To construct a 95% confidence interval for the mean amount of mercury in the tuna sushi samples, we can follow these steps:

Step 1: Calculate the Sample Mean ($\bar{x}$)

The sample data is:

• 0.59
• 0.75
• 0.10
• 0.98
• 1.25
• 0.51
• 0.85

Calculating the mean: $\bar{x} = \frac{0.59 + 0.75 + 0.10 + 0.98 + 1.25 + 0.51 + 0.85}{7} = \frac{4.93}{7} \approx 0.7043$

Step 2: Calculate the Sample Standard Deviation ($s$)

To find the sample standard deviation: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$ Where $n = 7$.

Calculating each squared deviation:

• $(0.59 - 0.7043)^2 \approx 0.0131$
• $(0.75 - 0.7043)^2 \approx 0.0021$
• $(0.10 - 0.7043)^2 \approx 0.3643$
• $(0.98 - 0.7043)^2 \approx 0.0767$
• $(1.25 - 0.7043)^2 \approx 0.2970$
• $(0.51 - 0.7043)^2 \approx 0.0370$
• $(0.85 - 0.7043)^2 \approx 0.0215$

Summing these squared deviations: $\sum (x_i - \bar{x})^2 \approx 0.0131 + 0.0021 + 0.3643 + 0.0767 + 0.2970 + 0.0370 + 0.0215 \approx 0.8117$

Now calculate $s$: $s = \sqrt{\frac{0.8117}{6}} \approx \sqrt{0.1353} \approx 0.3686$

Step 3: Determine the Critical Value for 95% Confidence

For a 95% confidence level and $n-1 = 6$ degrees of freedom, we can find the critical t-value from a t-distribution table: $t_{0.025, 6} \approx 2.447$

Step 4: Calculate the Margin of Error (ME)

$ME = t^* \cdot \frac{s}{\sqrt{n}} = 2.447 \cdot \frac{0.3686}{\sqrt{7}} \approx 2.447 \cdot 0.1395 \approx 0.3422$

Step 5: Construct the Confidence Interval

$\text{Lower limit} = \bar{x} - ME \approx 0.7043 - 0.3422 \approx 0.3621$ $\text{Upper limit} = \bar{x} + ME \approx 0.7043 + 0.3422 \approx 1.0465$

Final Confidence Interval

Thus, the 95% confidence interval for the mean amount of mercury in the tuna sushi is approximately: $(0.3621, 1.0465)$

Summary

The mean amount of mercury in the sampled tuna sushi is estimated to be between 0.3621 ppm and 1.0465 ppm with 95% confidence.

Would you like more details on any part of this process? Here are some related questions to consider:

1. How do different sample sizes affect the width of a confidence interval?
2. What assumptions are made when calculating a confidence interval for the mean?
3. How would the confidence interval change if the sample contained more extreme values?
4. Why is it important to monitor mercury levels in fish?
5. What are some other common food safety guidelines regarding mercury levels?

Tip: Always check if your sample size is sufficient to make reliable inferences about the population.