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

a)

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

b)

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

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:

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:

Where and

We are interested on this probability

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

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

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

Part b

We are interested on this probability

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

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

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

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:

sorry i need the points

step-by-step explanation:

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a 4/6 = 9/12   not proportional

step-by-step explanation:

to find which are not proportional, we can use cross products

if it is true, they are proportional.   if it is not true, they are not proportional.

a 4/6 = 9/12

4* 12 = 6*9

48 = 54

not this is not proportional  

b 2/3 = 12/18  

2* 18 = 3 * 12

36 = 36

this is proportional

c 5/4 = 10/8  

5*8 = 4*10

40 = 40

this is proportional

d 7/5 = 35/25

7*25 = 5 * 35

175 = 175

is proportional