Statistics

Week 6 Discussion Example:

An inspector at a bottling plant is concerned that the 2 liter filling machine may be averaging less than 2

liters of product in the 2-liter bottles. She asks for proof that the 2 liter filling machine is averaging at

least 2 liters of product in the 2-liter bottles.

The Production Manager randomly pulls 30 2-liter bottles from the production line and measures their

content. He gets a sample mean of 2.027 liters. He claims that the standard deviation has historically been

0.054 liters. He asks you to construct a confidence interval on the mean fill of the 2-liter bottles to

convince the inspector that the filling machine averages at least 2 liters of product.

Construct a confidence interval for the mean using the �̅�, σ, and n provided.

Which of the three Confidence Interval equations should you use? Why?

I would use the equation for finding a confidence interval on the mean when σ is known.

𝐶𝐼𝐿𝑖𝑚𝑖𝑡𝑠 = �̅� ± 𝑍(𝛼 2

) ∗

𝜎

√𝑛

This is the correct equation to use when the population standard deviation (σ) is given in the

problem and the sample size is large enough to assume the sampling distribution of the mean is

approximately normal.

What Confidence Level should you use? Why?

Because the inspector has asked for “proof”, I would not use any confidence level less than 95%

or 99%. The choice of confidence level (certainty) should be balanced against the increased width

of the interval (loss of precision) with the higher confidence level. Given the width of the two

intervals found below, I would use the 99% confidence level.

What are limits on your confidence interval?

The 95% confidence interval, using the �̅� = 2.027 liters, σ = 0.054 liters, and n = 30 is 2.008 liters to 2.046 liters. The 99% confidence interval using the same values is 2.002 liters to 2.052 liters.

What do the limits mean statistically?

We can be 99% confident that the population mean (μ) is contained in the interval.

2.002 liter ≤ μ ≤ 2.052

What do the limits mean in terms of this problem?

We can be 99% confident that the average of all bottles filled on the 2-liter bottling machine is at

least 2.002 liters.

Was the inspector’s concern justified?

The inspector’s concern was not justified in this case. However, the results found are dependent

on the population standard deviation provided by the Production Manager. If the value for σ were

slightly larger, the conclusion could have been different.