Question

A coin-operated coffee machine made by BIG Corporation was designed to discharge a mean of 7.1 ounces of coffee per cup. If it dispenses more than that on average, the corporation may lose money, and if it dispenses less, the customers may complain. Believing that the mean amount of coffee dispensed by the machine, μ, is less than 7.1 ounces, BIG plans to do a statistical test of the claim that the machine is working as designed. Technicians gather a random sample of fill amounts and find that the mean of the sample is 6.6 ounces and that the standard deviation is 0.5 ounces.

Answer 1

To determine if the coffee machine is working as designed, a hypothesis test can be performed. If the test statistic is smaller than the critical value, the null hypothesis can be rejected, indicating that the machine is not working as designed.

Step-by-step explanation:

To determine if the machine is working as designed, we can perform a hypothesis test. The null hypothesis, H0, is that the mean amount of coffee dispensed, μ, is equal to 7.1 ounces. The alternative hypothesis, Ha, is that μ is less than 7.1 ounces.

We can use a one-sample t-test to test this hypothesis. We calculate the test statistic t by subtracting the population mean from the sample mean and dividing by the standard deviation divided by the square root of the sample size. In this case, the sample mean is 6.6 ounces, the population mean is 7.1 ounces, the standard deviation is 0.5 ounces, and the sample size is not given.

Once we have the test statistic, we can compare it to the critical value of t from a t-distribution with n-1 degrees of freedom and the desired level of significance. If the test statistic is smaller than the critical value, we reject the null hypothesis in favor of the alternative hypothesis, concluding that the machine is not working as designed.