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.