Answer: The mean is 556.4 ppm.
The standard deviation is 84.92 ppm.
Explanation:
Let X denotes the concentration of carbon (ppm) .
Given :
X p(x)
350 0.08
440 0.11
560 0.48
640 0.33
The mean(expected value) is given by :-
![E[x]=\sum x p(x)](https://img.qammunity.org/2020/formulas/mathematics/college/85ci95oipoupi2nanylsaeoutp1e0vnzoy.png)
![\Rightarrow\ E[x]=(350)(0.08)+(440)(0.11)+(560)(0.48)+(640)(0.33)=556.4](https://img.qammunity.org/2020/formulas/mathematics/college/gumademcsddj6bz39olkfi1bt7fd98chor.png)
Hence, the mean is 556.4 ppm.
Now,
![E[x^2]=\sum x^2 p(x)](https://img.qammunity.org/2020/formulas/mathematics/college/v5abmk42ea59pnoehwwet0p0psx4g6302y.png)
![\Rightarrow\ E[x]=(350)^2(0.08)+(440)^2(0.11)+(560)^2(0.48)+(640)^2(0.33)=316792](https://img.qammunity.org/2020/formulas/mathematics/college/b54ocoiy5xoq8r1ai54967s6iv86nissyn.png)
![\text{Var(x)}=E[x^2]-[E[x]]^2\\\\=316792-556.4^2\\\\=316792-309580.96=7211.04](https://img.qammunity.org/2020/formulas/mathematics/college/vacypapju7xr18b668w6mfv9ni6uk18ka9.png)
Standard deviation:

Hence, The standard deviation is 84.92 ppm.