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A Type II error occurs when we fail to reject the null hypothesis when the alternative hypothesis is actually true. In simpler terms, it's like sounding the all-clear when there's still a risk. Imagine a doctor telling a patient they are completely healthy when they're not; that's akin to a Type II error in statistical testing. This error is represented by the symbol \( \beta \).

The probability of a Type II error increases when the actual value of \( p \) is close to the null hypothesis value. Thus, if \( p = 0.35 \) and our null hypothesis \( H_0 \) asserts \( p = 1/3 \), it's challenging to distinguish between these close values, leading to a higher chance of making a Type II error. However, as the true value of \( p \) moves away from \( H_0 \), it becomes simpler to detect this deviation, and thus the probability of making this error declines.

In this context, finding \( \beta = 0.89 \) when \( p = 0.35 \) means there's an 89% chance of a Type II error, indicating how tough it is to spot the slight difference when \( p \) is close to \( H_0 \).

The significance level, denoted as \( \alpha \), is the threshold at which we decide whether to reject the null hypothesis. In hypothesis testing, it reflects the probability of making a Type I error, which occurs when we reject the null hypothesis \( H_0 \) when it is actually true—like believing a false alarm.

For the given exercise, \( \alpha \) is set at 0.05. This means we accept a 5% risk of incorrectly rejecting \( H_0 \). It acts as the cut-off point in deciding if our test results are statistically significant. If the calculated test statistic falls into the critical region determined by this significance level, we reject \( H_0 \). Otherwise, we fail to reject it.

Choosing a smaller \( \alpha \) reduces the chances of a Type I error but potentially increases the probability of a Type II error, and vice versa. Researchers must carefully determine the significance level based on the context of their study and the consequences of making errors.

When conducting hypothesis tests, especially with a large sample size, it is common to use a normal approximation to model the sampling distribution of a proportion. This is due to the Central Limit Theorem, which suggests that, for large enough \( n \), the distribution of sample means will be approximately normal, regardless of the distribution of the population.

In the given exercise, the sample size \( n = 116 \) is sufficiently large, allowing us to apply this approximation effectively. This aids in converting the discrete distribution of sample proportions into a continuous normal distribution, making it easier to calculate probabilities and critical values using standard normal tables.

The standard deviation of the sampling distribution is computed using the formula \( \sigma = \sqrt{\frac{p(1-p)}{n}} \), which helps in defining the z-value or test statistic. This transformation enables the use of a standard normal distribution (or z-distribution) to assess the test statistics derived from the sample data.

A test statistic is a standardized value calculated from sample data during a hypothesis test. It enables us to decide between the null and alternative hypotheses. Essentially, it measures how far the sample data diverges from what is expected under the null hypothesis.

In this exercise, we compute a z-value as the test statistic. The z-value is calculated using the formula: \[ z = \frac{\hat{p} - p_0}{\sigma} \] where \( \hat{p} \) is the sample proportion, \( p_0 \) is the hypothesized proportion under \( H_0 \), and \( \sigma \) is the standard deviation of the sampling distribution.

The test statistic is then compared to a critical value from the standard normal distribution. For a significance level \( \alpha = 0.05 \), a one-tailed test would use a critical value of approximately 1.645. If the test statistic exceeds this critical value, we reject the null hypothesis. This methodology helps determine whether the observed effect is significant or could have occurred by random chance, guiding decision-making in statistics.