P(X>8)$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we apply this formula to our probability we got this:

$P(X>8)=P(\frac{X-\mu}{\sigma}>\frac{8-\mu}{\sigma})=P(Z>\frac{8-7.6}{0.4})=P(Z>1)$

And we can find this probability using the complement rule and the standard normal table or excel:

$P(Z>1)=1-P(Z$

Part b

We are interested on this probability

$P(X$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we apply this formula to our probability we got this:

$P(X$

And we can find this probability using the standard normal table or excel:

$P(Z$

Part c

For this case we have a total of 868 cups and we know that the fraction of cups that will be overflow is 0.159, so then the expected number of cups overflow would be:

$N = 868*0.159= 138.012 \approx 138$