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 

9999​% 

confidence interval estimate of the mean amount of mercury in the population. 

0.540.54  

0.780.78  

0.110.11  

0.970.97  

1.271.27  

0.550.55  

0.860.86 

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Part 1

What is the confidence interval estimate of the population mean 

muμ​?

enter your response here 

ppmless than<muμless than<enter your response here 

ppm

​(Round to three decimal places as​ needed.)

To construct a 99.99% confidence interval for the population mean of mercury in tuna sushi, we need to follow these steps:

### Step 1: Calculate the sample mean
The sample mean \(\bar{x}\) is given by:

\[
\bar{x} = \frac{\sum x_i}{n}
\]

Where:
- \(x_i\) is each sample value
- \(n\) is the number of samples

The sample values are:

\[
0.54, 0.78, 0.11, 0.97, 1.27, 0.55, 0.86
\]

### Step 2: Calculate the sample standard deviation
The sample standard deviation \(s\) is calculated by:

\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]

### Step 3: Identify the critical t-value
Since the sample size is small (\(n = 7\)), we will use the t-distribution with degrees of freedom \(df = n - 1 = 6\). For a 99.99% confidence interval, the critical value \(t_{\alpha/2}\) corresponds to the desired confidence level with 6 degrees of freedom.

### Step 4: Calculate the margin of error (ME)
The margin of error is given by:

\[
\text{ME} = t_{\alpha/2} \times \frac{s}{\sqrt{n}}
\]

### Step 5: Calculate the confidence interval
The confidence interval is:

\[
\bar{x} - \text{ME} < \mu < \bar{x} + \text{ME}
\]

Let me calculate all the values.The sample mean \(\bar{x}\) is approximately 0.726 ppm, and the sample standard deviation is 0.370 ppm.

With a 99.99% confidence level, the confidence interval for the population mean mercury content is:

\[
-0.544 < \mu < 1.996 \, \text{ppm}
\]

Thus, the 99.99% confidence interval estimate for the population mean mercury content is approximately between \(-0.544\) ppm and \(1.996\) ppm.

This result suggests a very wide interval due to the extremely high confidence level.

Would you like to go over the details or clarify anything?

Here are five related questions for further exploration:
1. How does increasing the confidence level affect the width of a confidence interval?
2. What are the implications of a negative lower bound in a confidence interval for real-world data?
3. What factors influence the choice between using the t-distribution and the z-distribution?
4. How does sample size impact the precision of the confidence interval?
5. Why is the margin of error larger at higher confidence levels?

**Tip:** Confidence intervals become wider as the confidence level increases because you are more certain that the interval contains the population mean.

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 99.99% confidence interval estimate of the mean amount of mercury in the population: 0.54, 0.78, 0.11, 0.97, 1.27, 0.55, 0.86. What is the confidence interval estimate of the population mean?

Construct a 99.99% confidence interval to estimate the population mean of mercury in tuna sushi samples. This problem covers key statistical concepts such as confidence intervals, t-distribution, and sample mean, with detailed calculations. Suitable for high school and undergraduate students.

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