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This problem involves hypothesis testing and calculating the probability of a Type II error.

### Given Data:
- Sample mean \(\bar{x} = 16.8\)
- Sample size \(n = 15\)
- Population standard deviation \(\sigma = 2.6\)
- Significance level \(\alpha = 0.05\)

### Hypotheses:
- Null hypothesis \(H_0: \mu \geq 18\)
- Alternative hypothesis \(H_a: \mu < 18\)

We are asked to compute the probability of a **Type II error** (\(\beta\)) if the true population mean \(\mu = 16\). A Type II error occurs when we fail to reject the null hypothesis when it is actually false.

To solve this, we need to follow these steps:

1. **Find the critical value (z-critical)** at the 5% significance level for a one-tailed test.
2. **Calculate the test statistic** for the sample data.
3. **Determine the power of the test** (which is \(1 - \beta\)), and from that, find \(\beta\), the probability of a Type II error.

Let's work through these steps mathematically. I will calculate this now.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you through the manual steps to calculate the probability of a Type II error, or assist you with another query?

Here are five related questions for deeper understanding:
1. How is a Type II error different from a Type I error in hypothesis testing?
2. What is the role of sample size in reducing Type II errors?
3. How can power be calculated, and what does it indicate in hypothesis testing?
4. What does the z-critical value represent in a hypothesis test?
5. How does the significance level affect the outcome of a hypothesis test?

**Tip:** Increasing the sample size generally reduces the probability of making a Type II error, improving the power of a statistical test.

n Steekproef uit 'n normaalpopulasie lewer: / A sample from a normal population yields: x̄ = 16.8, n = 15, σ = 2.6 Die volgende hipotesetoets moet gedoen word by 'n betekenispeil van 5%: / The following hypothesis test must be done at a significance level of 5%: H0: μ ≥ 18 Ha: μ < 18 Indien die werklike waarde van μ = 16, is die waarskynlikheid van 'n Tipe II fout / If the true value of μ = 16, the probability of a Type II error is

Explore the step-by-step calculation of the probability of a Type II error in a hypothesis test involving a normal population. The test checks whether the population mean is less than 18, given a sample mean of 16.8, sample size of 15, and standard deviation of 2.6. This advanced topic is suitable for college-level statistics courses.

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