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To determine if Natalie met her goal, we need to conduct a one-sample t-test with the following hypotheses:

1. **Null Hypothesis (H₀)**: \( \mu = 10 \) (The average percentage of unpopped kernels is 10%).
2. **Alternative Hypothesis (Hₐ)**: \( \mu < 10 \) (The average percentage of unpopped kernels is less than 10%).

Since option **B** is selected, it correctly represents these hypotheses.

### Step 1: Calculate the sample mean (\( \bar{x} \))
The sample data consists of percentages of unpopped kernels from 8 bags:
\[
6.9, 3.4, 9.5, 4.3, 8.7, 10.9, 5.1, 5.3
\]

Calculate the mean (\( \bar{x} \)) of these values:
\[
\bar{x} = \frac{6.9 + 3.4 + 9.5 + 4.3 + 8.7 + 10.9 + 5.1 + 5.3}{8}
\]

### Step 2: Calculate the sample standard deviation (\( s \))
We find the standard deviation of the sample data to use in the t-test.

### Step 3: Calculate the t-statistic
The t-statistic is given by:
\[
t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
\]
where:
- \( \bar{x} \) is the sample mean,
- \( \mu = 10 \) (hypothesized mean),
- \( s \) is the sample standard deviation,
- \( n = 8 \) (sample size).

Let me know if you'd like a step-by-step calculation for these values or if you have further questions. 

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Here are five related questions:
1. What are the assumptions for conducting a t-test?
2. How does changing the significance level (e.g., from 0.05 to 0.01) affect the test result?
3. What is a Type I error in hypothesis testing?
4. How would the test change if we suspected the mean was higher than 10% instead?
5. How can we interpret the p-value obtained from this test?

**Tip**: Always check if the sample size is adequate for a t-test, especially for small samples where normality is assumed.

Natalie ran an experiment to test optimum power and time settings for microwave popcorn. Her goal was to deliver popcorn with fewer than 10% of the kernels left unpopped, on average. She determined that power 9 at 4 minutes was the best combination. To be sure that the method was successful, she popped 8 more bags of popcorn (selected at random) at this setting, and recorded the percentages of unpopped kernels for each bag: 6.9, 3.4, 9.5, 4.3, 8.7, 10.9, 5.1, 5.3. Does this provide evidence that she met her goal of an average of fewer than 10% unpopped kernels? Use 0.05 as the P-value cutoff level.

Analyze whether Natalie's method meets the goal of fewer than 10% unpopped popcorn kernels on average. This problem involves a one-sample t-test with a sample mean and sample standard deviation calculation. Suitable for college-level statistics students.

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