SOLUTION: A food safety guideline is that the mercury in fish should be below 1 part per million (ppm). Listed below are the amounts of mercury (ppm) found in tuna sushi sampled at different
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Question 1181381: A food safety guideline is that the mercury in fish should be below 1 part per million (ppm). Listed below are the amounts of mercury (ppm) found in tuna sushi sampled at different stores in a major city. Construct a 99% confidence interval estimate of the mean amount of mercury in the population. Does it appear that there is too much mercury in tuna sushi?
0.51
0.71
0.10
0.90
1.34
0.53
0.94
What is the confidence interval estimate of the population mean μ?
nothing ppm<μ< nothing ppm
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Here's how to construct the 99% confidence interval and interpret the results:
**1. Calculate the Sample Mean (x̄):**
Sum the mercury levels and divide by the number of samples:
(0.51 + 0.71 + 0.10 + 0.90 + 1.34 + 0.53 + 0.94) / 7 = 0.72 ppm
**2. Calculate the Sample Standard Deviation (s):**
This measures the spread of the sample data. Here's the formula, but it's easiest to use a calculator or spreadsheet software:
s = sqrt[ Σ(xi - x̄)² / (n - 1) ]
Where:
* xi are the individual mercury levels
* x̄ is the sample mean
* n is the sample size
s ≈ 0.39 ppm
**3. Find the t-critical Value:**
Since the sample size is small (n < 30) and the population standard deviation is unknown, we use the t-distribution. We want a 99% confidence level. The degrees of freedom (df) are n - 1 = 7 - 1 = 6.
Using a t-table or calculator, the t-critical value for a 99% confidence level and 6 degrees of freedom is approximately 3.707.
**4. Calculate the Margin of Error (E):**
E = t-critical * (s / sqrt(n))
E = 3.707 * (0.39 / sqrt(7))
E ≈ 0.55 ppm
**5. Construct the Confidence Interval:**
Lower Bound = x̄ - E = 0.72 - 0.55 = 0.17 ppm
Upper Bound = x̄ + E = 0.72 + 0.55 = 1.27 ppm
**Confidence Interval:**
0.17 ppm < μ < 1.27 ppm
**Interpretation:**
We are 99% confident that the true mean mercury level in tuna sushi in this city is between 0.17 ppm and 1.27 ppm.
**Does it appear there is too much mercury?**
Yes, because the *upper bound* of the confidence interval (1.27 ppm) is *above* the recommended limit of 1 ppm. This suggests that some tuna sushi samples in this city may have mercury levels exceeding the safe guideline.
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