Question

1. the dotplot shows the number of wins for major league baseball teams in the 2019 season:

a) find the percentile for the seattle mariners, who won 68 games.
b) the washington nationals’ number of wins is at the 70th percentile of the distribution.
i) interpret this value.
ii) how many games did the washington nationals win?

Explanation:

Step1: Count the total number of data points

Count all the dots in the dot - plot. There are \(n = 30\) teams (since there are 30 dots).

Step2: For part (a)

For the Seattle Mariners with \(x = 68\) wins.
Count the number of data points less than or equal to \(68\). There are \(4\) data points less than \(68\) and \(0\) data points equal to \(68\). The formula for the percentile \(P=\frac{\text{Number of data points}\leq x}{n}\times100\). So \(P=\frac{4}{30}\times 100=\frac{400}{30}\approx13.33\)

Step3: For part (b)(i)

The interpretation of the \(70^{th}\) percentile: Approximately \(70\%\) of the teams won fewer games than the Washington Nationals, and approximately \(30\%\) of the teams won more games than the Washington Nationals.

Step4: For part (b)(ii)

We know that \(L=\frac{p}{100}\times n\), where \(p = 70\) and \(n=30\). So \(L=\frac{70}{100}\times30 = 21\).
Order the data (from left - to - right on the dot - plot). The \(21^{st}\) data point.
Counting the dots:
The first \(1\) dot (at \(45\)), then \(2\) dots (at \(55\)), \(1\) dot (at \(60\)), \(5\) dots (at \(70\)), \(1\) dot (at \(75\)), \(3\) dots (at \(80\)), \(3\) dots (at \(85\)), \(2\) dots (at \(90\)), \(3\) dots (at \(95\)), \(2\) dots (at \(100\)), \(2\) dots (at \(105\)).
Cumulative counts:

• Up to \(45\): \(1\)
• Up to \(55\): \(1 + 2=3\)
• Up to \(60\): \(3+1 = 4\)
• Up to \(70\): \(4 + 5=9\)
• Up to \(75\): \(9+1 = 10\)
• Up to \(80\): \(10+3 = 13\)
• Up to \(85\): \(13+3 = 16\)
• Up to \(90\): \(16+2 = 18\)
• Up to \(95\): \(18+3=21\)

The value at the \(21^{st}\) position is \(95\)

Answer:

a) The percentile for the Seattle Mariners is approximately \(13.3\)
b) i) Approximately \(70\%\) of the teams won fewer games than the Washington Nationals, and approximately \(30\%\) of the teams won more games than the Washington Nationals.
ii) The Washington Nationals won \(95\) games.