Question 1178779: Suppose that the duration of a particular type of criminal trial is known to be normally distributed with a mean of 15 days and a standard deviation of 5 days. Let X be the number of days for a randomly selected trial. Round all answers to 4 decimal places where possible.
a. What is the distribution of X? X ~ N(
,
)
b. If one of the trials is randomly chosen, find the probability that it lasted at least 16 days.
c. If one of the trials is randomly chosen, find the probability that it lasted between 14 and 19 days.
d. 87% of all of these types of trials are completed within how many days? (Please enter a whole number)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this step-by-step:
**a. What is the distribution of X?**
* X follows a normal distribution with a mean (μ) of 15 days and a standard deviation (σ) of 5 days.
* Therefore, X ~ N(15, 5²) which is X ~ N(15, 25).
**b. If one of the trials is randomly chosen, find the probability that it lasted at least 16 days.**
1. **Calculate the z-score:**
* z = (X - μ) / σ
* z = (16 - 15) / 5
* z = 1 / 5
* z = 0.2
2. **Find the probability:**
* P(X ≥ 16) = P(z ≥ 0.2)
* Using a z-table or calculator:
* P(z < 0.2) ≈ 0.5793
* P(z ≥ 0.2) = 1 - P(z < 0.2) ≈ 1 - 0.5793 ≈ 0.4207
**c. If one of the trials is randomly chosen, find the probability that it lasted between 14 and 19 days.**
1. **Calculate the z-scores:**
* z₁₄ = (14 - 15) / 5 = -1 / 5 = -0.2
* z₁₉ = (19 - 15) / 5 = 4 / 5 = 0.8
2. **Find the probability:**
* P(14 ≤ X ≤ 19) = P(-0.2 ≤ z ≤ 0.8)
* Using a z-table or calculator:
* P(z ≤ 0.8) ≈ 0.7881
* P(z ≤ -0.2) ≈ 0.4207
* P(-0.2 ≤ z ≤ 0.8) = P(z ≤ 0.8) - P(z ≤ -0.2) ≈ 0.7881 - 0.4207 ≈ 0.3674
**d. 87% of all of these types of trials are completed within how many days?**
1. **Find the z-score for the 87th percentile:**
* Using a z-table or calculator, the z-score corresponding to the 87th percentile is approximately 1.1264.
2. **Calculate the value of X:**
* X = μ + z * σ
* X = 15 + 1.1264 * 5
* X = 15 + 5.632
* X = 20.632
* Rounded to the nearest whole number, X = 21.
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