Answered

IDNLearn.com offers a seamless experience for finding and sharing knowledge. Find the information you need quickly and easily with our comprehensive and accurate Q&A platform.

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.)

Sagot :

GinnyAnsweridn
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]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Find reliable answers at IDNLearn.com. Thanks for stopping by, and come back for more trustworthy solutions.