Answer:
a) Probability that exactly 1 fastener is defective, P(X = 1) = 0.144
b) Confidence interval for mean price, ![CI = [4.1058, 4.1202]](https://tex.z-dn.net/?f=CI%20%3D%20%5B4.1058%2C%204.1202%5D)
Step-by-step explanation:
a) Total number of fasteners = 120
Number of defective fasteners = 4
Probability of selecting a defective fastener, p = 4/120
p = 0.033
Probability of selecting an undefective fastener, q = 1 - p
q = 1 - 0.033
q = 0.967
5 fasteners were randomly selected, n =5
Probability that exactly one fastener is defective:
![P(X =r) = (nCr) p^r q^{n-r}\\P(X =1) = (5C1) 0.033^1 0.967^{5-1}\\P(X =1) = 0.144](https://tex.z-dn.net/?f=P%28X%20%3Dr%29%20%3D%20%28nCr%29%20p%5Er%20q%5E%7Bn-r%7D%5C%5CP%28X%20%3D1%29%20%3D%20%285C1%29%200.033%5E1%200.967%5E%7B5-1%7D%5C%5CP%28X%20%3D1%29%20%3D%200.144)
b) Number of gasoline outlets sampled, n = 900
Average gasoline price, ![\bar{x} = 4.113](https://tex.z-dn.net/?f=%5Cbar%7Bx%7D%20%3D%204.113)
Standard deviation, ![\sigma = 0.11](https://tex.z-dn.net/?f=%5Csigma%20%3D%200.11)
Confidence Level, CL = 95% = 0.95
Significance level, ![\alpha = 1 - 0.95 = 0.05](https://tex.z-dn.net/?f=%5Calpha%20%3D%201%20-%200.95%20%3D%200.05)
![\alpha/2 = 0.05/2 = 0.025](https://tex.z-dn.net/?f=%5Calpha%2F2%20%3D%200.05%2F2%20%3D%200.025)
From the standard normal table, ![z_{\alpha/2} = z_{0.025} = 1.96](https://tex.z-dn.net/?f=z_%7B%5Calpha%2F2%7D%20%3D%20z_%7B0.025%7D%20%3D%201.96)
error margin can be calculated as follows:
![e_{margin} = z_{\alpha/2} * \frac{\sigma}{\sqrt{n} } \\e_{margin} = 1.96 * \frac{0.11}{\sqrt{900} }\\e_{margin} = 0.0072](https://tex.z-dn.net/?f=e_%7Bmargin%7D%20%3D%20z_%7B%5Calpha%2F2%7D%20%2A%20%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%20%7D%20%5C%5Ce_%7Bmargin%7D%20%3D%201.96%20%2A%20%5Cfrac%7B0.11%7D%7B%5Csqrt%7B900%7D%20%7D%5C%5Ce_%7Bmargin%7D%20%3D%200.0072)
The confidence interval will be given as:
![CI = \bar{x} \pm e_{margin} \\CI = 4.113 \pm 0.0072\\CI = [(4.113-0.0072), (4.113+0.0072)]\\CI = [4.1058, 4.1202]](https://tex.z-dn.net/?f=CI%20%3D%20%5Cbar%7Bx%7D%20%5Cpm%20e_%7Bmargin%7D%20%20%5C%5CCI%20%3D%204.113%20%5Cpm%200.0072%5C%5CCI%20%3D%20%5B%284.113-0.0072%29%2C%20%284.113%2B0.0072%29%5D%5C%5CCI%20%3D%20%5B4.1058%2C%204.1202%5D)