To construct a 98% confidence interval estimate of the mean amount of mercury in the population, follow these steps:
1. List the sample data and determine sample size:
[tex]\[
0.55, 0.79, 0.10, 0.96, 1.30, 0.57, 0.87
\][/tex]
The sample size is:
[tex]\[
n = 7
\][/tex]
2. Calculate the sample mean ([tex]\(\bar{x}\)[/tex]):
[tex]\[
\bar{x} = 0.734
\][/tex]
3. Calculate the sample standard deviation ([tex]\(s\)[/tex]):
[tex]\[
s = 0.378
\][/tex]
4. Determine the t-critical value for a 98% confidence level with [tex]\(n-1\)[/tex] degrees of freedom:
With [tex]\(df = 6\)[/tex] (since [tex]\(df = n - 1\)[/tex]), we find:
[tex]\[
t_{\alpha/2, df} = 3.143
\][/tex]
5. Calculate the standard error of the mean (SEM):
[tex]\[
SEM = \frac{s}{\sqrt{n}} = \frac{0.378}{\sqrt{7}} = 0.143
\][/tex]
6. Calculate the margin of error (ME):
[tex]\[
ME = t_{\alpha/2} \times SEM = 3.143 \times 0.143 = 0.448
\][/tex]
7. Construct the confidence interval:
The lower limit of the confidence interval is:
[tex]\[
\bar{x} - ME = 0.734 - 0.448 = 0.286
\][/tex]
The upper limit of the confidence interval is:
[tex]\[
\bar{x} + ME = 0.734 + 0.448 = 1.183
\][/tex]
Thus, the 98% confidence interval estimate of the population mean [tex]\(\mu\)[/tex] is:
[tex]\[
0.286 \ \text{ppm} < \mu < 1.183 \ \text{ppm}
\][/tex]
Now, we need to determine if there is too much mercury in tuna sushi. According to the food safety guideline, the mercury level should be below 1 ppm. Since the upper limit of our confidence interval (1.183 ppm) is greater than 1 ppm, it suggests that there is a possibility that the mean mercury level in tuna sushi is above the safety guideline. Therefore, it appears that there could be too much mercury in the tuna sushi sampled.
