– Make an estimate of the 95% confidence interval for the average mercury content of the population. Does tuna sushi seem to have too much mercury?
Figure 1
– What is the confidence interval estimate of the population mean?
The question aims to find confidence interval estimates given sample mean and percentage confidence interval. The confidence interval estimate (CI) is a range of values for the population parameters based on the sample mean and percentage.
Expert Answer
We need sample mean and standard deviation to find confidence intervals for the population.
Step 1: Calculate sample mean and standard deviation:
Figure 2
\[ \text{Total samples},\ n = 7 \]
\[ \sum x = 4.34\]
The sample mean is calculated as follows:
\[\bar x = \dfrac{\sum x}{n} = \dfrac{4.34}{7}=0.62\]
Figure 3
Now, we will find the standard deviation by using the formula:
\[S.D=\sqrt {\dfrac{\sum (x-\bar x)^2}{n-1}} \]
\[S.D=\sqrt{\dfrac{1.1716}{7-1}}=0.4419\]
The standard deviation is $0.4419$.
Step 2: The confidence level is given as $95\%$.
Significance level is calculated as:
\[\sigma=(100-95)\% =0.05\]
We can find the degree of freedom as follows:
\[d.f = n-1=7-1=6\]
The critical value is given as:
\[ t = 2.44469 \]
The standard error is calculated as:
\[S.E=\dfrac{S.D}{\sqrt n}=\dfrac{0.4419}{\sqrt 7}=0.167\]
The margin of error can be found as:
\[M.E=t\ast S.E = 0.40868\]
Lower and Upper limit are calculated as:
\[L.L=(\bar x-M.E)=0.62-0.40868\]
\[L.L=0.211\]
\[U.L=(\bar x+M.E)=0.62+0.40868\]
\[U.L=1.02868\]
Numerical Result
The sample mean is given as:
\[\bar x=0.62\]
Standard deviation is given as:
\[S.D = 0.4419\]
Lower limit for the confidence interval is $L.L = 0.211$.
Upper limit for the confidence interval is $U.L = 1.02868$.
The $95\%$ confidence interval is $(0.211, 1.02868)$.
The upper limit of the confidence interval is greater than $1 ppm$ and the mercury must be less than $1 ppm$. That’s why there is too much mercury in tuna sushi.
Example
Food safety guidelines stipulate that fish mercury must be less than one part per million (ppm). Below is the amount of mercury (ppm) in tuna sushi tasted at various stores in major cities. Make an estimate of the $95\%$ confidence interval for the average mercury content of the population. Does it seem like there is too much mercury in tuna sushi?
Figure 4
The total number of samples is $7$.
The sample mean for seven samples is calculated as:
\[\bar x=0.714\]
Standard deviation is calculated as:
\[S.D=0.3737\]
The confidence level is given as $95\%$.
After calculating standard error and margin of error, lower and upper limits are calculated as:
\[L.L=(\bar x-margin\:of \:error)=0.3687\]
\[U.L=(\bar x+margin\: of \:error)=1.0599\]