The volume of beverage in a 12-ounces can is normally distributed with mean 12.07 ounces and standard deviation 0.03 ounces.(a) What is the probability that a randomly selected can will contain more than 12.09 ounces?(b) What is the probability that a randomly selected can will contain between 12 and 12.05 ounces?(c) Is it unusual for a can to be under filled (contain less than 12 ounces)?
Question
The volume of beverage in a 12-ounces can is normally distributed with mean 12.07 ounces and standard deviation 0.03 ounces.(a) What is the probability that a randomly selected can will contain more than 12.09 ounces?(b) What is the probability that a randomly selected can will contain between 12 and 12.05 ounces?(c) Is it unusual for a can to be under filled (contain less than 12 ounces)?
Solution
(a) To find the probability that a randomly selected can will contain more than 12.09 ounces, we first need to calculate the z-score. The z-score is calculated as follows:
Z = (X - μ) / σ
where X is the value we are interested in, μ is the mean, and σ is the standard deviation.
So, Z = (12.09 - 12.07) / 0.03 = 0.67
We then look up this z-score in a standard normal distribution table, or use a calculator or software that can do this. The table will give us the probability that a value is less than our value. But we want the probability that a value is more than our value, so we subtract the table value from 1.
The value for 0.67 in the z-table is 0.7486. So, the probability that a randomly selected can will contain more than 12.09 ounces is 1 - 0.7486 = 0.2514 or 25.14%.
(b) To find the probability that a randomly selected can will contain between 12 and 12.05 ounces, we need to calculate the z-scores for both these values and find the probability between them.
Z1 = (12 - 12.07) / 0.03 = -2.33 Z2 = (12.05 - 12.07) / 0.03 = -0.67
The value for -2.33 in the z-table is 0.0099 and for -0.67 is 0.2514. The probability that a randomly selected can will contain between 12 and 12.05 ounces is 0.2514 - 0.0099 = 0.2415 or 24.15%.
(c) A can is considered to be under filled if it contains less than 12 ounces. To find out if this is unusual, we calculate the z-score for 12 ounces and see if it falls within the usual range.
Z = (12 - 12.07) / 0.03 = -2.33
A z-score of -2.33 is more than 2 standard deviations away from the mean. In a normal distribution, about 95% of values are within 2 standard deviations of the mean. So, a can containing less than 12 ounces is unusual.
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