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The sample mean, denoted as \(\bar{x}\), is the average of the sampled values. To find the sample mean, sum all the mercury amounts and divide by the number of samples. This provides a central value around which the data points are distributed. For instance, in the given problem, we calculate the sample mean as follows:
\(\bar{x} = \frac{0.56 + 0.75 + 0.10 + 0.95 + 1.25 + 0.54 + 0.88}{7} \)
The result represents the average mercury level in the sampled tuna sushi. This measure is crucial as it provides an estimate of the central tendency of the data. Testing whether the sample mean is below or above the guideline helps in assessing compliance with the FDA guideline of 1 ppm.

The sample standard deviation (\(s\)) quantifies the amount of variation or dispersion in the sample data. It shows how much individual data points differ from the sample mean. Using the formula:
\( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \)
where \( x_i \) are the sample values, \( \bar{x} \) is the sample mean, and \( n \) is the sample size.
In our example, calculate the deviations of each value from the mean, square these deviations, sum them up, then divide by \( n-1 \) and finally take the square root. This results in a numerical value that indicates the spread of the mercury content around the mean. A smaller \( s \) implies that the mercury levels are closely packed around the mean, whereas a larger \( s \) indicates more variability.

When estimating population parameters from a small sample size, the t-distribution is used. This distribution is similar to the normal distribution but has thicker tails, which account for the increased variability inherent in small samples. The shape of the t-distribution changes based on the sample size, denoted by degrees of freedom (df).
To construct a confidence interval for the mean mercury level, you need to determine the t-critical value (\( t* \)). For a 98% confidence level and 6 degrees of freedom (\( n-1 \)), use a t-distribution table or calculator. This critical value helps define the range within which the true population mean is expected to lie. The t-distribution thus corrects for the additional uncertainty in the estimate when the sample size is small (less than 30).

The margin of error (\( E \)) represents the range above and below the sample mean within which the true population mean is expected to lie with a certain level of confidence. It accounts for the variability in the data and the confidence level chosen. The formula for the margin of error is:
\( E = t* \times \frac{s}{\sqrt{n}} \)
where \( t* \) is the t-critical value, \( s \) is the sample standard deviation, and \( n \) is the sample size.
By calculating the margin of error, we can then construct a confidence interval around the sample mean. For example, for a 98% confidence level, insert the appropriate \( t* \) value, the sample standard deviation, and the sample size into the formula to get \( E \). This margin of error tells us the interval's width, giving us information on how precise our estimate is. A smaller margin of error indicates a more precise estimate.