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In hypothesis testing, the significance level, often denoted by \( \alpha \), represents the probability of rejecting the null hypothesis when it is actually true. This is the threshold we set to decide whether the evidence is strong enough to support an alternative hypothesis. For example, if you set \( \alpha = 0.01 \), you are allowing a 1% risk of incorrectly rejecting a true null hypothesis, which is generally considered very stringent. The choice of significance level affects the likelihood of making type errors:

• If \( \alpha \) is too high, you increase the risk of committing a Type I error (false positive).
• If \( \alpha \) is too low, you might miss a true effect, increasing the risk of a Type II error (false negative).

A significance level of 0.01 is common in studies where highly accurate results are critical, such as in medical trials."

The power of a test is a fundamental concept in hypothesis testing. It is defined as the probability that the test correctly rejects a false null hypothesis. In simpler terms, power measures a test's ability to detect an effect when there is one. It is calculated as:\[ \text{Power} = 1 - \beta \]where \( \beta \) is the probability of committing a Type II error.A higher power means a higher probability of detecting true differences, which is crucial for researchers who want reliable results. Typically, a power of 0.80 or 0.90 is considered desirable. In the given exercise, the researcher has designed the test with a power of 0.90, meaning there is a 90% chance of detecting a true effect.

A Type I error occurs when the null hypothesis is true, but our test erroneously rejects it. This is known as a "false positive." Consider it like seeing a "ghost" when none exists.The probability of committing a Type I error is directly linked to the significance level \( \alpha \). For instance, if \( \alpha = 0.01 \), there is a 1% chance of making a Type I error in the test. Therefore, setting a lower \( \alpha \) reduces the chances of a false positive. Reducing Type I errors is crucial in situations where the consequences of a false positive are severe. In hypothesis testing, balancing Type I and Type II errors helps achieve reliable outcomes.

A Type II error occurs when a test fails to reject a false null hypothesis. This is referred to as a "false negative," akin to missing an actual pattern or effect.The probability of making a Type II error is denoted by \( \beta \). It is related to the power of the test by the equation:

• \( \beta = 1 - \text{Power} \)

This means if a test has a power of 0.90, the probability of making a Type II error is 0.10, as calculated in our exercise.Understanding and controlling both Type I and Type II error probabilities is key to effective hypothesis testing, ensuring results are both valid and impactful.