The average commute to work (one way) is 25 minutes.  If we assume that commuting times are normally distributed and that the standard deviation is 6.1 minutes, what is the probability that a randomly selected commuter spends more than 30 minutes commuting one way? (round to 4 decimal places)

To find the probability that a randomly selected commuter spends more than 30 minutes commuting, we need to standardize the data using the **Z-score** formula. The Z-score helps to convert a raw value (in this case, 30 minutes) into a standard normal distribution.

The Z-score formula is:

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

Where:
- \( X = 30 \) (the value we're interested in),
- \( \mu = 25 \) (the mean of the commute time),
- \( \sigma = 6.1 \) (the standard deviation).

### Step 1: Calculate the Z-score

\[
Z = \frac{30 - 25}{6.1} = \frac{5}{6.1} \approx 0.8197
\]

### Step 2: Find the corresponding probability

Now that we have the Z-score of approximately \( 0.8197 \), we need to find the probability that \( Z \) is greater than this value. Using standard normal distribution tables (or a calculator), we find the cumulative probability for \( Z = 0.8197 \).

The cumulative probability \( P(Z < 0.8197) \approx 0.7939 \).

Since we are interested in the probability of spending **more than** 30 minutes commuting (i.e., \( P(Z > 0.8197) \)):

\[
P(Z > 0.8197) = 1 - 0.7939 = 0.2061
\]

### Final Answer:

The probability that a randomly selected commuter spends more than 30 minutes commuting one way is approximately **0.2061** (rounded to four decimal places).

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Would you like further details or have any questions? Here are some related questions for you to explore:

1. What is the probability that a commuter spends less than 20 minutes commuting one way?
2. How long do the top 5% of commuters spend commuting one way?
3. If a commuter spends 40 minutes commuting, what is their Z-score?
4. What is the probability that a commuter spends between 20 and 30 minutes commuting?
5. How does increasing the standard deviation affect the probability of commuting longer than 30 minutes?

**Tip**: Z-scores are a powerful way to compare values from different normal distributions!

The average commute to work (one way) is 25 minutes. If we assume that commuting times are normally distributed and that the standard deviation is 6.1 minutes, what is the probability that a randomly selected commuter spends more than 30 minutes commuting one way? (round to 4 decimal places)

Find out the probability that a randomly selected commuter spends more than 30 minutes commuting, assuming the average commute is 25 minutes with a standard deviation of 6.1 minutes. This problem covers the use of the Z-score formula and normal distribution properties to calculate probabilities, perfect for Grades 10-12 students.

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