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In hypothesis testing, the significance level, denoted by \(\alpha\), is a critical concept. It represents the probability of rejecting the null hypothesis when it is actually true. Common significance levels are 0.01, 0.05, or 0.10, which correspond to 1%, 5%, and 10% probabilities of making a Type I error.

- A smaller \(\alpha\) indicates stronger evidence is needed to reject the null hypothesis.
- In this exercise, with \(\alpha = 0.01\), the test is very stringent, requiring strong evidence to conclude that the true average percentage differs from the hypothesized 5.5.

When determining critical values for the test, this significance level helps define the rejection region. For a two-tailed test at \(\alpha = 0.01\), we look for critical Z-values from the standard normal distribution, approximately \(\pm 2.576\). If the test statistic falls into this rejection region, the null hypothesis is rejected, as seen in this case with a test statistic of \(-3.33\). The decision to reject confirms that the observed sample mean significantly deviates from the expected value.

The test statistic is a standardized value that is used to decide whether to reject the null hypothesis. It measures how far the sample statistic is from the expected value under the null hypothesis, labeled as \(H_0\).

The calculation for the test statistic in this scenario uses the formula:
\[Z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}\]
where \(\bar{x}\) is the sample mean, \(\mu_0\) the population mean under \(H_0\), \(\sigma\) the standard deviation, and \(n\) the sample size.

For this exercise:

• \(\bar{x} = 5.25\), the observed sample mean.
• \(\mu_0 = 5.5\), the hypothesized population mean.
• \(\sigma = 0.3\), indicating the variability.
• \(n = 16\), reflecting the sample size.

Substituting these values gives a \(Z\) score of \(-3.33\). This score tells us how many standard deviations the observed mean is from the hypothesized mean.

In summary, a larger absolute value of the test statistic indicates that the sample results significantly deviate from \(H_0\), leading to a decision that likely the true average percentage is different from 5.5.

Determining the right sample size is crucial for designing experiments that are both efficient and have satisfactory statistical power. Sample size affects the reliability of hypothesis tests, particularly influencing Type I and Type II errors.

To achieve a desired significance level \(\alpha\) and a specific \(\beta\) (Type II error rate), we use the sample size formula:
\[n = (Z_{\alpha/2} + Z_{\beta})^2 \times \frac{\sigma^2}{(\mu - \mu_0)^2}\]

• \(Z_{\alpha/2}\) is the Z-value corresponding to the desired significance level's half. For \(\alpha = 0.01\), it is 2.576.
• \(Z_{\beta}\) is the Z-value corresponding to the allowable Type II error. For \(\beta = 0.01\), it is 2.33.
• \(\sigma\) is the population standard deviation.
• \(\mu - \mu_0\) is the minimum detectable difference.

In this case, calculating with given values, the required sample size \(n\) is around 84, ensuring the test has the power to detect the specified effects at given significance and power thresholds. Thus, this thorough calculation prepares the experiment to reliably conclude differences in the cement's \(\mathrm{SiO}_2\) percentage.