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.

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.