91Ó°ÊÓ

To find the average amount of mercury in tuna sushi from the data given, we need to calculate the sample mean. This can be done by summing all the mercury levels from the sampled stores and then dividing by the number of stores. The formula for the sample mean is given by:
\(\bar{x} = \frac{\text{\textbackslash sum{x_i}}}{n}\)
Where:
\(\bar{x}\) = sample mean,
\(\text{\textbackslash sum{x_i}}\) = sum of all sample values,
\({n}\) = number of samples.
In our example, the mercury levels are: 0.8, 0.9, 0.9, 0.8, 1.1, 0.7, and 0.6. Using the formula, we find:
\(\bar{x} = \frac{0.8 + 0.9 + 0.9 + 0.8 + 1.1 + 0.7 + 0.6}{7} \approx 0.8286\)
This gives us the sample mean of about 0.8286 ppm.

The sample standard deviation helps us understand the variability or spread of mercury levels in our sampled sushi. It is calculated by measuring how much each sample value deviates from the sample mean.
The formula for the sample standard deviation is:
\[ s = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n-1}} \]
Where:
\(s\) = sample standard deviation,
\(x_i\) = each sample value,
\(\bar{x}\) = sample mean,
\(n\) = number of samples.
Steps involved:
1. Subtract the sample mean from each sample value to find the deviation.
2. Square each deviation.
3. Sum all squared deviations.
4. Divide by the degrees of freedom (which is \(n-1\)).
5. Take the square root of the result.
In this case, we would compute each of these steps to find the standard deviation of the mercury levels.

The t-score is a critical value used in statistics when constructing confidence intervals, especially for smaller sample sizes. It comes from the t-distribution, which is similar to the normal distribution but with heavier tails.
To find the t-score for a given confidence interval, we use a t-distribution table based on our sample size and desired confidence level.
For a 98% confidence interval and 6 degrees of freedom (df = n-1), we look up the t-score in the table. In this problem, the t-score is approximately 2.447.
The t-score helps to adjust the margin of error, accounting for the uncertainty inherent in a smaller sample size.

The margin of error quantifies the range within which we expect the true population mean to lie, given our sample data. It accounts for the variability in the sample and the confidence level we choose.
The formula for margin of error is:
\[ E = t \times \frac{s}{\sqrt{n}} \]
Where:
\(E\) = margin of error,
\(t\) = t-score,
\(s\) = sample standard deviation,
\(n\) = number of samples.
By substituting the values we have from earlier steps, we calculate the margin of error as:
\(E = 2.447 \times \frac{s}{\sqrt{7}}\)
This result allows us to construct the confidence interval by adding and subtracting the margin of error from the sample mean.