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In hypothesis testing, a Type I error occurs when we wrongly reject a true null hypothesis. You can think of it as a "false alarm." We conclude there is an effect when there actually isn't one. For example, imagine a fire alarm going off when there is no fire. The probability of making a Type I error is represented by \( \alpha \), also known as the significance level. This value is predefined by the researcher before conducting the experiment, typically set at 0.05, 0.01, or 0.10. By setting \( \alpha = 0.05 \), we are accepting a 5% risk of incorrectly rejecting the null hypothesis. This allows us to maintain control over the amount of Type I errors we are willing to tolerate.

• Occurs when the null hypothesis is true, but is rejected.
• Probability is denoted by \( \alpha \).
• Commonly set at levels like 0.05.

A Type II error happens when we fail to reject a false null hypothesis. Essentially, it's a "missed detection." Think of it like ignoring a fire alarm when a fire indeed exists. This error is less emphasized compared to Type I but is equally important. The probability of committing a Type II error is denoted by \( \beta \). In the context of the exercise, the power of the test was given as 0.78. Since statistical power is defined as \( 1-\beta \), it means the probability \( \beta \) of making a Type II error is 0.22 (calculated using \( 1-0.78 \)). As \( \beta \) decreases, the power of the test increases, making the test more reliable.

• Occurs when the null hypothesis is false, but is not rejected.
• Probability is represented by \( \beta \).
• Lower \( \beta \) leads to higher test power.

The significance level, \( \alpha \), is a critical threshold used to determine if the results of a hypothesis test are statistically significant. This predetermined threshold sets the probability of committing a Type I error. In many fields, a commonly used significance level is \( \alpha = 0.05 \). This means that there's a 5% chance of rejecting the null hypothesis when it is actually true. The choice of \( \alpha \) affects the balance between Type I and Type II errors, making it essential to choose an appropriate level for the specific research context. A smaller \( \alpha \) reduces the risk of a Type I error but increases the risk of a Type II error unless the sample size or test sensitivity is adjusted accordingly.

• Represents the risk level for false positive results.
• Commonly set at 0.05 to control Type I errors.
• Influences balance between Type I and Type II errors.

Statistical power is an essential concept that refers to the probability of correctly rejecting a false null hypothesis. It is the ability of a test to detect an effect if there truly is one. High statistical power means a test is more likely to highlight a true effect. It is quantified as \( 1- \beta \), where \( \beta \) is the probability of a Type II error. In the exercise, the statistical power is given as 0.78. This translates to a 78% probability of successfully identifying when the null hypothesis is false. Increasing the power of a test can be achieved by increasing the sample size or using a more sensitive test design. Adequate power is crucial for ensuring that research findings are reliable and valid.

• Probability of detecting a true effect.
• Calculated as \( 1- \beta \).
• Higher power indicates a more sensitive test.