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.

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.