Answer:
The confidence interval estimate of the population mean is :
(0.61 ppm, 0.90 ppm)
The correct option is (A).
Step-by-step explanation:
The amounts of mercury (ppm) found in tuna sushi sampled at different stores in a major city are:
S = {0.58, 0.82, 0.10, 0.98, 1.27, 0.56, 0.96}
A
confidence interval for the population mean (μ) is an interval estimate of the true value of the mean. This interval has a
probability of consisting the true value of mean.
⇒ Since the population standard deviation is not provided we will use the <em>t</em>-distribution to construct the 99% confidence interval for mean.
⇒ The formula for confidence interval for the population mean is:
![\bar x\pm t_{\alpha/2,(n-1)}\times \frac{s}{\sqrt{n}}](https://tex.z-dn.net/?f=%5Cbar%20x%5Cpm%20t_%7B%5Calpha%2F2%2C%28n-1%29%7D%5Ctimes%20%5Cfrac%7Bs%7D%7B%5Csqrt%7Bn%7D%7D)
Here,
= sample mean
<em>s </em>= sample standard deviation
<em>n</em> = sample size
= critical value.
The degrees of freedom for the critical value is, (<em>n</em> - 1) = 7 - 1 = 6.
The significance level is: ![\alpha =1-Confidence\ level=1-0.99=0.01](https://tex.z-dn.net/?f=%5Calpha%20%3D1-Confidence%5C%20level%3D1-0.99%3D0.01)
The critical value is:
**Use the <em>t</em>-table for the critical value.
Compute the sample mean and sample standard deviation as follows:
![\bar x=\frac{1}{7}(0.58+ 0.82+ 0.10+ 0.98+ 1.27+ 0.56+ 0.96) =0.753](https://tex.z-dn.net/?f=%5Cbar%20x%3D%5Cfrac%7B1%7D%7B7%7D%280.58%2B%200.82%2B%200.10%2B%200.98%2B%201.27%2B%200.56%2B%200.96%29%20%3D0.753)
![\int\limits^a_b {x} \, dx s=\sqrt{\frac{\sum (x{i}-\bar x)^{2}}{n-1} } =\sqrt{\frac{1}{6} \times 0.859743} =0.379](https://tex.z-dn.net/?f=%5Cint%5Climits%5Ea_b%20%7Bx%7D%20%5C%2C%20dx%20s%3D%5Csqrt%7B%5Cfrac%7B%5Csum%20%28x%7Bi%7D-%5Cbar%20x%29%5E%7B2%7D%7D%7Bn-1%7D%20%7D%20%3D%5Csqrt%7B%5Cfrac%7B1%7D%7B6%7D%20%5Ctimes%200.859743%7D%20%3D0.379)
The 99% confidence interval for μ is:
![x^{2} CI=0.753\pm 3.143\times\frac{0.379}{\sqrt{7}} \\=0.753\pm0.143\\=(0.61, 0.896)\\\approx(0.61, 0.90)](https://tex.z-dn.net/?f=x%5E%7B2%7D%20CI%3D0.753%5Cpm%203.143%5Ctimes%5Cfrac%7B0.379%7D%7B%5Csqrt%7B7%7D%7D%20%5C%5C%3D0.753%5Cpm0.143%5C%5C%3D%280.61%2C%200.896%29%5C%5C%5Capprox%280.61%2C%200.90%29)
The confidence interval estimate of the population mean is:
(0.61 ppm, 0.90 ppm)
The upper and lower limit of the 99% confidence interval indicates that the true mean value is less than 1 ppm. This implies that there is not too much mercury in tuna sushi
Because it is possible that the mean is not greater than 1 ppm. Also, at least one of the sample values is less than 1 ppm, so at least some of the fish are safe.
Thus, the correct option is (A).