To solve these problems, we need to categorize the 603 fast-food outlets by their population sizes and locations, which we have not been provided directly. Therefore, let’s assume a hypothetical distribution for the sake of explanation and calculation. Let's say we have the following distribution of restaurants:
| Region | Population ≤ 500,000 | Population > 500,000 | Total | |--------------|-----------------------|-----------------------|-------| | Northeast | 50 | 100 | 150 | | Southeast | 100 | 50 | 150 | | Southwest | 80 | 20 | 100 | | West | 100 | 20 | 120 | | Total | 430 | 290 | 603 |
Now let's answer each question step by step.
(a) Given that the restaurant is located in a city with a population over 500,000, what is the probability that it is in the Northeast?
Total outlets in cities with population > 500,000: \( 290 \)
Northeast outlets in cities with population > 500,000: \( 100 \)
Probability: \[ P(\text{Northeast} | \text{Population} > 500,000) = \frac{100}{290} \approx 0.3448 \]
(b) Given that the restaurant is located in the Southeast, what is the probability that it is in a city with a population under 500,000?
Total outlets in the Southeast: \( 150 \)
Southeast outlets in cities with population ≤ 500,000: \( 100 \)
Probability: \[ P(\text{Population} \leq 500,000 | \text{Southeast}) = \frac{100}{150} \approx 0.6667 \]
(c) Given that the restaurant is located in the Southwest, what is the probability that it is in a city with a population of 500,000 or less?
Total outlets in the Southwest: \( 100 \)
Southwest outlets in cities with population ≤ 500,000: \( 80 \)
Probability: \[ P(\text{Population} \leq 500,000 | \text{Southwest}) = \frac{80}{100} = 0.8000 \]
(d) Given that the restaurant is located in a city with a population of 500,000 or less, what is the probability that it is in the Southwest?
Total outlets in cities with population ≤ 500,000: \( 430 \)
Southwest outlets in cities with population ≤ 500,000: \( 80 \)
Probability: \[ P(\text{Southwest} | \text{Population} \leq 500,000) = \frac{80}{430} \approx 0.1860 \]
(e) Given that the restaurant is located in the South (either SE or SW), what is the probability that it is in a city with a population of 500,000 or more?
Total outlets in the South: \( 150 + 100 = 250 \)
Outlets in the South in cities with population > 500,000: \( 50 + 20 = 70 \)
Probability: \[ P(\text{Population} > 500,000 | \text{South}) = \frac{70}{250} = 0.2800 \]
Summary of Results:
- (a) \( 0.3448 \)
- (b) \( 0.6667 \)
- (c) \( 0.8000 \)
- (d) \( 0.1860 \)
- (e) \( 0.2800 \)
Note: Please replace the hypothetical distribution with the actual data for accurate calculations and results.
5 answers