Answered

Thirty-two people were chosen at random from employees of a large company. Their commute times (in hours) were recorded in a table (shown below).

\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
0.5 & 0.9 & 0.5 & 0.4 & 0.7 & 1.3 & 1.0 & 0.7 \\
\hline
0.6 & 0.4 & 0.8 & 1.0 & 0.9 & 1.4 & 0.4 & 1.1 \\
\hline
0.9 & 1.1 & 0.7 & 0.2 & 0.6 & 1.0 & 0.7 & 1.2 \\
\hline
0.4 & 1.2 & 0.7 & 0.6 & 1.3 & 0.8 & 0.4 & 0.9 \\
\hline
\end{tabular}

1. What is the probability that a person chosen at random from the sample will have a commuting time of at least an hour? (Let [tex]n[/tex] represent the commuting time.)
[tex]\[P(n \geq 1) = \frac{5}{16}\][/tex] (Type a simplified fraction.)

2. What is the probability that a person chosen at random from the sample will have a commuting time of at most half an hour? (Let [tex]n[/tex] represent the commuting time.)
[tex]\[P(n \leq 0.5) = \square \][/tex] (Type a simplified fraction.)



Answer :

GinnyAnswer
Certainly! Let's break down the problem and solve it step by step:

### Part 1: Probability of Commuting Time of at least an Hour

We are given a table with the commute times of 32 employees. We need to find the probability of a person having a commute time of at least one hour (i.e., [tex]\( n \geq 1 \)[/tex]).

1. Count the number of employees with commute times of at least one hour:
- The commute times are: 1.3, 1.0, 1.0, 1.4, 1.1, 1.1, 1.0, 1.2, 1.2, 1.3
- There are 10 such commute times.

2. Calculate the probability:
- Since there are 32 total employees, the probability is given by the ratio of the number of favorable outcomes to the total number of outcomes.
- Thus, [tex]\( P(n \geq 1) = \frac{\text{Number of employees with } n \geq 1}{\text{Total number of employees}} = \frac{10}{32} \)[/tex].
- Simplify the fraction: [tex]\( \frac{10}{32} = \frac{5}{16} \)[/tex].

Therefore, the probability that a person chosen at random will have a commuting time of at least an hour is [tex]\( \frac{5}{16} \)[/tex].

### Part 2: Probability of Commuting Time of at most Half an Hour

Next, we need to find the probability of a person having a commute time of at most half an hour (i.e., [tex]\( n \leq 0.5 \)[/tex]).

1. Count the number of employees with commute times of at most half an hour:
- The commute times are: 0.5, 0.5, 0.4, 0.4, 0.4, 0.2, 0.4
- There are 8 such commute times.

2. Calculate the probability:
- Since there are 32 total employees, the probability is given by the ratio of the number of favorable outcomes to the total number of outcomes.
- Thus, [tex]\( P(n \leq 0.5) = \frac{\text{Number of employees with } n \leq 0.5}{\text{Total number of employees}} = \frac{8}{32} \)[/tex].
- Simplify the fraction: [tex]\( \frac{8}{32} = \frac{1}{4} \)[/tex].

Therefore, the probability that a person chosen at random will have a commuting time of at most half an hour is [tex]\( \frac{1}{4} \)[/tex].

### Final Answers:
- [tex]\( P(n \geq 1) = \frac{5}{16} \)[/tex]
- [tex]\( P(n \leq 0.5) = \frac{1}{4} \)[/tex]