Final answer:
The correct test for the claim that the standard deviation of bottled water is less than half an ounce is the chi-square (χ²) test for the standard deviation. This test is appropriate for samples less than 30 and assumes a normally distributed population.
Step-by-step explanation:
The question at hand involves testing a claim about the variability of a bottled water company's product, specifically whether the standard deviation of the amount of water in their bottles is less than half an ounce. To do this, we must perform a hypothesis test for the population standard deviation.
In this case, because we are testing a claim about the standard deviation (and not the mean), and because the population standard deviation is not known, we would use a chi-square (χ²) test for the standard deviation, not a z-test or a t-test. The chi-square test is appropriate when the following conditions are met: the sample is randomly selected, the population is normally distributed, and the sample size is less than 30.
Since the sample size is 24 (which is less than 30) and the population is assumed to be normally distributed, the chi-square test is the correct choice. The formula for the chi-square statistic in this context is: χ² = (n - 1)s² / σ², where n is the sample size, s is the sample standard deviation, and σ is the population standard deviation. Using the chi-square distribution with a significance level (α) of 0.005, we would compare the calculated chi-square statistic to the critical value to determine whether the claim that the standard deviation is less than half an ounce is supported.