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Let's go through the problem step-by-step.

### Problem Summary
1. **Context**: The FDA regulates the number of insect parts in wheat flour. The allowed mean limit is 75 insect parts per 100 grams.
2. **Sample Data**: A sample of 49 bags of flour is found to contain an average of 79.5 insect parts per 100 grams.
3. **Claim**: Test if the entire shipment of wheat is unfit for human consumption based on this sample, using a significance level of 10% (0.10).

### Hypotheses
- **Null Hypothesis \( H_0 \)**: The mean number of insect parts in the wheat flour is 75 per 100 grams.
  \[
  H_0: \mu = 75
  \]
- **Alternative Hypothesis \( H_1 \)**: The mean number of insect parts in the wheat flour is greater than 75 per 100 grams.
  \[
  H_1: \mu > 75
  \]

### Test Statistic
The **Z-test** is used here, as the population standard deviation (12) is known. The formula for the Z-test statistic in this case is:
\[
Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}
\]
where:
- \( \bar{x} = 79.5 \) (sample mean),
- \( \mu = 75 \) (population mean under \( H_0 \)),
- \( \sigma = 12 \) (population standard deviation),
- \( n = 49 \) (sample size).

Plugging in the values:
\[
Z = \frac{79.5 - 75}{12 / \sqrt{49}} = \frac{4.5}{12 / 7} = \frac{4.5 \times 7}{12} = 2.625
\]

The test statistic here is approximately **2.625**.

### P-Value
The p-value corresponds to the probability of observing a test statistic at least as extreme as the one computed, assuming \( H_0 \) is true.

In the provided image, the p-value is approximately **0.0069**.

### Decision
Since the p-value (0.0069) is less than the significance level (0.10), we **reject the null hypothesis**. This implies there is significant evidence to suggest the mean insect parts are greater than 75, supporting the claim that the shipment may not be fit for human consumption.

### Type I Error
A **Type I error** would occur if we reject \( H_0 \) when it is actually true. In this context, it means concluding that the flour is contaminated (unfit for human consumption) when it actually meets the acceptable limit.

Let me know if you need further explanation or have questions. Here are five related questions to deepen understanding:

1. How does changing the significance level affect the decision in hypothesis testing?
2. What is the difference between a one-tailed and two-tailed test?
3. Why is the population standard deviation used instead of the sample standard deviation here?
4. How would a Type II error differ in this situation, and what are its implications?
5. How does increasing the sample size influence the test statistic and p-value?

**Tip**: When interpreting p-values, remember that a smaller p-value means stronger evidence against the null hypothesis.

The Food and Drug Administration (FDA) regulates the allowable insect parts in foods. The mean number of insect parts allowed in 100 grams of wheat flour is 75. A sample of 49 bags of flour contains an average of 79.5 insect parts per 100 grams. Test if the entire shipment of wheat is unfit for human consumption at a 10% significance level.

This page covers a hypothesis testing problem based on FDA regulations. A sample of wheat flour contains 79.5 insect parts per 100 grams, exceeding the mean allowable limit of 75. Learn to apply Z-test for hypothesis testing at a 10% significance level to evaluate contamination.

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