Question

the heights of the starting players on each of two basketball teams are shown in the table below. jacob found that the mean height of team a is 71 and the mean height of team b is 72. he believes that because team b has a greater mean, it also has a greater mean absolute deviation. which explains jacobs error? one of the means is incorrect, but the reasoning is correct. both of the means are incorrect, and the reasoning is also incorrect. both of the means are correct, but the reasoning is incorrect. one of the means is incorrect, and the reasoning is also incorrect.

Explanation:

Step1: Calculate the mean of Team A

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). For Team A, \(n = 5\), \(x_1=70\), \(x_2 = 72\), \(x_3=75\), \(x_4=68\), \(x_5 = 70\).
\(\bar{x}_A=\frac{70 + 72+75+68+70}{5}=\frac{355}{5}=71\)

Step2: Calculate the mean of Team B

For Team B, \(n = 5\), \(x_1=71\), \(x_2 = 73\), \(x_3=71\), \(x_4=72\), \(x_5 = 73\).
\(\bar{x}_B=\frac{71+73 + 71+72+73}{5}=\frac{360}{5}=72\)

Step3: Calculate the mean absolute deviation (MAD)

The formula for MAD is \(MAD=\frac{\sum_{i = 1}^{n}|x_{i}-\bar{x}|}{n}\)

• For Team A:

\(|70 - 71|=1\), \(|72-71| = 1\), \(|75 - 71|=4\), \(|68-71|=3\), \(|70 - 71|=1\)
\(MAD_A=\frac{1+1 + 4+3+1}{5}=\frac{10}{5}=2\)

• For Team B:

\(|71-72| = 1\), \(|73-72|=1\), \(|71 - 72|=1\), \(|72-72|=0\), \(|73-72|=1\)
\(MAD_B=\frac{1+1+1+0+1}{5}=\frac{4}{5} = 0.8\)

The means of Team A (\(71\)) and Team B (\(72\)) are correct. But the mean absolute deviation of Team A (\(2\)) is greater than that of Team B (\(0.8\)). So, Jacob's reasoning (greater mean implies greater MAD) is incorrect.

Answer:

Both of the means are correct, but the reasoning is incorrect.