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The t-test is a statistical tool commonly used to determine if there is a significant difference between the means of two datasets. In this scenario, we want to know whether the average reading from our radon detectors is different from a hypothesized population mean. Here, our sample of radon detectors shows a sample mean reading, and we hypothesize it to deviate from the standard 100 units.In hypothesis testing, you first need to set up two hypotheses:

• The null hypothesis (\(H_0\)): shows no effect or no difference, meaning our mean is equal to 100.
• The alternative hypothesis (\(H_a\)): indicates a difference, suggesting the mean is not equal to 100.

Performing a t-test involves calculating a test statistic that considers the sample mean, hypothesized mean, and the sample's standard deviation. If this statistic is greater than a critical value from the t-distribution table, we reject the null hypothesis. We'll explore critical values and their importance in the decision-making below.

The sample mean is a crucial part of hypothesis testing. It gives us an average value based on the sample data available. Calculating the sample mean involves summing up all observed values and dividing by the number of observations.In our exercise, the radon detectors' readings provide a sample mean calculated as follows:\[ \bar{x} = \frac{105.6 + 90.9 + 91.2 + \ldots + 92.4}{12} = 98.8 \]This calculated mean contrasts the hypothesized population mean of 100. Noticing such differences prompts us to conduct the hypothesis test to statistically verify if this observed difference is significant or just due to sample variability.

Standard deviation is another critical metric. It measures how much the data points deviate from the sample mean. A smaller standard deviation indicates that data points are close to the sample mean, while a larger one suggests that they are more spread out.For our t-test, the standard deviation (\(s\)) helps quantify variability in the sample readings:

• A high standard deviation means more dispersion, potentially affecting the test's outcome.
• A low standard deviation indicates tightly clustered data, which often leads to more precise inferences.

When computing the test statistic for the t-test, the standard deviation is used to standardize the difference between the sample mean and the hypothesized population mean. This standardization allows us to employ the t-distribution in making our statistical decision.

Critical values serve as thresholds in hypothesis testing. These values determine when to reject the null hypothesis. For a given significance level (\(\alpha\)), they mark the boundary of the acceptance region.In a two-tailed test at \(\alpha = 0.05\) and 11 degrees of freedom, we look for critical values in the t-distribution table:

• If the absolute value of our t-statistic exceeds the critical value, we reject \(H_0\).
• If it does not, we do not have enough evidence to reject \(H_0\).

Thus, critical values are vital to the decision-making process. They help us understand whether observed differences in sample data point to a genuine departure from the hypothesized population mean or are consistent with random sampling variability.

Determining sample size is essential when planning experiments. It ensures that the test is powerful enough to detect meaningful differences. In our example, we want to choose the right number of radon detectors to achieve a specific power level, considering a known population standard deviation and an assumed alternative mean.The formula for calculating the required sample size \(n\) is:\[ n = \left(\frac{Z_{\alpha/2} \cdot \sigma + Z_{\beta} \cdot \sigma}{\mu - \mu_0}\right)^2 \]Where:

• \(Z_{\alpha/2}\) and \(Z_{\beta}\) are values from the standard normal distribution, reflecting the preferred confidence level and power, respectively.
• \(\sigma\) is the assumed standard deviation.
• \(\mu\) and \(\mu_0\) represent the alternative and null hypothesis means.

Choosing an adequate sample size helps ensure the test can distinguish between true and false hypotheses, ultimately strengthening the reliability and accuracy of your conclusions.