Let X, X2, and X, represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal rv's with expected values u. lg, and ug and variances o,, o,, and o, respectively. (Round your answers to four decimal places.) A USE SALT

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Please solve (b), (c), and (d) only! Thanks!

 

Let \( X_1, X_2, \) and \( X_3 \) represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal random variables with expected values \(\mu_1, \mu_2,\) and \(\mu_3\) and variances \(\sigma_1^2, \sigma_2^2,\) and \(\sigma_3^2\) respectively. (Round your answers to four decimal places.)

---

### (a)
If \(\mu_1 = \mu_2 = \mu_3 = 70\) and \(\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = 18\), calculate \( P(T_0 \le 228) \) and \( P(174 \le T_0 \le 228) \).

- \( P(T_0 \le 228) = \) [Input box]
- \( P(174 \le T_0 \le 228) = \) [Input box]

---

### (b)
Using the \(\mu_i\) s and \(\sigma_i\) s given in part (a), calculate both \( P(64 \le \bar{X} \) and \( P(68 \le \bar{X} \le 72) \).

- \( P(64 \le \bar{X}) = \) [Input box]
- \( P(68 \le \bar{X} \le 72) = \) [Input box]

---

### (c)
Using the \(\mu_i\) s and \(\sigma_i\) s given in part (a), calculate \( P(-12 \le X_1 - 0.5X_2 - 0.5X_3 \le 6) \).

\[ P(-12 \le X_1 - 0.5X_2 - 0.5X_3 \le 6) = \] [Input box]

#### Interpret the quantity \( X_1 - 0.5X_2 - 0.5X_3 \).

- [ ] The quantity represents the probability that the difference between \( X_3 \) and the sum of \( X_1 \) and \( X_2 \) is between \(-12\)
Transcribed Image Text:Let \( X_1, X_2, \) and \( X_3 \) represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal random variables with expected values \(\mu_1, \mu_2,\) and \(\mu_3\) and variances \(\sigma_1^2, \sigma_2^2,\) and \(\sigma_3^2\) respectively. (Round your answers to four decimal places.) --- ### (a) If \(\mu_1 = \mu_2 = \mu_3 = 70\) and \(\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = 18\), calculate \( P(T_0 \le 228) \) and \( P(174 \le T_0 \le 228) \). - \( P(T_0 \le 228) = \) [Input box] - \( P(174 \le T_0 \le 228) = \) [Input box] --- ### (b) Using the \(\mu_i\) s and \(\sigma_i\) s given in part (a), calculate both \( P(64 \le \bar{X} \) and \( P(68 \le \bar{X} \le 72) \). - \( P(64 \le \bar{X}) = \) [Input box] - \( P(68 \le \bar{X} \le 72) = \) [Input box] --- ### (c) Using the \(\mu_i\) s and \(\sigma_i\) s given in part (a), calculate \( P(-12 \le X_1 - 0.5X_2 - 0.5X_3 \le 6) \). \[ P(-12 \le X_1 - 0.5X_2 - 0.5X_3 \le 6) = \] [Input box] #### Interpret the quantity \( X_1 - 0.5X_2 - 0.5X_3 \). - [ ] The quantity represents the probability that the difference between \( X_3 \) and the sum of \( X_1 \) and \( X_2 \) is between \(-12\)
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