Let's address the problem step by step.
### Problem Restatement:
We need to find the probability that a commuter train, which has a mean arrival time of 8:47 a.m. and a standard deviation of 4 minutes, arrives between 8:35 a.m. and 8:59 a.m.
### Step 1: Convert Times to Minutes
First, let's convert the given times into minutes past midnight for easier calculations.
- Mean arrival time: 8:47 a.m.
- Lower bound: 8:35 a.m.
- Upper bound: 8:59 a.m.
Calculating in minutes past midnight:
- 8:47 a.m. is [tex]\(8 \times 60 + 47 = 480 + 47 = 527\)[/tex] minutes.
- 8:35 a.m. is [tex]\(8 \times 60 + 35 = 480 + 35 = 515\)[/tex] minutes.
- 8:59 a.m. is [tex]\(8 \times 60 + 59 = 480 + 59 = 539\)[/tex] minutes.
### Step 2: Calculate the Z-Scores
Next, we'll calculate the z-scores for the lower and upper bounds using the mean and the standard deviation.
The formula for z-score is:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
Where:
- [tex]\( X \)[/tex] is the time in minutes.
- [tex]\( \mu \)[/tex] is the mean time.
- [tex]\( \sigma \)[/tex] is the standard deviation.
For the lower bound (515 minutes):
[tex]\[ z_{\text{lower}} = \frac{515 - 527}{4} = \frac{-12}{4} = -3.0 \][/tex]
For the upper bound (539 minutes):
[tex]\[ z_{\text{upper}} = \frac{539 - 527}{4} = \frac{12}{4} = 3.0 \][/tex]
### Step 3: Use the Empirical Rule
The empirical rule (or the 68-95-99.7 rule) states that for a normal distribution:
- About 68% of values lie within 1 standard deviation of the mean.
- About 95% of values lie within 2 standard deviations of the mean.
- About 99.7% of values lie within 3 standard deviations of the mean.
### Step 4: Determine Probability Between Z-Scores ±3
From the empirical rule, we know that the probability of a value falling within 3 standard deviations of the mean is approximately 99.7%. This corresponds to the probability that the train arrives between 8:35 a.m. and 8:59 a.m.
Hence, the probability that, on a randomly chosen day, the train arrives between 8:35 a.m. and 8:59 a.m. is:
[tex]\[ \text{Probability} \approx 99.7\% = 0.997 \][/tex]
Therefore, the probability is 0.997 (or 99.7%).
