To determine the probability that a random person from the sample has a commute time of at least an hour, we need to follow these steps:
1. Count the Number of People with Commute Times of at Least 1 Hour:
First, let's identify how many people in the sample have commute times that are greater than or equal to 1 hour. By examining the commute times:
- 1.3, 1.0, 1.0, 1.4, 1.1, 1.1, 1.0, 1.2, 1.2, 1.3
From this, we can see that there are 10 instances where the commute time is at least 1 hour.
2. Determine the Total Number of People in the Sample:
According to the problem, there are 32 recorded commute times.
3. Calculate the Probability:
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of outcomes. In this case, the number of favorable outcomes is the number of people with commute times of at least 1 hour, and the total number of outcomes is the total number of people sampled.
So, the probability [tex]\( P(n \geq 1) \)[/tex] is:
[tex]\[
P(n \geq 1) = \frac{\text{Number of people with commute times} \geq 1 \text{ hour}}{\text{Total number of people}}
\][/tex]
Substituting the numbers we found:
[tex]\[
P(n \geq 1) = \frac{10}{32}
\][/tex]
4. Simplify the Fraction:
To simplify [tex]\( \frac{10}{32} \)[/tex], we find the greatest common divisor (GCD) of 10 and 32. The GCD of 10 and 32 is 2. Dividing both the numerator and the denominator by their GCD:
[tex]\[
\frac{10}{32} = \frac{10 \div 2}{32 \div 2} = \frac{5}{16}
\][/tex]
Therefore, the simplified fraction representing the probability that a randomly chosen person will have a commute time of at least 1 hour is:
[tex]\[
P(n \geq 1) = \frac{5}{16}
\][/tex]
