Question 1182183
Here's how to solve this problem:

**a) Probability of time before charging being greater than 127 hours:**

1. **Calculate the z-score:**
   z = (x - μ) / σ
   z = (127 - 100) / 15
   z = 1.8

2. **Find the probability:** Use a standard normal distribution table (z-table) or a calculator to find the probability associated with a z-score of 1.8.  We want the probability of the time being *greater* than 127 hours, so we look for the area to the *right* of z = 1.8.
   P(x > 127) = P(z > 1.8) = 1 - P(z < 1.8) ≈ 1 - 0.9641 ≈ 0.0359

**b) Find the 10th percentile:**

1. **Find the z-score:** The 10th percentile is the value below which 10% of the data falls. Look up 0.10 in the *body* of the z-table (or use a calculator) to find the corresponding z-score. The z-score corresponding to 0.10 is approximately -1.28.

2. **Convert the z-score to hours:**
   x = μ + zσ
   x = 100 + (-1.28 * 15)
   x ≈ 80.8 hours

**c) Probability of not needing charging during a 6-hour trip, given it was charged 127 hours ago:**

This is a conditional probability problem.  We know the phone lasted 127 hours, and we want to know the probability it will last *at least* an additional 6 hours (127 + 6 = 133 hours).

1. **Calculate the conditional probability:** We want P(x > 133 | x > 127). This can be written as:

   P(x > 133 and x > 127) / P(x > 127)

   Since if x>133, it is automatically true that x>127, then P(x > 133 and x > 127) simplifies to P(x>133)

   So we have P(x > 133) / P(x > 127)

2. **Calculate P(x > 133):**
   z = (133 - 100) / 15
   z = 2.2
   P(x > 133) = P(z > 2.2) = 1 - P(z < 2.2) ≈ 1 - 0.9861 ≈ 0.0139

3. **Calculate the conditional probability:**
   P(x > 133 | x > 127) = P(x > 133) / P(x > 127) ≈ 0.0139 / 0.0359 ≈ 0.387

**Answers:**

*   a) The probability that the time before charging is greater than 127 hours is approximately 0.0359 or 3.59%.
*   b) The 10th percentile is approximately 80.8 hours.
*   c) The probability that the mobile will not need charging during the 6-hour trip, given it was charged 127 hours ago, is approximately 0.387 or 38.7%.