A soft drink distributor is considering two new models of...

A soft drink distributor is considering two new models of dispensing machines. Both the Harvard Company machine and the Fizzit machine can be adjusted to fill the cups to a certain mean amount. However, the variation in the amount dispensed from cup to cup is a primary concern. Ten cups dispensed from the Harvard machine showed a variance of $0.065,$ whereas 15 cups dispensed from the Fizzit machine showed a variance of $0.033 .$ The factory representative from the Harvard Company maintains that his machine had no more variability than the Fizzit machine. Assume the amount dispensed is normally distributed. At the 0.05 level of significance, does the sample refute the representative's assertion? a. Solve using the $p$ -value approach. b. Solve using the classical approach.

A soft drink distributor is considering two new models of dispensing machines. Both the Harvard Company machine and the Fizzit machine can be adjusted to fill the cups to a certain mean amount. However, the variation in the amount dispensed from cup to cup is a primary concern. Ten cups dispensed from the Harvard machine showed a variance of $0.065,$ whereas 15 cups dispensed from the Fizzit machine showed a variance of $0.033 .$ The factory representative from the Harvard Company maintains that his machine had no more variability than the Fizzit machine. Assume the amount dispensed is normally distributed.
At the 0.05 level of significance, does the sample refute the representative's assertion?
a. Solve using the $p$ -value approach.
b. Solve using the classical approach.

Key Concepts

Classical hypothesis testing

The classical approach in hypothesis testing involves setting up a null hypothesis and an alternative hypothesis, calculating a test statistic, and then comparing this statistic to a critical value from the relevant reference distribution. If the test statistic falls into the rejection region defined by the critical value, the null hypothesis is rejected.

Null and alternative hypotheses

In hypothesis testing, the null hypothesis (H0) typically represents a default state of no effect or no difference, while the alternative hypothesis (Ha) represents the specific claim or difference to be tested. Defining these hypotheses clearly is crucial for setting up the correct test and interpreting the results.

F-test for comparing variances

The F-test is a statistical method used to compare the variances of two independent samples drawn from normally distributed populations. By taking the ratio of the two sample variances, the test statistic follows an F-distribution, which allows for determining whether the difference between the variances is statistically significant.

p-value approach

The p-value approach is a method in hypothesis testing where the probability of obtaining an observed test statistic or one more extreme is calculated under the null hypothesis. If this p-value is less than the predefined significance level, it provides evidence to reject the null hypothesis in favor of the alternative.

Significance level

The significance level, often denoted as alpha (such as 0.05), is the threshold used in hypothesis testing to determine whether a result is statistically significant. It represents the maximum probability of committing a Type I error, which is the error of rejecting a true null hypothesis.

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