Answer:
The value of the constant C is 0.01 .
Step-by-step explanation:
Given:
Suppose X, Y, and Z are random variables with the joint density function,
The value of constant C can be obtained as:
![C\int\limits^\infty_0 {e^{-0.5x}(\int\limits^\infty_0{e^{-0.2y}([\frac{-e^{-0.1z} }{0.1} ]\limits^\infty__0 }) \, dy }) \, dx = 1](https://tex.z-dn.net/?f=C%5Cint%5Climits%5E%5Cinfty_0%20%7Be%5E%7B-0.5x%7D%28%5Cint%5Climits%5E%5Cinfty_0%7Be%5E%7B-0.2y%7D%28%5B%5Cfrac%7B-e%5E%7B-0.1z%7D%20%7D%7B0.1%7D%20%5D%5Climits%5E%5Cinfty__0%20%7D%29%20%5C%2C%20dy%20%20%7D%29%20%5C%2C%20dx%20%3D%201)
![C\int\limits^\infty_0 {e^{-0.5x}(\int\limits^\infty_0 {e^{-0.2y}([\frac{-e^{-0.1(\infty)} }{0.1}+\frac{e^{-0.1(0)} }{0.1} ]) } \, dy }) \, dx = 1](https://tex.z-dn.net/?f=C%5Cint%5Climits%5E%5Cinfty_0%20%7Be%5E%7B-0.5x%7D%28%5Cint%5Climits%5E%5Cinfty_0%20%7Be%5E%7B-0.2y%7D%28%5B%5Cfrac%7B-e%5E%7B-0.1%28%5Cinfty%29%7D%20%7D%7B0.1%7D%2B%5Cfrac%7Be%5E%7B-0.1%280%29%7D%20%7D%7B0.1%7D%20%5D%29%20%20%7D%20%5C%2C%20dy%20%20%7D%29%20%5C%2C%20dx%20%3D%201)
![C\int\limits^\infty_0 {e^{-0.5x}(\int\limits^\infty_0 {e^{-0.2y}[0+\frac{1}{0.1}] } \, dy }) \, dx =1](https://tex.z-dn.net/?f=C%5Cint%5Climits%5E%5Cinfty_0%20%7Be%5E%7B-0.5x%7D%28%5Cint%5Climits%5E%5Cinfty_0%20%7Be%5E%7B-0.2y%7D%5B0%2B%5Cfrac%7B1%7D%7B0.1%7D%5D%20%20%7D%20%5C%2C%20dy%20%20%7D%29%20%5C%2C%20dx%20%3D1)
![10C\int\limits^\infty_0 {e^{-0.5x}([\frac{-e^{-0.2y} }{0.2}]^\infty__0 }) \, dx = 1](https://tex.z-dn.net/?f=10C%5Cint%5Climits%5E%5Cinfty_0%20%7Be%5E%7B-0.5x%7D%28%5B%5Cfrac%7B-e%5E%7B-0.2y%7D%20%7D%7B0.2%7D%5D%5E%5Cinfty__0%20%20%7D%29%20%5C%2C%20dx%20%3D%201)
![10C\int\limits^\infty_0 {e^{-0.5x}([\frac{-e^{-0.2(\infty)} }{0.2}+\frac{e^{-0.2(0)} }{0.2}] } \, dx = 1](https://tex.z-dn.net/?f=10C%5Cint%5Climits%5E%5Cinfty_0%20%7Be%5E%7B-0.5x%7D%28%5B%5Cfrac%7B-e%5E%7B-0.2%28%5Cinfty%29%7D%20%7D%7B0.2%7D%2B%5Cfrac%7Be%5E%7B-0.2%280%29%7D%20%7D%7B0.2%7D%5D%20%20%20%7D%20%5C%2C%20dx%20%3D%201)
![10C\int\limits^\infty_0 {e^{-0.5x}[0+\frac{1}{0.2}] } \, dx = 1](https://tex.z-dn.net/?f=10C%5Cint%5Climits%5E%5Cinfty_0%20%7Be%5E%7B-0.5x%7D%5B0%2B%5Cfrac%7B1%7D%7B0.2%7D%5D%20%20%7D%20%5C%2C%20dx%20%3D%201)
![50C([\frac{-e^{-0.5x} }{0.5}]^\infty__0}) = 1](https://tex.z-dn.net/?f=50C%28%5B%5Cfrac%7B-e%5E%7B-0.5x%7D%20%7D%7B0.5%7D%5D%5E%5Cinfty__0%7D%29%20%3D%201)
![50C[\frac{-e^{-0.5(\infty)} }{0.5} + \frac{-0.5(0)}{0.5}] =1](https://tex.z-dn.net/?f=50C%5B%5Cfrac%7B-e%5E%7B-0.5%28%5Cinfty%29%7D%20%7D%7B0.5%7D%20%2B%20%5Cfrac%7B-0.5%280%29%7D%7B0.5%7D%5D%20%3D1)
![50C[0+\frac{1}{0.5} ] =1](https://tex.z-dn.net/?f=50C%5B0%2B%5Cfrac%7B1%7D%7B0.5%7D%20%5D%20%3D1)
⇒ 
C = 0.01
Answer:
(A) Yes, since the test statistic is in the rejection region defined by the critical value, reject the null. The claim is the alternative, so the claim is supported.
Step-by-step explanation:
Null hypothesis: The wait time before a call is answered by a service representative is 3.3 minutes.
Alternate hypothesis: The wait time before a call is answered by a service representative is less than 3.3 minutes.
Test statistic (t) = (sample mean - population mean) ÷ sd/√n
sample mean = 3.24 minutes
population mean = 3.3 minutes
sd = 0.4 minutes
n = 62
degree of freedom = n - 1 = 62 - 1 = 71
significance level = 0.08
t = (3.24 - 3.3) ÷ 0.4/√62 = -0.06 ÷ 005 = -1.2
The test is a one-tailed test. The critical value corresponding to 61 degrees of freedom and 0.08 significance level is 1.654
Conclusion:
Reject the null hypothesis because the test statistic -1.2 is in the rejection region of the critical value 1.654. The claim is contained in the alternative hypothesis, so it is supported.
Answer:
yes
Step-by-step explanation:
