To construct a 95% confidence interval estimate for the mean amount of mercury in the population, we can follow these steps:
1. Collect the Sample Data:
The amounts of mercury in ppm found in the sampled tuna sushi are:
[tex]\[
0.60, 0.82, 0.11, 0.91, 1.30, 0.52, 0.91
\][/tex]
2. Calculate the Sample Mean ([tex]$\bar{x}$[/tex]):
The sample mean is calculated by summing all the sample values and dividing by the number of samples.
[tex]\[
\bar{x} = 0.739 \quad (\text{rounded to three decimal places})
\][/tex]
3. Calculate the Sample Standard Deviation ([tex]$s$[/tex]):
The standard deviation measures the amount of variation or dispersion of the sample values.
[tex]\[
s = 0.375 \quad (\text{rounded to three decimal places})
\][/tex]
4. Determine the Sample Size ([tex]$n$[/tex]):
The sample size is the number of observations in our dataset.
[tex]\[
n = 7
\][/tex]
5. Find the t-Critical Value ([tex]$t^*$[/tex]):
For a 95% confidence level, with degrees of freedom ([tex]$df = n - 1 = 6$[/tex]), the t-critical value can be found using a t-distribution table or calculator.
[tex]\[
t^* = 2.447 \quad (\text{rounded to three decimal places})
\][/tex]
6. Calculate the Margin of Error (E):
The margin of error is given by the formula:
[tex]\[
E = t^* \left(\frac{s}{\sqrt{n}}\right) = 2.447 \left(\frac{0.375}{\sqrt{7}}\right) = 0.347 \quad (\text{rounded to three decimal places})
\][/tex]
7. Construct the Confidence Interval:
The 95% confidence interval for the population mean [tex]$\mu$[/tex] is:
[tex]\[
\bar{x} - E < \mu < \bar{x} + E
\][/tex]
Substituting the values we have:
[tex]\[
0.739 - 0.347 < \mu < 0.739 + 0.347
\][/tex]
Simplifying this, we get:
[tex]\[
0.392 < \mu < 1.085 \quad (\text{rounded to three decimal places})
\][/tex]
Therefore, the 95% confidence interval estimate for the population mean amount of mercury in tuna sushi is [tex]\( 0.392 \)[/tex] ppm to [tex]\( 1.085 \)[/tex] ppm.
To determine if there is too much mercury, we compare this interval with the food safety guideline of 1 ppm. Since the upper limit of our confidence interval (1.085 ppm) is above the safety guideline of 1 ppm, it appears that there is a potential for too much mercury in the tuna sushi sampled, as the mean amount of mercury could exceed the safe limit.
### Confidence Interval:
[tex]\[
0.392 \, \text{ppm} < \mu < 1.085 \, \text{ppm}
\][/tex]
