Trucks in a delivery fleet travel a mean of 110
 miles per day with a standard deviation of 35
 miles per day. The mileage per day is distributed normally. Find the probability that a truck drives between 61
 and 169
 miles in a day. Round your answer to four decimal places.

To find the probability that a truck drives between 61 and 169 miles in a day, we need to standardize these values and use the properties of the standard normal distribution.

We are given:
- Mean \( \mu = 110 \) miles
- Standard deviation \( \sigma = 35 \) miles
- We need to find \( P(61 \leq X \leq 169) \), where \( X \) is the daily mileage.

The formula to standardize a value \( X \) to a \( Z \)-score is:

\[
Z = \frac{X - \mu}{\sigma}
\]

### Step 1: Convert the lower bound (61 miles) to a Z-score
\[
Z_1 = \frac{61 - 110}{35} = \frac{-49}{35} \approx -1.4
\]

### Step 2: Convert the upper bound (169 miles) to a Z-score
\[
Z_2 = \frac{169 - 110}{35} = \frac{59}{35} \approx 1.686
\]

### Step 3: Find the cumulative probabilities
Using the standard normal distribution table or a calculator:
- The probability corresponding to \( Z_1 = -1.4 \) is approximately \( 0.0808 \).
- The probability corresponding to \( Z_2 = 1.686 \) is approximately \( 0.9544 \).

### Step 4: Calculate the probability between 61 and 169 miles
The desired probability is the difference between these two cumulative probabilities:

\[
P(61 \leq X \leq 169) = P(Z_2) - P(Z_1) = 0.9544 - 0.0808 = 0.8736
\]

Thus, the probability that a truck drives between 61 and 169 miles in a day is approximately **0.8736** or **87.36%**.

Would you like more details or have any questions?

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Here are 5 related questions to further your understanding:
1. How would the probability change if the standard deviation were smaller?
2. What is the probability that a truck drives more than 169 miles in a day?
3. How do you interpret a negative Z-score in terms of the normal distribution?
4. What percentage of trucks drive less than 61 miles in a day?
5. How do we calculate the probability if the mileage distribution were not normal?

**Tip:** When dealing with normally distributed data, always check if you need to standardize values using Z-scores for accurate probability calculations.

Trucks in a delivery fleet travel a mean of 110 miles per day with a standard deviation of 35 miles per day. The mileage per day is distributed normally. Find the probability that a truck drives between 61 and 169 miles in a day. Round your answer to four decimal places.

Discover how to find the probability that a delivery truck drives between 61 and 169 miles in a day, given a normal distribution with a mean of 110 miles and a standard deviation of 35 miles. This detailed solution covers the concepts of probability, normal distribution, and Z-scores. Ideal for high school students in Grades 10-12.

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