1 Verify through calculations whether the difference between the IQR (Interquartile Range) of the basketball players and the baseball players is 3 inches.

The difference is 0 inches, not 3 inches.
Therefore, the statement is incorrect.

From the basketball box plot, Q3 is 76 inches and the maximum is 84 inches. This means 25% of the players are between 76 and 84 inches tall. The value 79 inches falls within this top quartile. A box plot does not provide enough information to determine the exact number of players for a specific height within a quartile unless that height is a quartile boundary or the min/max. However, in the context of such problems, if a specific value within a quartile is given, and no further distribution information is provided, the question often implicitly refers to the entire quartile. Given the phrasing, we can only state the range. If an exact number is required, we must assume a uniform distribution within the quartile, which is not always accurate.
However, if we consider the question to be asking for the number of players in the top quartile (i.e., taller than Q3 = 76 inches), this is a common interpretation.

The top quartile represents 25% of the data.
Number of players = $0.25 \times 84 = 21$ players.
Since 79 inches is within the top quartile (76 to 84 inches), the number of players taller than 79 inches must be less than or equal to 21. Without more information on the distribution within this quartile, we cannot determine the exact number. If we assume the question is asking for the number of players in the top quartile, then the answer is 21. If we assume a uniform distribution within the top quartile (76 to 84 inches), which contains 21 players, then the proportion of players from 79 to 84 inches is $\frac{84-79}{84-76} = \frac{5}{8}$. So, $\frac{5}{8} \times 21 = 13.125$. Since the number of players must be an integer, this indicates that a precise answer cannot be determined from the box plot alone.
Given the limitations, the most reasonable interpretation is that the question is testing the understanding of quartiles. The number of players taller than 76 inches (i.e., in the top quartile) is 21. The number of players taller than 79 inches cannot be precisely determined from the box plot. If forced to choose the closest interpretable value, it would be the number of players in the top quartile.

Let's assume the question is asking for the number of players in the top quartile, as 79 inches falls within this range.
Number of players = $0.25 \times 84 = \boxed{\text{21 players}}$

• 9.2 (b): A data analyst states that 75% of the baseball players are shorter than 50% of the basketball players. Validate this by referring to the quartiles of the different data sets.

This means X is the third quartile ($Q3$) of the baseball players.
$Q3_{baseball} = 78$ inches.

This refers to the median ($Q2$) of the basketball players.
$Q2_{basketball} = 72$ inches.

The statement is: "75% of baseball players are shorter than 72 inches".
This implies that $Q3_{baseball} < 72$ inches.
However, $Q3_{baseball} = 78$ inches.
Since $78 \text{ inches} \not< 72 \text{ inches}$, the statement is incorrect.

For basketball players, Q1 = 68 inches.

Using the method where Q1 is the value at position $0.25(n)$, Q2 at $0.5(n)$, and Q3 at $0.75(n)$:

• Q1 position: $0.25 \times 84 = 21$. So, Q1 is the 21st value.
• Q2 position: $0.5 \times 84 = 42$. So, Q2 is the 42nd value.
• Q3 position: $0.75 \times 84 = 63$. So, Q3 is the 63rd value. However, if Q1 is an average of two values, it's often calculated as the average of the values at positions $n/4$ and $n/4+1$ if $n/4$ is an integer. Or, using the $(n+1)$ method, Q1 is at position $0.25(n+1) = 0.25(85) = 21.25$, meaning it's interpolated between the 21st and 22nd values.

The bold values are 67.5 and 68.5.
The box plot states $Q1 = 68$.
If Q1 is calculated as the average of two consecutive values, say $x_k$ and $x_{k+1}$, then $Q1 = \frac{x_k + x_{k+1}}{2}$.
If we take $x_k = 67.5$ and $x_{k+1} = 68.5$, then $\frac{67.5 + 68.5}{2} = \frac{136}{2} = 68$.
This matches the Q1 value from the box plot.
Therefore, 67.5 and 68.5 represent the two data values that are averaged to determine the first quartile (Q1). Specifically, 67.5 is the 21st value and 68.5 is the 22nd value in the ordered data set.

The bold values 67.5 and 68.5 represent the 21st and 22nd values in the ordered data set, whose average defines the first quartile (Q1).

$Q2_{baseball}$ (Median) = 74 inches.
$Q3_{baseball}$ = 78 inches.
Maximum = 86 inches.

76 inches is between the median (74 inches) and Q3 (78 inches).

• 50% of players are taller than 74 inches (Median).

• 25% of players are taller than 78 inches (Q3). Since 76 inches is exactly halfway between 74 and 78, and assuming a relatively even distribution within this quartile, we can estimate the percentage. The range from 74 to 78 inches contains 25% of the data. The value 76 inches is $\frac{76-74}{78-74} = \frac{2}{4} = \frac{1}{2}$ of the way from Q2 to Q3. So, 50% of the data is above Q2 (74 inches). The data between 74 and 76 inches represents half of the second quartile (from Q2 to Q3), which is $0.5 \times 25\% = 12.5\%$. The percentage of players taller than 76 inches is the percentage of players between 76 and 78 inches (12.5%) plus the percentage of players taller than 78 inches (25%). Percentage = $12.5\% + 25\% = \boxed{37.5\%}$

• 9.4 (b): Shorter than quartile 2, but taller than quartile 1 (as a simplified fraction).

$Q1_{baseball} = 70$ inches.
$Q2_{baseball}$ (Median) = 74 inches.

The data between Q1 and Q2 represents the second quartile.
This range contains 25% of the data.