A brewery's filling machine is adjusted to fill bottles with a mean of 33.8 oz. of ale and a variance of 0.005. Periodically, a bottle is checked and the amount of ale noted.
(b) Let's say you buy 93 bottles of this ale for a party. How many bottles would you expect to find containing more than 33.91 oz. of ale? (Round your answer up to the nearest whole number.)
_____ bottles
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Multiply by 93.
To find the number of bottles that would contain more than 33.91 oz. of ale, we need to use the concept of the normal distribution.
First, we need to calculate the standard deviation (σ) which is the square root of the variance. In this case, the variance is 0.005, so the standard deviation is √0.005 = 0.0707107.
Next, we calculate the z-score, which is a measure of how many standard deviations a value is from the mean. The formula for the z-score is:
z = (x - μ) / σ
Where:
- x is the value we want to find the probability for (33.91 oz. in this case)
- μ is the mean (33.8 oz.)
- σ is the standard deviation (0.0707107)
Substituting the values into the formula, we get:
z = (33.91 - 33.8) / 0.0707107
= 0.11 / 0.0707107
≈ 1.5565
We can now use a standard normal distribution table or a calculator to find the probability associated with the z-score of 1.5565. We're interested in the probability of getting a value greater than this z-score, so we need to look up the area under the normal curve to the right of the z-score.
Using a standard normal distribution table or a calculator, we find that the probability is approximately 0.0606.
Finally, to find the number of bottles containing more than 33.91 oz. of ale, we multiply the probability by the total number of bottles (93):
Number of bottles = probability * total number of bottles
≈ 0.0606 * 93
≈ 5.63
Rounding up to the nearest whole number, we would expect to find approximately 6 bottles containing more than 33.91 oz. of ale.