Answer:
a. the control limits should be set at (10.72, 11.28)
b. ![\mathbf{P(10.72](https://tex.z-dn.net/?f=%5Cmathbf%7BP%2810.72%3Cx%3C11.28%29%20%3D%200.4526%7D)
c. ![\mathbf{P(10.72](https://tex.z-dn.net/?f=%5Cmathbf%7BP%2810.72%3Cx%3C11.28%29%20%3D%200.0065%7D)
Step-by-step explanation:
Given that:
population mean μ = 11
standard deviation
= 1.0
sample size n = 35
5% of the sample means will be greater than the upper control limit, and 5% of the sample means will be less than the lower control limit.
Therefore, level of significance ∝ = 0.05+0.05 = 0.10
Critical value for ![z_{1-\alpha/2} =z_{1-0.10 /2}](https://tex.z-dn.net/?f=z_%7B1-%5Calpha%2F2%7D%20%3Dz_%7B1-0.10%20%2F2%7D)
![\implies z_{1-0.05} = z_{0.95}](https://tex.z-dn.net/?f=%5Cimplies%20z_%7B1-0.05%7D%20%20%3D%20z_%7B0.95%7D)
Using the EXCEL FORMULA: = NORMSINV (0.95)
z = 1.64
The lower control limit and the upper control limit can be determined by using the respective formulas:
Lower control limit = ![\mathtt{\mu - z_{1-\alpha/2} \times \dfrac{\sigma}{\sqrt{n}}}](https://tex.z-dn.net/?f=%5Cmathtt%7B%5Cmu%20-%20z_%7B1-%5Calpha%2F2%7D%20%5Ctimes%20%5Cdfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D)
Upper control limit = ![\mathtt{\mu + z_{1-\alpha/2} \times \dfrac{\sigma}{\sqrt{n}}}](https://tex.z-dn.net/?f=%5Cmathtt%7B%5Cmu%20%2B%20z_%7B1-%5Calpha%2F2%7D%20%5Ctimes%20%5Cdfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D)
For the lower control limit = ![11-1.64 \times \dfrac{1.0}{\sqrt{35}}](https://tex.z-dn.net/?f=11-1.64%20%5Ctimes%20%5Cdfrac%7B1.0%7D%7B%5Csqrt%7B35%7D%7D)
For the lower control limit = ![11-0.27721](https://tex.z-dn.net/?f=11-0.27721)
For the lower control limit = 10.72279
For the lower control limit
10.72
For the upper control limit = ![11+1.64 \times \dfrac{1.0}{\sqrt{35}}](https://tex.z-dn.net/?f=11%2B1.64%20%5Ctimes%20%5Cdfrac%7B1.0%7D%7B%5Csqrt%7B35%7D%7D)
For the upper control limit = 11 + 0.27721
For the upper control limit = 11.27721
For the upper control limit
11.28
Therefore , the control limits should be set at (10.72, 11.28)
b. If the population mean shifts to 10.7, what is the probability that the change will be detected?
i.e
![\mathtt{P(10.72](https://tex.z-dn.net/?f=%5Cmathtt%7BP%2810.72%3Cx%3C11.28%29%20%3D%20P%28%20%5Cdfrac%7BX-%20%5Cmu%7D%7B%5Cdfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D%3Cz%20%3C%20%5Cdfrac%7BX-%20%5Cmu%7D%7B%5Cdfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D%7D%29)
![\mathtt{P(10.72](https://tex.z-dn.net/?f=%5Cmathtt%7BP%2810.72%3Cx%3C11.28%29%20%3D%20P%28%20%5Cdfrac%7B10.72-%2010.7%7D%7B%5Cdfrac%7B1.0%7D%7B%5Csqrt%7B35%7D%7D%7D%3Cz%20%3C%20%5Cdfrac%7B11.28-%2010.7%7D%7B%5Cdfrac%7B1.0%7D%7B%5Csqrt%7B35%7D%7D%7D%7D%29)
![\mathtt{P(10.72](https://tex.z-dn.net/?f=%5Cmathtt%7BP%2810.72%3Cx%3C11.28%29%20%3D%20P%28%20%5Cdfrac%7B0.02%7D%7B%5Cdfrac%7B1.0%7D%7B5.916%7D%7D%3Cz%20%3C%20%5Cdfrac%7B0.58%7D%7B%5Cdfrac%7B1.0%7D%7B5.916%7D%7D%7D%29)
![\mathtt{P(10.72](https://tex.z-dn.net/?f=%5Cmathtt%7BP%2810.72%3Cx%3C11.28%29%20%3D%20P%280.1183%3Cz%20%3C%203.4313%7D%29)
![\mathtt{P(10.72](https://tex.z-dn.net/?f=%5Cmathtt%7BP%2810.72%3Cx%3C11.28%29%20%3D%20P%28z%3C%203.4313%29%20-%20P%28z%3C%200.1183%29%20%7D)
Using the EXCEL FORMULA: = NORMSDIST (3.4313) - NORMSDIST (0.118 ); we have:
![\mathbf{P(10.72](https://tex.z-dn.net/?f=%5Cmathbf%7BP%2810.72%3Cx%3C11.28%29%20%3D%200.4526%7D)
c If the population mean shifts to 11.7, what is the probability that the change will be detected?
i.e
![\mathtt{P(10.72](https://tex.z-dn.net/?f=%5Cmathtt%7BP%2810.72%3Cx%3C11.28%29%20%3D%20P%28%20%5Cdfrac%7BX-%20%5Cmu%7D%7B%5Cdfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D%3Cz%20%3C%20%5Cdfrac%7BX-%20%5Cmu%7D%7B%5Cdfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D%7D%29)
![\mathtt{P(10.72](https://tex.z-dn.net/?f=%5Cmathtt%7BP%2810.72%3Cx%3C11.28%29%20%3D%20P%28%20%5Cdfrac%7B10.72-%2011.7%7D%7B%5Cdfrac%7B1.0%7D%7B%5Csqrt%7B35%7D%7D%7D%3Cz%20%3C%20%5Cdfrac%7B11.28-%2011.7%7D%7B%5Cdfrac%7B1.0%7D%7B%5Csqrt%7B35%7D%7D%7D%7D%29)
![\mathtt{P(10.72](https://tex.z-dn.net/?f=%5Cmathtt%7BP%2810.72%3Cx%3C11.28%29%20%3D%20P%28%20%5Cdfrac%7B-0.98%7D%7B%5Cdfrac%7B1.0%7D%7B5.916%7D%7D%3Cz%20%3C%20%5Cdfrac%7B-0.42%7D%7B%5Cdfrac%7B1.0%7D%7B5.916%7D%7D%7D%29)
![\mathtt{P(10.72](https://tex.z-dn.net/?f=%5Cmathtt%7BP%2810.72%3Cx%3C11.28%29%20%3D%20P%28-5.7978%3Cz%20%3C%20-2.48472%7D%29)
![\mathtt{P(10.72](https://tex.z-dn.net/?f=%5Cmathtt%7BP%2810.72%3Cx%3C11.28%29%20%3D%20P%28z%3C%20-2.48472%29%20-%20P%28z%3C%20-5.7978%29%20%7D)
Using the EXCEL FORMULA: = NORMSDIST (-2.48472) - NORMSDIST (-5.7978); we have:
![\mathbf{P(10.72](https://tex.z-dn.net/?f=%5Cmathbf%7BP%2810.72%3Cx%3C11.28%29%20%3D%200.0065%7D)