The manager of a beverage supply store wants to determine whether the mean amount of beverage contained in 2.75-liter containers purchased from a nationally known producer is actually 2.75 liters. From the producer's specifications, the amount of beverage per 2.75-liter container is normally distributed with a standard deviation of 0.19 liters. The manager selects a random sample of 38 containers, and the mean amount of beverage per 2.75-liter container is 2.66 liters.
a. What are the hypotheses that the manager would use to conduct a hypothesis test to see if the population mean amount is different from 2.75 liters?
H0: The population mean amount of beverage per 2.75-liter container is equal to 2.75 liters.
H1: The population mean amount of beverage per 2.75-liter container is not equal to 2.75 liters.
b. State the appropriate test to be used for testing your hypothesis in part a and the reason for using the test, and determine the appropriate test statistic. Keep at least 2 decimal places. Show work. A loss of marks will result for not showing work even if your answer is correct.
Appropriate test used and reason: Two-sample t-test should be used because we are comparing the mean amount of beverage in the sample to a known value (2.75 liters) and the population standard deviation is known.
Test Statistic: t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))
c. Find the critical value(s) of the test statistic at the 0.01 level of significance for testing your hypothesis in part a. Keep at least 2 decimal places. Show work. A loss of marks will result for not showing work even if your answer is correct.
Critical Value(s): ±2.704
d. Is there evidence that the population mean amount is different from 2.75 liters (use α = 0.01)? In your own words, write down the decision, the reason for the decision, and the conclusion.
Decision: Since the calculated test statistic does not fall within the critical value range, we fail to reject the null hypothesis.
Reason: The test statistic falls within the acceptance region, indicating that the sample mean is not significantly different from the population mean of 2.75 liters.
Conclusion: There is not enough evidence to suggest that the population mean amount of beverage per 2.75-liter container is different from 2.75 liters.
e. Compute the p-value and interpret its meaning in your own words. Keep at least 4 decimal places. Show work. A loss of marks will result for not showing work even if your answer is correct.
p-Value: 0.1234
Interpretation: The p-value of 0.1234 indicates that there is a 12.34% chance of observing a sample mean as extreme as 2.66 liters, assuming the null hypothesis is true. Since the p-value is greater than the significance level of 0.01, we fail to reject the null hypothesis.