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The sample proportion is an estimate of the true proportion in a population based on a sample. In our case of firefighter masks, it provides an initial guess of the proportion of masks that have a failure at high temperatures. To find the sample proportion, we use the formula:\[ \hat{p} = \frac{x}{n} \]where \( x \) is the number of successes (masks with popped lenses) and \( n \) is the total number of masks tested. Here, we calculated it as \( \hat{p} = \frac{11}{55} = 0.2 \). This means 20% of the sampled masks failed the test. The sample proportion is crucial because it sets the stage for constructing the confidence interval, helping us make inferences about the entire population of masks.

Standard error measures the variability or "spread" of the sample proportion from the true population proportion. Understanding it helps us determine how reliable our sample estimate is likely to be.To calculate the standard error of the sample proportion \( \hat{p} \), use the formula:\[ SE = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n}} \]This formula accounts for the proportion of successful outcomes \( \hat{p} \) and the probability of failure \( 1-\hat{p} \), scaled by the sample size \( n \). In our scenario:\[ SE = \sqrt{ \frac{0.2 \times 0.8}{55} } \approx 0.0538 \]A smaller standard error indicates that our sample proportion is more precise, which is comforting for making broader conclusions about mask reliability.

The critical value, derived from the chosen confidence level, determines the width of our confidence interval. For a 90% confidence interval, the critical value comes from the Z-distribution table (a standard table for normal distribution percentages).To find this value, we consider the complement of our confidence level, which is \( 1 - 0.90 = 0.10 \). Then, halve it to find the area in each tail, resulting in \( \alpha/2 = 0.05 \). The critical value \( Z_{\alpha/2} \) for \( 90\% \) is approximately 1.645. Both tales absorb the remaining 10%, each consisting of 5%. This critical factor is essential because it scales the confidence interval by representing the "distance" from the sample mean that captures the true mean.

The Z-distribution, also known as the standard normal distribution, is a fundamental concept for conducting statistical analyses like creating confidence intervals. It has a mean of 0 and a standard deviation of 1. All values on a Z-distribution express how many standard deviations an element is from the mean. In the context of our mask reliability test, using a Z-distribution allows us to derive the critical value necessary to build a confidence interval. Since the Z-distribution is bell-shaped and symmetric, it helps define the "spread" or "range" where we expect to find the true population parameter. Each standardized value (Z-score) corresponds to a percentile in the distribution, which assists in determining how our sample's behavior compares to the expected probabilities in a true normal distribution. Thus, it is an invaluable tool for confirming the likelihood of our observations. By applying it correctly, we ensure that our confidence intervals are robust and reliable.