The P-value for the one-sample t-test is approximately 0.0232. At a 0.02 significance level, we reject the null hypothesis.
To test whether the bag filling machine is underfilling at the 409 gram setting, we can use a one-sample t-test. The null hypothesis (H0) is that the mean filling weight is 409 grams, and the alternative hypothesis (H1) is that the mean is less than 409 grams.
1. Formulate Hypotheses:
- Null Hypothesis (H0): μ = 409 grams
- Alternative Hypothesis (H1): μ < 409 grams
2. Calculate the Test Statistic:
- The formula for the t-test statistic is:
![\[ t = \frac{{\bar{X} - \mu}}{{s/√(n)}} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/wppfx873nhczwywuc5iou9br814tu2v1wz.png)
where
is the sample mean,
is the population mean, s is the sample standard deviation, and n is the sample size.
Substituting the given values, we get
![\[ t = \frac{{399 - 409}}{{21/√(20)}} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/d9op9u23nogfezu5dxqx0a4tbe1svhlpnp.png)
3. Determine the P-value:
- Using the t-distribution table or statistical software, find the P-value associated with the calculated t-statistic.
4. Make a Decision:
- Compare the P-value to the significance level (0.02). If P-value < 0.02, reject the null hypothesis.
For the P-value calculation, you can use statistical software or online calculators. The P-value is the probability of observing a sample mean as extreme as 399 grams (or more extreme) if the true population mean is 409 grams. Report the P-value as a decimal rounded to four decimal places or as an interval if needed.