A brewery's filling machine is adjusted to fill bottles with a mean of 32.6 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 32.68 oz? (Give your answer correct to four decimal places.)
(b) Let's say you buy 97 bottles of this ale for a party. How many bottles would you expect to find containing more than 32.68 oz. of ale? (Round your answer up to the nearest whole number.)
how many bottles?
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 the Z score.
b. Multiply answer in a by 97.
To answer this question, we can use the properties of the normal distribution.
(a) To find the probability that the next bottle contains more than 32.68 oz, we need to calculate the area under the normal curve to the right of 32.68.
First, we need to calculate the standard deviation (σ) from the given variance. Since variance equals the square of the standard deviation, we have:
variance = 0.004 = σ^2.
Therefore, the standard deviation is σ = sqrt(0.004) = 0.0632.
Next, we need to calculate the z-score for 32.68 oz using the formula:
z = (x - μ) / σ,
where x is the value (32.68 oz), μ is the mean (32.6 oz), and σ is the standard deviation (0.0632 oz).
Plugging in the values, we get:
z = (32.68 - 32.6) / 0.0632 = 1.2658.
Now, we can use a standard normal table or a calculator to find the probability corresponding to this z-score. The probability of getting a value greater than 32.68 oz is the same as finding the area to the right of the z-score (1.2658). Using a standard normal table or calculator, we find that this probability is 0.1038 (rounded to four decimal places).
Therefore, the probability that the next randomly checked bottle contains more than 32.68 oz is 0.1038.
(b) To determine the number of bottles you would expect to find containing more than 32.68 oz out of 97 bottles, we can multiply the probability from part (a) by the total number of bottles:
Number of bottles = probability * total number of bottles
= 0.1038 * 97
= 10.0776.
Rounding to the nearest whole number, you would expect to find 10 bottles containing more than 32.68 oz out of the 97 bottles.