Question

The State Department of Weights and Measures is responsible for making sure that commercial weighing and measuring devices, such as scales, are accurate so customers and businesses are not cheated. Periodically, employees of the department go to businesses and test their scales. For example, a dairy bottles milk in 1-gallon containers. Suppose that if the filling process is working correctly, the mean volume of all gallon containers is 1.00 gallon with a standard deviation equal to 0.10 gallon. The department's test process requires that they select a random sample of n = 9 containers. If the sample mean is less than 0.97 gallons, the department will fine the dairy. Based on this information, what is the probability that the dairy will get fined even when its filling process is working correctly?

A. 0.90
B. Approximately 0.3159
C. About 0.1841
D. Approximately 0.382

Answer 1

Option C

Explanation:

Given

Probability of Mean value less than 0.97

= Mean value -1/(0.1/sqrt (9))

Substituting the given values, we get -

Probability of Mean value less than 0.97

= 0.97 -1/(0.1/sqrt (9))

= 0.97-1/(0.1/3)

= 0.1841

Hence, option C is correct

Answer 2

B. Approximately 0.3159

Explanation:

To solve this question, we need to understand the normal probability distribution, and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

The mean volume of all gallon containers is 1.00 gallon with a standard deviation equal to 0.10 gallon.

This means that
`\mu = 1, \sigma = 0.1`

Sample of 9:

This means that
`n = 9, s = (0.1)/(√(9))`

If the sample mean is less than 0.97 gallons, the department will fine the dairy. Based on this information, what is the probability that the dairy will get fined even when its filling process is working correctly?

This is the pvalue of Z when X = 0.97. So

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (0.97 - 1)/((0.1)/(√(9)))`

`Z = -0.9`

`Z = -0.9` has a pvalue of 0.1841.

The correct answer is given by option B.