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This problem involves testing whether the average percentage of unpopped kernels is below 10%, based on the provided sample data. This is a one-sample t-test, as we have a sample of data and are comparing it to a known population mean (10%).

### Steps to Solve

1. **Identify the Hypotheses:**
   - Null hypothesis (\( H_0 \)): The average percentage of unpopped kernels is 10% (or more). Mathematically, \( H_0: \mu \geq 10 \).
   - Alternative hypothesis (\( H_a \)): The average percentage of unpopped kernels is less than 10%. Mathematically, \( H_a: \mu < 10 \).

2. **Calculate the Test Statistic:**
   The t-test statistic is given by:
   \[
   t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
   \]
   where:
   - \( \bar{x} \) is the sample mean,
   - \( \mu \) is the population mean under \( H_0 \) (10% in this case),
   - \( s \) is the sample standard deviation,
   - \( n \) is the sample size.

   Given in the image:
   - Sample mean (\( \bar{x} \)) = 8.05 (assuming this from the t value),
   - Sample standard deviation (\( s \)) = 3.42 (approx.),
   - \( n = 8 \).

3. **Compute the p-value:**
   The p-value represents the probability of observing a sample mean as extreme as, or more extreme than, the observed sample mean under the null hypothesis.

4. **Decision Rule:**
   - If \( p \)-value < 0.05, we reject \( H_0 \).
   - If \( p \)-value ≥ 0.05, we fail to reject \( H_0 \).

Since the image already provides the \( t \)-value as -1.37, we can proceed by finding the corresponding p-value for this t-statistic in a one-tailed test with \( n - 1 = 7 \) degrees of freedom.

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Would you like further assistance on calculating the exact p-value or verifying the sample mean and standard deviation?

Here are some related questions to consider:

1. What is the interpretation of a p-value in hypothesis testing?
2. How do degrees of freedom impact the shape of the t-distribution?
3. What conditions must be met to use a one-sample t-test?
4. Why is a one-tailed test used in this scenario?
5. What might be some sources of error in this experimental setup?

**Tip:** Always check the assumptions of your test, especially normality for small samples when using t-tests.

Pat ran an experiment to test the optimum power and time settings for microwave popcorn. His goal was to deliver popcorn with fewer than 10% of the kernels left unpopped, on average. He popped 8 bags of popcorn at a specific setting, and the percentages of unpopped kernels were recorded as follows: 2.2, 4.6, 2.7, 12.1, 12.6, 9.6, 13.7, and 3.8. Does this provide evidence that he met his goal of an average of fewer than 10% unpopped kernels? Use a 0.05 significance level.

Explore a hypothesis testing problem where we determine if the average percentage of unpopped kernels in microwave popcorn is below 10%. Using a one-sample t-test with data from 8 popcorn bags, we calculate the test statistic and p-value at a 0.05 significance level to test the hypothesis.

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