Question

think about the process the top 25% in a talent competition move on to the final round. here are the scores of the 11 contestants: 77, 82, 73, 74, 71, 79, 75, 76, 66, 72, 68. why would a box plot be the best data display for the results? what are the scores of the top 25%? ... why would a box plot be the results? a. a box plot shows the clusters and gaps in data. b. a box plot shows frequencies of data values. c. a box plot shows the five boundary values of a data set. d. a box plot shows every data value.

Explanation:

First, let's address the multiple - choice question about why a box - plot is the best data display:

• Option A: A box - plot is not mainly used to show clusters and gaps in data. Tools like dot - plots or scatter - plots are better for that.
• Option B: A box - plot does not show the frequencies of data values. Histograms or frequency tables are used for frequency representation.
• Option C: A box - plot shows the five - number summary which includes the minimum, the first quartile ($Q_1$), the median ($Q_2$), the third quartile ($Q_3$), and the maximum. These values act as boundary values that divide the data into four parts (each representing approximately 25% of the data). In the context of finding the top 25% of the scores, we can use the third quartile ($Q_3$) as a boundary. So this option is correct.
• Option D: A box - plot does not show every data value. It shows summary statistics (the five - number summary) and the spread of the data.

Step 1: Sort the data

First, we arrange the scores from the smallest to the largest. The scores are 66, 68, 71, 72, 72, 73, 74, 75, 76, 77, 79, 82? Wait, wait, the number of contestants is 11? Wait, the problem says "11 contestants". Wait, let's check the data again: 77, 82, 73, 74, 71, 79, 75, 76, 66, 72, 68. Let's count: 77, 82, 73, 74, 71, 79, 75, 76, 66, 72, 68. That's 11 data points.

Sorting them in ascending order: 66, 68, 71, 72, 73, 74, 75, 76, 77, 79, 82. Wait, no, 72 appears twice? Wait, original data: 77, 82, 73, 74, 71, 79, 75, 76, 66, 72, 68. Let's list all: 66, 68, 71, 72, 73, 74, 75, 76, 77, 79, 82. Wait, 72 is once? Wait, maybe I made a mistake. Let's re - list:

Given scores: 77, 82, 73, 74, 71, 79, 75, 76, 66, 72, 68.

Let's sort:

66 (1st), 68 (2nd), 71 (3rd), 72 (4th), 73 (5th), 74 (6th), 75 (7th), 76 (8th), 77 (9th), 79 (10th), 82 (11th).

Step 2: Find the position of the third quartile ($Q_3$)

For a data set with $n$ values, the position of the $k$ - th quartile is given by $i=\frac{k(n + 1)}{4}$, where $k = 1,2,3$ for the first, second, and third quartiles respectively.

Here, $n = 11$ and $k = 3$ (for the third quartile, which separates the top 25% from the rest).

So, $i=\frac{3(11 + 1)}{4}=\frac{3\times12}{4}=9$.

Step 3: Determine the scores of the top 25%

The third quartile is at the 9th position (when the data is sorted). The values from the 9th position to the end of the data set represent the top 25% (since the data is divided into four parts of approximately 25% each by the quartiles).

The 9th value is 77, the 10th is 79, and the 11th is 82. Wait, but let's check the formula for quartiles again. Another way to calculate quartiles is:

For $n$ data points, the first quartile ($Q_1$) is the median of the lower half, the second quartile ($Q_2$) is the median of the whole data, and the third quartile ($Q_3$) is the median of the upper half.

The whole data set (sorted): 66, 68, 71, 72, 73, 74, 75, 76, 77, 79, 82.

The median ($Q_2$) is the 6th value (since $n = 11$, the middle value is at position $\frac{11 + 1}{2}=6$), which is 74.

The upper half of the data (values above the median) is: 75, 76, 77, 79, 82 (these are the values from position 7 to 11). The median of this upper half is the 3rd value in this sub - set (since there are 5 values, the middle one is at position $\frac{5+1}{2}=3$). The 3rd value in the upper half (75, 76, 77, 79, 82) is 77. So the third quartile ($Q_3$) is 77.

The top 25% of the data are the values greater than or equal to $Q_3$. So the scores of the top 25% are 77, 79, 82.

Answer:

C. A box plot shows the five boundary values of a data set.

Now, let's find the scores of the top 25% of the contestants:

First, we need to sort the data set. The scores are: 68, 71, 72, 72, 73, 74, 75, 76, 77, 79, 82. Wait, let's re - list and sort the given scores: 68, 71, 72, 72, 73, 74, 75, 76, 77, 79, 82. Wait, the original data is 77, 82, 73, 74, 71, 79, 75, 76, 66, 72, 68. Let's sort them in ascending order: