A brewery's filling machine is adjusted to fill bottles with a mean of 31.9 oz. of ale and a variance of 0.004. Periodically, a bottle is checked and the amount of ale noted.
(a) Assuming the amount of fill is normally distributed, what is the probability that the next randomly checked bottle contains more than 32.00 oz? (Give your answer correct to four decimal places.)
(b) Let's say you buy 109 bottles of this ale for a party. How many bottles would you expect to find containing more than 32.00 oz. of ale? (Round your answer up to the nearest whole number.)
a) Z = (score-mean)/SD
Variance = SD^2
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to Z.
b) Multiply probability by 109.
To answer these questions, we need to use the concept of the standard normal distribution. Given that the amount of fill is normally distributed, we can convert the problem into a standard normal distribution using the z-score formula:
z = (x - μ) / σ
Where:
z = z-score
x = value of interest
μ = mean
σ = standard deviation
(a) To find the probability that the next randomly checked bottle contains more than 32.00 oz, we need to find the area under the standard normal curve to the right of 32.00 oz.
First, we calculate the z-score using the given values:
z = (32.00 - 31.9) / √0.004
Next, we look up the cumulative probability associated with this z-score. We can use a standard normal distribution table or a statistical calculator to find this value. Assuming a z-score of 0.5, the cumulative probability is 0.6915.
However, we are looking for the probability to the right of 32.00 oz, which is 1 - 0.6915 = 0.3085.
So, the probability that the next randomly checked bottle contains more than 32.00 oz is approximately 0.3085.
(b) To find out how many bottles you would expect to find containing more than 32.00 oz out of the 109 bottles, we can use the concept of expected value. The expected value is calculated by multiplying the probability of an event by the number of occurrences:
Expected value = Probability × Number of occurrences
In this case, we have already calculated the probability to be 0.3085. Now we multiply this probability by the total number of bottles (109) to get the expected number of bottles:
Expected number of bottles = 0.3085 × 109 = 33.65765
Rounding up to the nearest whole number, you would expect to find approximately 34 bottles containing more than 32.00 oz.