mr. smith measured the heights of some students in his math class. students heights (in.) 56 61 55 63 66 63…

Answer:

We need to first find the five - number summary (minimum, first quartile $Q_1$, median, third quartile $Q_3$, maximum) of the data set ${56,61,55,63,66,63,52,56,64,64}$ to determine the correct box - plot.

1. Sort the data set:
   • Sort the data in ascending order: ${52,55,56,56,61,63,63,64,64,66}$.
2. Find the minimum and maximum:
   • The minimum value is $52$.
   • The maximum value is $66$.
3. Find the median:
   • Since there are $n = 10$ data points, the median is the average of the $\frac{n}{2}=5$th and $\frac{n}{2}+ 1 = 6$th ordered data points.
   • Median $M=\frac{61 + 63}{2}=62$.
4. Find the first quartile $Q_1$:
   • The lower half of the data is ${52,55,56,56,61}$. Since there are $n_1=5$ data points in the lower half, the first quartile $Q_1$ is the 3rd ordered data point in the lower half. So $Q_1 = 56$.
5. Find the third quartile $Q_3$:
   • The upper half of the data is ${63,63,64,64,66}$. Since there are $n_2 = 5$ data points in the upper half, the third quartile $Q_3$ is the 3rd ordered data point in the upper half. So $Q_3=64$.

The box - plot should have a minimum at $52$, $Q_1$ at $56$, median at $62$, $Q_3$ at $64$, and maximum at $66$. Based on these values, we can match the box - plot to the correct one among the given options (not fully shown in the question but can be determined by these values).

Explanation:

Step1: Sort the data

${52,55,56,56,61,63,63,64,64,66}$

Step2: Find minimum

Minimum = $52$

Step3: Find maximum

Maximum = $66$

Step4: Calculate median

$M=\frac{61 + 63}{2}=62$

Step5: Calculate $Q_1$

$Q_1 = 56$

Step6: Calculate $Q_3$

$Q_3=64$