To determine what percentage of adults take longer than 27 minutes to walk a mile, we will follow these steps:
### Step 1: Understand the Parameters
1. The average (mean) walking time is 22 minutes.
2. The standard deviation of the walking times is 6 minutes.
3. We want to find the probability of an individual taking longer than 27 minutes.
### Step 2: Calculate the Z-Score
First, we need the z-score which standardizes the value. The z-score indicates how many standard deviations a data point (walking time of 27 minutes) is from the mean.
The formula for the z-score [tex]\( z \)[/tex] is:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
where:
- [tex]\( X \)[/tex] is the value we are looking at (27 minutes)
- [tex]\( \mu \)[/tex] is the mean (22 minutes)
- [tex]\( \sigma \)[/tex] is the standard deviation (6 minutes)
Substituting the values in:
[tex]\[ z = \frac{27 - 22}{6} \][/tex]
[tex]\[ z = \frac{5}{6} \][/tex]
[tex]\[ z \approx 0.833 \][/tex]
### Step 3: Use the Z-Score to Find the Cumulative Probability
Using the z-score, we can determine the cumulative probability from the standard normal distribution table.
The z-score we found is 0.833. Referring to the closest values in the provided table:
- For [tex]\( z = 0.83 \)[/tex], it shows the probability to the left of this z-score is approximately 0.7967.
### Step 4: Find the Probability of Taking Longer than 27 Minutes
The cumulative probability provides the proportion of the data to the left of the z-score. To find the percentage of adults who take longer than 27 minutes, we must subtract this probability from 1 (since the total probability is 1).
[tex]\[
P(X > 27) = 1 - P(X \le 27) = 1 - 0.7967 = 0.2033
\][/tex]
### Step 5: Convert the Probability to a Percentage
Finally, convert the probability to a percentage:
[tex]\[
0.2033 \times 100 \approx 20.33\%
\][/tex]
### Conclusion
Approximately [tex]\( 20.33\% \)[/tex] of adults take longer than 27 minutes to walk a mile.
