A manufacturer of potato chips would like to know whether its bag filling machine works correctly at the 434 gram setting.  Based on a 14 bag sample where the mean is 437 grams and the variance is 121, is there sufficient evidence at the 0.05 level that the bags are overfilled?  Assume the population distribution is approximately normal. Find the value of the test statistic. 

To determine if there is sufficient evidence at the 0.05 level that the bags are overfilled, we will perform a hypothesis test using the sample data. Specifically, we will conduct a one-sample t-test because we are working with a sample (14 bags) and we do not know the population standard deviation.

### Step 1: Define the hypotheses

- **Null Hypothesis (H₀):** The mean weight of the bags is 434 grams. \( \mu = 434 \)
- **Alternative Hypothesis (H₁):** The mean weight of the bags is greater than 434 grams. \( \mu > 434 \) (this is a one-tailed test for overfilling).

### Step 2: Set the significance level

- The significance level \( \alpha \) is 0.05.

### Step 3: Calculate the test statistic

We will use the t-test statistic formula:

\[
t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}
\]

Where:
- \( \bar{x} \) is the sample mean (437 grams),
- \( \mu_0 \) is the hypothesized population mean (434 grams),
- \( s \) is the sample standard deviation, and
- \( n \) is the sample size (14 bags).

#### First, calculate the sample standard deviation:
The variance is given as 121, so the standard deviation \( s \) is:

\[
s = \sqrt{121} = 11
\]

#### Now, calculate the t-statistic:
\[
t = \frac{437 - 434}{11/\sqrt{14}} = \frac{3}{11/\sqrt{14}} = \frac{3}{2.940} \approx 1.02
\]

### Step 4: Conclusion

The value of the test statistic is approximately **1.02**. 

This value would need to be compared against the critical value from the t-distribution with \( n - 1 = 13 \) degrees of freedom for a one-tailed test at the 0.05 significance level. 

Would you like to continue with the next step to compare this with the critical value, or do you have any questions?

### Related Questions:
1. What is the critical t-value for a one-tailed test at the 0.05 significance level?
2. How do we determine whether to reject the null hypothesis using the p-value?
3. How would this test differ if the sample size were larger?
4. What assumptions are we making about the population in this test?
5. How would the conclusion change if we were testing for underfilling rather than overfilling?

**Tip:** Always check if your sample size is large enough; for small samples, use the t-distribution, but for large samples (n > 30), the z-distribution may be more appropriate.

A manufacturer of potato chips would like to know whether its bag filling machine works correctly at the 434 gram setting. Based on a 14 bag sample where the mean is 437 grams and the variance is 121, is there sufficient evidence at the 0.05 level that the bags are overfilled? Assume the population distribution is approximately normal. Find the value of the test statistic.

Explore a hypothesis test to determine if potato chip bags are overfilled at the 434 gram setting. Based on a sample of 14 bags with a mean weight of 437 grams and a variance of 121, we calculate the test statistic using the t-distribution. Ideal for Grades 11-12 students studying statistics.

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