Question

A company that produces carbonated beverage has to check the shelf life of their product periodically for quality control. They collected a random sample of 10 bottles and measured their shelf life in days, and obtained the following results:

108, 124, 124, 106, 115, 138, 163, 159, 134, 139

They would like to demonstrate that the mean shelf life of their product exceeds 120 days.

Set up the appropriate hypotheses. Use mathematical notation, and explain the symbols that you are using.

Show the formula for the test statistic and compute its value.

What is distribution of the test statistic under the null hypothesis?

Using a=0.01, what is your conclusion?

Compute the p-value.

What are the assumptions that you made in order to perform this analysis? Conduct the appropriate steps in order to check if the assumptions hold in this case. If not, suggest what you would do.

Answer 1

Let denotes the mean shelf life of their product.

To test against

Here

sample mean

sample standard deviation

and sample size

The test statistic can be written as

which under H₀ follows a t distribution with n-1 df.

We reject H₀ at 1% level of significance if P-value < 0.01

The value of the test statistic

and P-value

  

Since  , so we fail to reject H₀ at 1% level of significance and we can conclude that the mean shelf life of their product is not significantly greater than 120

The assumptions that one will make  in order to perform this analysis : the shelf life of the products follow a normal distribution.