Question

the number of earned medals for 17 countries who participated in the 2012 olympics is listed in the table. find the number of medals that separates the lowest 25% from the highest 75%. country medals argentina 4 turkey 5 lithuania 5 ireland 5 north korea 6 south africa 6 columbia 8 czech republic 10 azerbaijan 10 poland 10 belarus 12 hungary 18 canada 18 south korea 28 japan 38 great britain 65 united states 104 4.5 6 5.5 6.5

Explanation:

Step1: Arrange data in ascending order

The data is already in ascending - order of the number of medals.

Step2: Calculate the position of the first - quartile

The formula to find the position of the $p$ - th percentile ($L_p$) in a data set of size $n$ is $L_p=(n + 1)\times\frac{p}{100}$. Here, $n = 17$ and $p = 25$. So, $L_{25}=(17 + 1)\times\frac{25}{100}=18\times\frac{25}{100}=4.5$.

Step3: Interpolate to find the first - quartile

Since the position is $4.5$, the first - quartile $Q_1$ is the value that is $0.5$ between the 4th and 5th ordered data values. The 4th value is 5 and the 5th value is 5. Using the interpolation formula $Q_1=x_4+(L_{25}-\lfloor L_{25}
floor)(x_5 - x_4)$, where $\lfloor L_{25}
floor$ is the greatest integer less than or equal to $L_{25}$. Here, $x_4 = 5$, $x_5 = 5$, and $L_{25}=4.5$, so $Q_1=5+(4.5 - 4)(5 - 5)=5$.

Answer:

5.5