A manufacturer of banana chips would like to know whether its bag filling machine works correctly at the 419419 gram setting. It is believed that the machine is underfilling the bags. A 99 bag sample had a mean of 417417 grams with a standard deviation of 2525. A level of significance of 0.0250.025 will be used. Assume the population distribution is approximately normal. State the null and alternative hypotheses.

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.

A manufacturer of banana chips would like to know whether its bag filling machine works correctly at the 419 gram setting. It is believed that the machine is underfilling the bags. A 99 bag sample had a mean of 417 grams with a standard deviation of 25. A level of significance of 0.025 will be used. Assume the population distribution is approximately normal. State the null and alternative hypotheses.

This statistical problem involves conducting a hypothesis test to determine if a banana chip manufacturer's bag filling machine is underfilling bags. Given a sample of 99 bags with a mean of 417 grams and a standard deviation of 25 grams, we test whether the machine is underperforming at a significance level of 0.025. Key concepts include normal distribution, t-tests, and one-tailed tests.

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