or
Soda is often packaged in cans that are supposed to contain 12 ounces. However, no
manufacturing plant is perfect and so there might be slight errors. For example, Sam's Splendid
Soda company has verified that the amount of soda in their cans has a normal distribution with
a mean of 12 ounces and a standard deviation of 0.7 ounces. Although this is made up, it's not
completely divorced from the truth.
1. You open a can of Sam's and realize there are only 11.6 oz in the can. What is the
probability that a single can will contain 11.6 ounces or less of soda? (2 points)
2. Troubled by the under-filled soda, you decide to empty out all the cans in a six pack of Sam's
Soda and find that the mean amount of soda in all the cans is 11.6 ounces. What is the
probability that six pack will have a mean of 11.6 ounces or less of soda? (2 points)
3. Not satisfied with the information you figured out in #2, you take a case (36 cans) and
empty out all the cans of Sam's Soda and find that the mean amount of soda in all the cans is
11.6 ounces. What is the probability that case will have a mean of 11.6 ounces or less of soda?
(2 points)
4. Draw three normal distributions on the same set of axes or with the same scale to show
how the probabilities decrease from one can to six cans to 36 cans even though we're looking
at "less than 11.6 ounces." (2 points)
5. Use the graphs and your own understanding of the Central Limit Theorem to write a few
sentences explaining what is happening here. (2)

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MrRoyal MrRoyal · Answer 1 · 2022-07-15T14:18:32+02:00

The probability that a single can will contain 11.6 ounces or less of soda is 0.2843

The given parameters are:

x = 11.6

Mean = 12

Standard deviation = 0.7

Calculate the z value using:

[tex]z = \frac{x - \bar x}{\sigma}[/tex]

This gives

[tex]z = \frac{11.6-12}{0.7}[/tex]

z = -0.57

The probability is then calculated as:

P(x ≤ 11.6) = P(z ≤ -0.57)

Using the z table of probabilities, we have:

P(x ≤ 11.6) = 0.2843

In (a), the probability that a can contains 11.6 ounces or less is 0.2843

The probability that all cans in a pack contains 11.6 ounces or less is

P(6) = 0.2843^6

P(6) = 0.00053

In (a), the probability that a can contains 11.6 ounces or less is 0.2843

The probability that all cans in a case contains 11.6 ounces or less is

P(36) = 0.2843^36

P(36) ≈ 0

See attachment for the normal distributions

The summary of the graph is that, as the sample size increases the probability decreases