5.22) A survey of employees at a large company found the following relative frequencies for the one-way distances they had to travel to arrive at work:
Number of Miles (One-way)
A B C D E F
¡Ü5 6-10 11-15 16-20 21-30 ¡Ý31
Relative Frequency 0.38 0.25 0.16 0.09 0.07 0.05
a. What is the probability that a randomly selected individual will have to travel 11 or more miles to work?
b. What is the probability that a randomly selected individual will have to travel between 6 and 15 miles to work?
c. Draw a Venn diagram that includes the relative frequency probabilities in the table.
d. Using the letter identifications provided, calculate the following probabilities: P(A or B or E); P(A or F); P(A¡¯ or B); or C¡¯).
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To answer these questions, we need to understand the relative frequencies and use them to calculate probabilities.
a. To find the probability that a randomly selected individual will have to travel 11 or more miles to work, we need to sum up the relative frequencies for distances 11-15, 16-20, 21-30, and ≥31. Adding up the relative frequencies for these categories, we get:
0.16 + 0.09 + 0.07 + 0.05 = 0.37
Therefore, the probability of a randomly selected individual having to travel 11 or more miles to work is 0.37 or 37%.
b. To find the probability that a randomly selected individual will have to travel between 6 and 15 miles to work, we need to sum up the relative frequencies for distances 6-10 and 11-15. Adding up the relative frequencies for these categories, we get:
0.25 + 0.16 = 0.41
Therefore, the probability of a randomly selected individual having to travel between 6 and 15 miles to work is 0.41 or 41%.
c. To draw a Venn diagram, we need to represent the different distances and their corresponding relative frequencies in the diagram. We can divide the diagram into circles labeled A, B, C, D, E, and F, representing the different distance ranges. Then, we can place the relative frequencies inside each circle. A Venn diagram helps visualize the overlap and relationships between different categories.
d. To calculate the probabilities using the letter identifications provided, we can use the principle of addition and subtract the overlapping probabilities.
P(A or B or E): The probability of selecting A or B or E can be calculated by adding their respective relative frequencies:
0.38 + 0.25 + 0.07 = 0.70 or 70%
P(A or F): The probability of selecting A or F can be calculated by adding their respective relative frequencies:
0.38 + 0.05 = 0.43 or 43%
P(A' or B): The probability of selecting A' (not A) or B can be calculated by subtracting the probability of A from 1 and then adding the probability of B:
1 - 0.38 + 0.25 = 0.87 or 87%
P(C'): The probability of not selecting C can be calculated by subtracting the probability of C from 1:
1 - 0.16 = 0.84 or 84%
I hope this helps! Let me know if you have any further questions.