1, 1, 5, 5, 7, 8, 8, 10, 12\nwhich box plot correctly represents the data?

1, 1, 5, 5, 7, 8, 8, 10, 12\nwhich box plot correctly represents the data?

1, 1, 5, 5, 7, 8, 8, 10, 12\nwhich box plot correctly represents the data?

Answer

Answer:

To determine the correct box - plot, we need to find the five - number summary (minimum, first quartile $Q_1$, median, third quartile $Q_3$, maximum) of the data set $1,1,5,5,7,8,8,10,12$.

  1. Minimum value:
    • The minimum value of the data set is $1$.
  2. Median ($Q_2$):
    • Since there are $n = 9$ data points, the median is the $\left(\frac{n + 1}{2}\right)$-th value.
    • $\frac{9+1}{2}=5$ - th value. So the median $Q_2=7$.
  3. First quartile ($Q_1$):
    • The lower half of the data set is $1,1,5,5$. There are $n_1 = 4$ data points in the lower half.
    • The median of the lower half (first quartile) is the average of the $\frac{4}{2}$-th and $\left(\frac{4}{2}+1\right)$-th ordered values.
    • The $\frac{4}{2}=2$ - nd and $\left(\frac{4}{2}+1\right)=3$ - rd values are $1$ and $5$. So $Q_1=\frac{1 + 5}{2}=3$.
  4. Third quartile ($Q_3$):
    • The upper half of the data set is $8,8,10,12$. There are $n_2 = 4$ data points in the upper half.
    • The median of the upper half (third quartile) is the average of the $\frac{4}{2}$-th and $\left(\frac{4}{2}+1\right)$-th ordered values.
    • The $\frac{4}{2}=2$ - nd and $\left(\frac{4}{2}+1\right)=3$ - rd values are $8$ and $10$. So $Q_3=\frac{8 + 10}{2}=9$.
  5. Maximum value:
    • The maximum value of the data set is $12$.

The box - plot should have the minimum at $1$, $Q_1$ at $3$, the median at $7$, $Q_3$ at $9$, and the maximum at $12$.

We need to visually inspect the box - plots to find the one that has the left - most whisker at $1$, the left side of the box at $3$, the line inside the box at $7$, the right side of the box at $9$, and the right - most whisker at $12$.

(Without seeing the actual visual details of the box - plots to select from, we can't give a specific option. But the above is the process to find the correct box - plot.)

Explanation:

Step1: Find minimum

The smallest value is $1$.

Step2: Calculate median

For $n = 9$, median is 5 - th value, $7$.

Step3: Find first quartile

Median of lower half ($1,1,5,5$) is $\frac{1 + 5}{2}=3$.

Step4: Find third quartile

Median of upper half ($8,8,10,12$) is $\frac{8+10}{2}=9$.

Step5: Find maximum

The largest value is $12$.