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A Type I Error occurs when we wrongly reject a true null hypothesis. It is also known as a 'false positive'. Imagine you're testing to see if a new drug is effective, and you conclude that it is, when in fact it isn't. This mistake can lead to incorrect actions based on bad evidence.

In hypothesis testing, the probability of making a Type I Error is denoted by \( \alpha \). An \( \alpha \) value of 0.05, for example, means there's a 5% risk of making a Type I Error. Therefore, the lower the \( \alpha \), the less likely we are to wrongly reject a true hypothesis.

In our crankshaft bearing study, if we choose \( \alpha = 0.05 \), it indicates that there's a 5% chance we might wrongly conclude that the proportion of bearings with excess roughness exceeds 0.10 when, in fact, it does not.

Type II Error, often referred to as a 'false negative', happens when we fail to reject a false null hypothesis. In layman's terms, it's when we mistakenly think there is no effect or difference when there truly is one.

The probability of a Type II Error is denoted by \( \beta \). Higher \( \beta \) values mean higher chances of making this error. Reducing \( \beta \) is crucial when we want to confidently identify real effects in experiments.

For instance, failing to identify that 15% of bearings exceed roughness due to an unusually low sample size or other factors leads to a Type II Error. In our study, \( \beta = 0.242 \) suggests there's a 24.2% chance of not detecting that more than 10% of bearings are rough, even if \( p = 0.15 \).

Determining the appropriate sample size is critical to achieving reliable results in hypothesis testing. Having too small a sample might lead to inaccurate conclusions, while too large a sample can be costly and time-consuming.

When calculating sample size, consider both Type I and Type II Error rates as well as the minimum detectable difference you wish to identify. The formula for sample size takes into account these factors along with expected proportions.

In our example with crankshaft bearings, to achieve a power of 0.90, which is the probability of correctly rejecting the null hypothesis, the sample size needed is 238. This ensures that if \( p = 0.15 \), there's a 90% chance of the test correctly identifying this, reducing the likelihood of a Type II Error to 10%.

The normal approximation condition is key when dealing with sample proportions. It's used to ensure that the sample size is large enough to approximate the sampling distribution to a normal distribution.

This condition requires both \( np \) and \( n(1-p) \) to be greater than 5. This ensures the central limit theorem applies, thereby allowing us to use normal distribution assumptions in our calculations.

For the crankshaft bearing study, with \( n = 85 \) and \( p = 0.10 \), we calculated \( np = 8.5 \) and \( n(1-p) = 76.5 \), satisfying the normal approximation condition. This validation is crucial for the reliability of our Z-test results.

In hypothesis testing, the test statistic is a standardized value that helps determine whether to reject the null hypothesis. It measures how far the sample statistic is from the hypothesized population parameter in terms of standard error.

The test statistic used for proportions is the Z-score, calculated as:\[Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]where \( \hat{p} \) is the sample proportion, \( p_0 \) is the hypothesized proportion, and \( n \) is the sample size.

In our crankshaft bearing example, the Z-score came out to be approximately 0.651. This score is compared to a critical value from the standard normal distribution (1.645 for a 0.05 significance level). Since 0.651 is less than 1.645, we do not have enough evidence to reject the null hypothesis that only 10% of bearings exceed roughness standards.