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The sample mean, often represented as \(\bar{x}\) or mu, is the average value of a set of data points. It's calculated by adding together all of the values in the sample and then dividing by the number of values. For example, for the arsenic data \(5.4, 5.6, 8.4, 7.3, 4.5, 7.5, 1.5, 5.5, 9.1 \text{and} 8.7\), you sum all these values and divide by 10 (since there are 10 data points). The formula is: \[\text{Mean} = \frac{5.4 + 5.6 + 8.4 + 7.3 + 4.5 + 7.5 + 1.5 + 5.5 + 9.1 + 8.7}{10} = 6.35\] This gives the average amount of arsenic in the brown rice samples from California. The sample mean is a critical piece because it serves as the center point for constructing the confidence interval.

The standard deviation measures how spread out the data points are around the mean. It indicates the amount of variation or dispersion of a set of values. For a sample, the formula to calculate the standard deviation is: \[s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}\] In this formula, \(n\) is the sample size, \(x_i\) are the individual data points, and \(\bar{x}\) is the sample mean. High standard deviation means that the data points are spread out over a larger range of values, whereas a low standard deviation indicates that the data points are close to the mean. In our case, once we calculate the sample deviation, we have more insight into how the arsenic levels vary in the brown rice samples.

The t-distribution is a type of probability distribution that is symmetric and bell-shaped, much like the normal distribution, but has heavier tails. This means it is more prone to producing values that fall further from its mean. It is particularly useful when dealing with small sample sizes (typically n < 30). The t-distribution is used instead of the normal distribution to estimate population parameters when the sample size is small and the population standard deviation is unknown. For our purpose, we use the t-distribution to find the critical value when creating the confidence interval. The degrees of freedom for the t-distribution are determined by the sample size minus one (n - 1). In this case, with 10 samples, we use 9 degrees of freedom.

The confidence level represents the percentage of all possible samples that can be expected to include the true population parameter. A 90% confidence level means that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect that 90 of the intervals will contain the population mean. It does not imply that the probability that our specific calculated interval contains the population mean is 90%. To construct a 90% confidence interval for the mean arsenic level in brown rice, we will use the sample mean, the standard error, and the critical value from the t-distribution with 9 degrees of freedom. Combining these elements appropriately helps estimate the range within which the true mean of arsenic levels in California brown rice samples lies, giving us confidence in the precision of our results.