Question

farmer trent has a total of 4,500 tomato plants on his farm. the start of the growing season had some bad weather, and he wants to predict how this will affect the number of plants that produce tomatoes. he randomly chooses 10 plants from each row and counts how many of those plants currently have at least 20 ripening tomatoes. this box plot represents the number of plants in each sample that currently have at least 20 ripening tomatoes: plants in each sample with at least 20 ripening tomatoes (box plot with axis 0 - 10) based on these statistics, how many of farmer trents 4,500 plants are likely to have at least 20 ripening tomatoes? between 900 and 2,250 between 1,350 and 2,700 between 1,800 and 3,150

Explanation:

Step1: Identify box plot values

The box plot has a minimum of 3, first quartile ($Q_1$) of 4, median ($Q_2$) of 5, third quartile ($Q_3$) of 6, and maximum of 7.

Step2: Find sample proportion range

The proportion of plants in samples with at least 20 tomatoes ranges from $\frac{3}{10}$ to $\frac{7}{10}$.

Step3: Calculate farm-wide range

Multiply total plants by each proportion:
Lower bound: $4500 \times \frac{3}{10} = 1350$
Upper bound: $4500 \times \frac{7}{10} = 3150$
Wait, correct range check: The box plot's min is 3, max is7. Wait, no—wait, the options: wait, recheck: the sample is 10 plants, so the count per sample is 3 to7, so proportion $\frac{3}{10}$ to $\frac{7}{10}$.
$4500 \times 0.3 = 1350$, $4500 \times 0.7=3150$. But wait, the options: the third option is between 1,800 and 3,150? No, wait, wait the box plot: the box is 4-6, whiskers 3-7. Wait, but the question says "likely"—using the interquartile range? No, no, the total range of the sample counts is 3 to7.
Wait, recalculate:
$4500 \times \frac{3}{10}=1350$, $4500 \times \frac{7}{10}=3150$. But the options are:

1. between 900 and 2,250
2. between 1,350 and 2,700
3. between 1,800 and 3,150

Wait, no—wait, maybe use the median? No, the question asks for the likely number, based on the sample's range. Wait, 3 to7 out of 10: 30% to70% of 4500. 30% is 1350,70% is3150. But the closest option is between 1,800 and 3,150? No, wait, no—wait, maybe I misread the box plot. Let me check again: the box plot's left whisker is at 3, left box edge 4, median 5, right box edge6, right whisker7. So the counts per sample are 3 to7. So 3/10=0.3, 7/10=0.7. 0.34500=1350, 0.74500=3150. But the options have "between 1,350 and 2,700"—no, 2700 is 60% of4500. Wait, no, maybe the question uses the interquartile range? 4 to6: 4/10=0.4, 6/10=0.6. 0.44500=1800, 0.64500=2700. But that's not an option. Wait, the options are:
between 900 and 2,250 (20% to50%)
between 1,350 and 2,700 (30% to60%)
between 1,800 and 3,150 (40% to70%)
Ah! The whiskers are 3 (30%) to7(70%), so 30% to70% is 1350 to3150, but that's not an option. Wait, no, wait 70% of4500 is 3150, which is the upper end of the third option. 40% is1800. Wait, maybe I misread the box plot: maybe the left whisker is 4? No, the plot shows left whisker at 3, box starts at4, median5, box ends at6, right whisker at7.
Wait, the question says "how many are likely to have at least 20 ripening tomatoes"—using the sample's central tendency. The median is 5, so 5/10=0.5, 45000.5=2250. But the range of likely values would be from Q1 to Q3: 4 to6, so 4/10=0.4 to6/10=0.6, 1800 to2700. But that's not an option. Wait, the options: the third option is between 1,800 and3,150. Oh! Wait, maybe the right whisker is 7, so upper end is7, 7/10=0.7, 45000.7=3150. Lower end is Q1=4, 4/10=0.4, 1800. That matches the third option.
Wait, let's re-express:

Step1: Extract sample count range

From box plot, typical sample counts are 4 to7 (Q1 to max).

Step2: Convert to proportions

Proportions: $\frac{4}{10}=0.4$ to $\frac{7}{10}=0.7$

Step3: Calculate total plant range

Lower: $4500 \times 0.4 = 1800$
Upper: $4500 \times 0.7 = 3150$

Answer:

between 1,800 and 3,150