Question

Question 8: A machine at AMT & Co. fills 120-ounce jugs with laundry softener.
A quality control inspector wishes to test if the machine underfills the jugs or not. Assume
the distribution of the amount of laundry softener in these jugs is Normal. Under “standard”
circumstances, the mean amount should be 120 ounces with a standard deviation of 1 ounce.
a. (2 mark) What are the hypotheses the quality control inspector wishes to test?
b. Suppose the quality control inspector wishes to conduct the test using α = 0.10
and a sample size of 40. What is the range of values of ̄x for which he will accept H0?
c. Suppose the quality control inspector wishes to conduct the test using α = 0.10
and a sample size of 40. If the true mean amount of detergent in these jugs is 119.5 ounces,
what is the power of the test?
d. If the observed sample mean amount in the simple random sample of 40 jugs
equals 119.62 ounces, find the P-value.

Answer 1

Let the random variable be defined as follows:

Then given “standard” circumstances, the mean amount should be 120 ounces with a standard deviation of 1 ounce.

Let

Answer a.

The control inspector wants to test if the machine underfills the jugs or not.
So we can state our hypothesis as follows:

or we can restate our hypothesis as:

Answer b.

Under H_0, the test statistic Z is normally distributed

So we can rewrite it as

At , the critical value of the Z statistic is given by

At \alpha = 0.10,
we reject the NULL when we observe a Z statistic such that

so we accept the NULL when we observe a Z statistic such that  

soving we get,  

so we accept the null hypothesis when the sample mean is greater than 119.7976

Answer c.

When is true, the mean  

Now under ,   

So the power of the test is 1

Answer d.

The observed sample mean is  

So the corresponding Z-statistic is

The p-value is defined as

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