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In statistical hypothesis testing, a Type I error occurs when we reject the null hypothesis even though it is true. This is also known as a "false positive" finding. In the context of our exercise, Type I error is connected with our level of significance, denoted as \( \alpha \), which is 0.05 in this example.
The significance level of 0.05 implies there is a 5% risk of making a Type I error. This means there is a 5% probability of incorrectly rejecting the null hypothesis when the population mean \( p = 14 \) is actually true.
Managing Type I error is crucial in hypothesis testing. Researchers set the significance level depending on how much risk they are willing to accept. Lower \( \alpha \) levels reduce the risk of a Type I error but might increase the risk of a Type II error.

A Type II error happens when we fail to reject the null hypothesis when it is false. In our problem, we want the probability of a Type II error, denoted as \( \beta \), to be 0.1. This indicates there is a 10% chance of not detecting a difference when the true mean is actually different from 14 by 0.5.
Balancing Type I and Type II errors is essential in hypothesis testing. A low probability of Type II error enhances the test's power. Testing power is the ability of a test to correctly reject a false null hypothesis.

• Lower \( \beta \) implies higher power, meaning the test is more likely to detect true differences.
• In practice, achieving a good balance between \( \alpha \) and \( \beta \) is key.

Optimizing sample size is one way to manage Type II error and power, which we will explore in the next section.

Determining the appropriate sample size is crucial to minimize both Type I and Type II errors. In our exercise, we aim to find the necessary sample size for a given Type II error probability. The formula utilized is:
\[n = \left( \frac{\sigma \cdot Z}{|\mu - 14|} \right)^2\]where:

• \( \sigma = 1.25 \) is the estimated standard deviation.
• \( \mu = 14.5 \) is the true mean different from 14 by 0.5.
• \( Z \) is the critical value from the Z-distribution for the chosen \( \alpha \) and \( \beta \).

Using this formula ensures that the sample size is adequate to meet the desired probabilities for error. Calculating the exact number involves using statistical software or tables, and rounding up the result to a whole number.
Make sure to consider your context. Larger samples increase precision but also require more resources.

The t-statistic is a type of test statistic typically used in hypothesis tests when the population standard deviation is unknown, particularly with small sample sizes. For our study, it helps determine how far the sample mean is from the population mean, in units of standard error.
Here's the formula for computing a t-statistic:
\[Z = \frac{|\mu - 14|}{\sigma / \sqrt{n}}\]where:

• \( \mu \) is the hypothesized population mean.
• "14" is the null hypothesis value of the mean (as per the exercise).
• \( \sigma \) and \( n \) are the estimated standard deviation and sample size, respectively.

The calculated t-statistic helps estimate how likely the observed sample mean is under the null hypothesis. It allows us to decide whether to reject the null hypothesis by comparing it to a critical value from the t-distribution.
The freedom to adjust sample size based on t-statistics enhances test sensitivity, guiding decisions on whether changes in means are statistically significant.