Question

a basketball player recorded her points for 11 consecutive games. her points are listed below: 13, 20, 15, 22, 24, 21, 18, 20, 25, 17, 27. how many points would she need to score in her 12th game in order to decrease the interquartile - range by 1 point?

Explanation:

Step1: Arrange data in ascending order

13, 15, 17, 18, 20, 20, 21, 22, 24, 25, 27

Step2: Calculate the position of Q1 and Q3 for n = 11 data - points

The position of Q1 is $\frac{n + 1}{4}=\frac{11+ 1}{4}=3$. So Q1 is the 3rd - value in the ordered list, Q1 = 17. The position of Q3 is $\frac{3(n + 1)}{4}=\frac{3\times(11 + 1)}{4}=9$. So Q3 is the 9th - value in the ordered list, Q3 = 24. The current IQR = Q3 - Q1=24 - 17 = 7.

Step3: Let the new score be x. Re - calculate Q1, Q3 and IQR with n = 12 data - points

Arrange the new data set (13, 15, 17, 18, 20, 20, 21, 22, 24, 25, 27, x) in ascending order. For n = 12, the position of Q1 is $\frac{n}{4}=3$, and the position of Q3 is $\frac{3n}{4}=9$.
If x is less than or equal to 18:
The ordered data set is (x, 13, 15, 17, 18, 20, 20, 21, 22, 24, 25, 27). Q1 is the average of the 3rd and 4th values, $Q1=\frac{15 + 17}{2}=16$. Q3 is the average of the 9th and 10th values, $Q3=\frac{22+24}{2}=23$. We want IQR = 6 (since we want to decrease it by 1). So $23-16 = 7
eq6$.
If x is greater than or equal to 22:
The ordered data set is (13, 15, 17, 18, 20, 20, 21, 22, x, 24, 25, 27). Q1 is the average of the 3rd and 4th values, $Q1=\frac{17 + 18}{2}=17.5$. Q3 is the average of the 9th and 10th values, $Q3=\frac{x + 24}{2}$. We want $Q3 - Q1=6$, so $\frac{x + 24}{2}-17.5 = 6$.
Solve the equation $\frac{x + 24}{2}-17.5 = 6$:
First, add 17.5 to both sides: $\frac{x + 24}{2}=6 + 17.5=23.5$.
Then multiply both sides by 2: $x+24 = 47$.
Subtract 24 from both sides: $x = 23$.

Answer:

23