To solve this problem, we need to find the probability of a Type II error. A Type II error occurs when we fail to reject a false null hypothesis. In this case, we are testing the hypothesis that $p = 0.2$ (i.e., the proportion of defective items is 0.2), but we want to determine the probability of a Type II error if the true proportion $p$ is not equal to 0.2.

Step-by-step Approach:

1. Hypothesis Test Setup:

   • Null Hypothesis (H₀): $p = 0.2$ (the proportion of defective items is 0.2).
   • Alternative Hypothesis (H₁): $p \neq 0.2$ (the proportion is not 0.2).

   The test involves a random sample of 8 items, and we accept the null hypothesis if the number of defective items in the sample is 6 or fewer. This means if we observe 7 or 8 defective items, we reject the null hypothesis.

2. Type II Error:

   • A Type II error occurs when the null hypothesis is not rejected even though it is false. In this case, we would fail to reject $H₀$ when the actual proportion of defective items is different from 0.2.
   • For a given true value of $p$, the probability of a Type II error is the probability that we fail to reject $H₀$, which means having 6 or fewer defective items in the sample.

3. Binomial Distribution: Since we are sampling 8 items, the number of defective items in the sample follows a binomial distribution. If the true proportion of defectives is $p$, the number of defectives $X$ in the sample follows: $X \sim \text{Binomial}(n = 8, p)$ where $n = 8$ is the sample size and $p$ is the true proportion of defectives.

   The probability of a Type II error is the probability of observing 6 or fewer defective items when the true proportion of defectives is $p$. This can be expressed as: $P(\text{Type II Error}) = P(X \leq 6 \mid p)$

4. Calculating the Probability of Type II Error: We can calculate the probability of observing 6 or fewer defective items using the binomial cumulative probability for different values of $p$. Let's assume the true proportion $p$ is not 0.2 (since that's the null hypothesis), and calculate the Type II error probability for various values of $p$.

Let's proceed with the calculation for a specific value of $p$ (e.g., $p = 0.3$):

Step-by-Step Calculation for $p = 0.3$:

For $n = 8$ and $p = 0.3$, we can calculate $P(X \leq 6)$ using the cumulative distribution function (CDF) of the binomial distribution:

$P(X \leq 6 \mid p = 0.3) = \sum_{x=0}^{6} \binom{8}{x} (0.3)^x (0.7)^{8-x}$

You can compute this sum using a calculator or software (like Python, Excel, or a statistical calculator). For this problem, the answer will give the probability of a Type II error when the true proportion of defectives is $p = 0.3$.

Final Answer:

The probability of a Type II error depends on the true proportion of defective items, $p$. You can compute it using the formula above for different values of $p$.

Would you like me to proceed with the exact calculation for $p = 0.3$, or is there another specific value of $p$ you'd like to analyze?