To test whether the bag filling machine is underfilling the bags, we need to set up the null and alternative hypotheses. In this case:

• The null hypothesis ($H_0$) represents the belief that there is no significant difference in the mean weight of the bags from the machine's setting of 419 grams.
• The alternative hypothesis ($H_1$) represents the claim that the machine is underfilling the bags, meaning the mean weight is less than 419 grams.

Given:

• Sample size ($n$) = 99
• Sample mean ($\bar{x}$) = 417 grams
• Sample standard deviation ($s$) = 25 grams
• Population mean setting ($\mu_0$) = 419 grams
• Level of significance ($\alpha$) = 0.025

The hypotheses can be stated as:

1. Null Hypothesis ($H_0$): $\mu = 419$ grams
   (The machine is filling the bags correctly, on average.)

2. Alternative Hypothesis ($H_1$): $\mu < 419$ grams
   (The machine is underfilling the bags, on average.)

This is a one-tailed test since the manufacturer believes the machine is underfilling, so we are only interested in whether the mean is less than 419 grams, not if it is different in any direction.

Would you like further details on how to perform the hypothesis test?

Further Questions:

1. What is the next step to conduct the hypothesis test for the mean?
2. How do we calculate the test statistic for this hypothesis test?
3. How does the level of significance affect the conclusion of this test?
4. What are the potential outcomes of this test, and how do we interpret them?
5. How can the sample size impact the results of a hypothesis test?

Tip:

For hypothesis tests involving the mean, ensure the sample size is sufficiently large or the population is normally distributed to apply the test correctly.