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Solution ID:45181b7e

Problem-Solving and Data AnalysisStatistics (Center and Spread)Median from Grouped Frequency Distribution

Core Desmos Strategy

Compute cumulative frequencies to determine which interval contains the 9th restaurant (the median position).

Solved Result

Answer: D

Extracted Problem (Vision OCR)

Complete Stem & Options

The table shown summarizes the number of employees at each of the 17 restaurants in a town.

\begin{array}{|c|c|} \hline \text{Number of employees} & \text{Number of restaurants} \\ \hline 2 \text{ to } 7 & 2 \\ \hline 8 \text{ to } 13 & 4 \\ \hline 14 \text{ to } 19 & 2 \\ \hline 20 \text{ to } 25 & 7 \\ \hline 26 \text{ to } 31 & 2 \\ \hline \end{array}

A. 2

B. 9

C. 15

D. 21

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Desmos Calculator Operations

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EvaluationCalculate the median position for 17 data values.

$\dfrac{17+1}{2}$

$9$

EvaluationCalculate cumulative frequency up through the 3rd interval: 2 + 4 + 2 = 8. The 9th item has not been reached yet.

$2+4+2$

$8$

EvaluationAdd the 4th interval: 8 + 7 = 15. The 9th item falls in this interval (20 to 25).

$8+7$

$15$

Graph Inspection Tip:A graph is not strictly necessary; use Desmos as a scratchpad calculator to confirm cumulative frequencies.

Analytical & Algebraic Solution

Step-by-step mathematical derivation with KaTeX proofs

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Step 1: Identify the Total Count and Median Position

The problem specifies that there are $N = 17$ restaurants in total.

For an odd number of ordered observations $N$ , the median is the value at position: $Position = \frac{N + 1}{2} =$ Thus, the median is the 9th value when ordered from smallest to largest.

Alternative Solving Methods

Alternative SAT test-taking techniques & secondary verification strategies

Listing Interval Positions Directly

shortcut

Write down the cumulative index ranges for each group: positions 1-2 are in [2, 7], positions 3-6 are in [8, 13], positions 7-8 are in [14, 19], and positions 9-15 are in [20, 25]. Since the median is the 9th value, it must lie between 20 and 25, meaning 21 is the only valid possibility.

Desmos Median List Simulation

desmos_alternative

Construct a list in Desmos using representative midpoint values or endpoints repeated by their respective frequencies, then apply median(L) to identify the interval.

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Extracted Problem (Vision OCR)

Desmos Calculator Operations

Analytical & Algebraic Solution

Step 1: Identify the Total Count and Median Position

Alternative Solving Methods

Listing Interval Positions Directly

Desmos Median List Simulation

Ask a Follow-Up Question

Other Solved Questions

Find the known root w = -5, subtract it from the total sum 37/3 to find the second root, then solve for a directly in Desmos.

Pick small consecutive integers for a, b, and c in Desmos and directly evaluate the means and their difference.

Step 2: Track Cumulative Frequencies

Step 3: Match with the Choices

Graph y = x^2 + (q - 2)x + 55 with a slider for q, or graph the inequality (q - 2)^2 - 220 < 0 to visually find the largest integer upper bound.

Define the factored polynomial h(x) = (x + 5)(x - 6)(x - 7) in Desmos and evaluate h(0) to find the constant term c.

Use the axis of symmetry x = 10: since x = 8 and x = 12 are both 2 units away from the vertex price, the number of bowls sold at $8 must equal the number sold at $12 (5 bowls).

Identify the contextual definitions of the coordinates: x is unit price and y is revenue, meaning an x-intercept (a, 0) corresponds to a unit price producing $0 revenue.