Question

question 23 (5 points)
provide an appropriate response. express your answer as a simplified fraction unless otherwise noted.
a fast - food restaurant chain with 700 outlets in the united states describes the geographic location of its restaurants with the accompanying table of percentages. a restaurant is to be chosen at random from the 700 to test market a new style of chicken. given that the restaurant is located in the eastern united states, what is the probability it is located in a city with a population of at least 10,000? round your answer to the nearest thousandth.
region
ne se sw nw
population of city < 10,000 8% 6% 3% 0%
10,000 - 100,000 15% 9% 12% 5%

100,000 20% 4% 10% 8%

question 23 options:

Explanation:

Step1: Identify Eastern Regions

Eastern regions are NE and SE. Calculate total percentage for eastern US (NE + SE) for each population category.
- For <10,000: \( 6\% + 8\% = 14\% \) (Wait, no: NE is 6%? Wait table: first row <10,000: NE is 6%? Wait table: first row: <10,000: columns NE, SE, SW, NW: 6%, 8%? Wait no, table:

Wait the table:

Population of City:
- <10,000: NE 6%, SE 8%? Wait no, original table:

Wait the user's table:

First row: <10,000: 8% (first column, maybe "Total"? Wait no, the first column under "Population of City" is:

Wait the table structure:

Row 1: <10,000: columns (maybe first column is a category, then NE, SE, SW, NW). Wait the first column under "Population of City" has:

- <10,000
- 10,000 - 100,000
- >100,000

Then columns: NE, SE, SW, NW.

So for <10,000: NE is 6%, SE is 8%? Wait no, the first number in <10,000 row is 8% (maybe a total? No, the problem says "accompanying table of percentages" for the restaurant chain. So probably, the first column (leftmost) is the population category, then NE, SE, SW, NW are the regions. So:

- <10,000: NE 6%, SE 8%? Wait no, the numbers: <10,000 row: 8% (maybe first column is a label, then NE 6%, SE 8%? Wait no, the user's table:

Looking at the user's table:

"Population of City" has three rows: <10,000; 10,000 - 100,000; >100,000.

Columns: NE, SE, SW, NW.

So:

- <10,000: NE 6%, SE 8%? Wait no, the numbers: <10,000 row: 8% (first cell), then NE 6%, SE 8%? No, maybe the first cell (leftmost) is a total, but no. Wait the problem says "eastern United States" – NE (Northeast) and SE (Southeast) are eastern.

So for eastern US (NE + SE), we need to find:

- Total percentage of eastern US (NE + SE) for all population categories.

- Percentage of eastern US with population ≥10,000 (10,000 - 100,000 + >100,000)

First, calculate total percentage for eastern US (NE + SE) across all population:

For <10,000: NE 6% + SE 8% = 14%

For 10,000 - 100,000: NE 9% + SE 15% = 24%

For >100,000: NE 4% + SE 20% = 24%

Wait no, wait the table:

Wait the numbers:

<10,000 row: 8% (maybe a typo, but looking at the user's table:

First row: <10,000: 8% (first column), then NE 6%, SE 3%? No, the user's table as written:

"Population of City <10,000 8% 6% 3% 0%

10,000 - 100,000 15% 9% 12% 5%

>100,000 20% 4% 10% 8%"

Ah! Wait, the first column (leftmost) under "Population of City" is a column with percentages, maybe the "Total" for all regions? No, that can't be. Wait, no – the first number in <10,000 row is 8% (maybe a mistake, but actually, the columns are: first column (population category), then NE, SE, SW, NW. So:

- <10,000: NE 6%, SE 8%? No, the numbers are: <10,000 row: 8% (first cell), then NE 6%, SE 3%? No, the user's table:

Looking at the user's text:

"Population of City <10,000 8% 6% 3% 0%

10,000 - 100,000 15% 9% 12% 5%

>100,000 20% 4% 10% 8%"

Ah! So the first column (leftmost) is a column with percentages, maybe the "Total" for each population category across all regions? No, that would sum to more than 100%. So actually, the first column (leftmost) is a mistake, and the columns are NE, SE, SW, NW for each population category. So:

- <10,000: NE 6%, SE 8%? No, the numbers are: <10,000 row: 8% (first cell), then NE 6%, SE 3%? No, the user's table as written:

Row 1: <10,000: 8% (first cell), NE 6%, SE 3%, SW 0%? No, the numbers are: 8%, 6%, 3%, 0% for <10,000. Then 15%, 9%, 12%, 5% for 10,000-100,000. Then 20%, 4%, 10%, 8% for >100,000.

Ah! So the first column (leftmost) is a column with percentages, maybe the "Total" for each population category across all regions? No, that's not possible. Wait, the problem says "a fast-food restaurant chain with 700 outlets... table of percentages" – so the percentages are of the 700 outlets. So each cell is the percentage of the 700 outlets in that population category and region.

So:

- <10,000: NE: 6% of 700, SE: 8%? No, the numbers: <10,000 row: 8% (maybe a typo, but the numbers are 8%, 6%, 3%, 0% – so first number (8%) is maybe a total for <10,000 across all regions? Let's check: 8% + 6% + 3% + 0% = 17%? No, that's not 100%. So probably, the first column (leftmost) is the population category, and the next columns are NE, SE, SW, NW. So:

- <10,000: NE: 6%, SE: 8%? No, the numbers are: <10,000 row: 8% (first cell), NE 6%, SE 3%, SW 0%? No, the user's table:

Looking at the user's input:

"Population of City <10,000 8% 6% 3% 0%

10,000 - 100,000 15% 9% 12% 5%

>100,000 20% 4% 10% 8%"

Ah! Now I see: the first column (leftmost) is a column with the population category, and the next columns are NE, SE, SW, NW. Wait, no – the numbers:

- <10,000: first number (8%) is maybe a mistake, but the columns are NE (6%), SE (8%?), no. Wait, the first number in <10,000 row is 8% (maybe a total for <10,000 across all regions), then NE 6%, SE 8%? No, the numbers are 8%, 6%, 3%, 0% – so 8% (maybe a total), then NE 6%, SE 3%, SW 0%? No, that doesn't make sense.

Wait, the problem is about conditional probability: given the restaurant is in the eastern US (NE or SE), what's the probability it's in a city with population ≥10,000.

So eastern regions are NE and SE.

First, calculate the total percentage of outlets in eastern US (NE + SE) for each population category:

1. <10,000: NE (6%) + SE (8%)? Wait, the numbers: <10,000 row: 6% (NE), 8% (SE)? No, the user's table: <10,000 row: 8% (first cell), 6% (NE), 3% (SE)? No, I think the first column (leftmost) is a mistake, and the columns are NE, SE, SW, NW. So:

- <10,000: NE: 6%, SE: 8%? No, the numbers are: <10,000 row: 8% (maybe a typo), 6% (NE), 3% (SE), 0% (SW), 0% (NW)? No, the user's table has four columns: NE, SE, SW, NW. So:

Row 1 ( <10,000 ): NE 6%, SE 8%, SW 3%, NW 0%? No, the numbers are 8%, 6%, 3%, 0% – so first number (8%) is maybe a total for <10,000 across all regions: 6% (NE) + 8% (SE) + 3% (SW) + 0% (NW) = 17%? No, that's not 8%. So I think the first column (leftmost) is the population category, and the next columns are NE, SE, SW, NW. So:

- <10,000: NE: 6%, SE: 8%? No, the numbers are: <10,000 row: 8% (first cell), NE 6%, SE 3%, SW 0%? No, I'm confused. Wait, let's re-express the table:

Population of City | NE | SE | SW | NW
--- | --- | --- | --- | ---
<10,000 | 6% | 8% | 3% | 0%
10,000 - 100,000 | 9% | 15% | 12% | 5%
>100,000 | 4% | 20% | 10% | 8%

Ah! That makes sense. So:

- <10,000: NE 6%, SE 8%, SW 3%, NW 0%

- 10,000 - 100,000: NE 9%, SE 15%, SW 12%, NW 5%

- >100,000: NE 4%, SE 20%, SW 10%, NW 8%

Yes, that must be it. So the first column is the population category, then NE, SE, SW, NW.

Now, eastern regions are NE and SE.

Step2: Calculate Total Eastern Percentage (NE + SE) for Each Category

- <10,000: NE (6%) + SE (8%) = 14%

- 10,000 - 100,000: NE (9%) + SE (15%) = 24%

- >100,000: NE (4%) + SE (20%) = 24%

Step3: Calculate Total Eastern Percentage (Denominator)

Total percentage of eastern US (NE + SE) is the sum of all eastern percentages: \( 14\% + 24\% + 24\% = 62\% \)

Step4: Calculate Percentage of Eastern US with ≥10,000 (Numerator)

Percentage of eastern US with population ≥10,000 is \( 24\% + 24\% = 48\% \)

Step5: Apply Conditional Probability Formula

Conditional probability \( P(A|B) = \frac{P(A \cap B)}{P(B)} \)

Here, A: population ≥10,000; B: eastern US.

\( P(A \cap B) = 48\% \) (percentage of eastern US with ≥10,000)

\( P(B) = 62\% \) (percentage of eastern US)

So \( P(A|B) = \frac{48}{62} \approx 0.774 \) (rounded to nearest thousandth)

Wait, wait, no – wait the percentages are of the 700 outlets. So each cell is the percentage of the 700 outlets. So:

- Number of eastern outlets (NE + SE) in <10,000: \( 0.06 \times 700 + 0.08 \times 700 = 42 + 56 = 98 \)

- Number of eastern outlets in 10,000 - 100,000: \( 0.09 \times 700 + 0.15 \times 700 = 63 + 105 = 168 \)

- Number of eastern outlets in >100,000: \( 0.04 \times 700 + 0.20 \times 700 = 28 + 140 = 168 \)

Total eastern outlets: \( 98 + 168 + 168 = 434 \)

Number of eastern outlets with ≥10,000: \( 168 + 168 = 336 \)

Conditional probability: \( \frac{336}{434} \approx 0.774 \) (same as before)

Wait, let's recalculate:

Total eastern percentage: 6% + 8% + 9% + 15% + 4% + 20% = 62% (as before). So 62% of 700 is 0.62*700=434.

Percentage of eastern with ≥10,000: 9% + 15% + 4% + 20% = 48% of 700 is 0.48*700=336.

336 / 434 ≈ 0.7742, which rounds to 0.774.

Answer:

0.774