Question

1. the two sets of data below represent the number of runs scored by two different youth baseball teams over the course of a season.

team a: 4, 8, 5, 12, 3, 9, 5, 2
team b: 5, 9, 11, 4, 6, 11, 2, 7
which set of statements about the mean and standard deviation is true?
a. mean a > mean b
standard deviation a > standard deviation b
b. mean a < mean b
standard deviation a < standard deviation b
c. mean a > mean b
standard deviation a < standard deviation b
d. mean a < mean b
standard deviation a > standard deviation b

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: \(x=[4,8,5,12,3,9,5,2]\), \(n = 8\). \(\sum_{i=1}^{8}x_{i}=4 + 8+5+12+3+9+5+2=48\). \(\bar{x}_{A}=\frac{48}{8}=6\)

Step2: Calculate the mean of Team B

For Team B: \(x = [5,9,11,4,6,11,2,7]\), \(n = 8\). \(\sum_{i = 1}^{8}x_{i}=5 + 9+11+4+6+11+2+7=55\). \(\bar{x}_{B}=\frac{55}{8}=6.875\)

Step3: Calculate the standard deviation of Team A

The formula for the standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\).
\((4 - 6)^{2}+(8 - 6)^{2}+(5 - 6)^{2}+(12 - 6)^{2}+(3 - 6)^{2}+(9 - 6)^{2}+(5 - 6)^{2}+(2 - 6)^{2}\)
\(=(- 2)^{2}+2^{2}+(-1)^{2}+6^{2}+(-3)^{2}+3^{2}+(-1)^{2}+(-4)^{2}\)
\(=4 + 4+1+36+9+9+1+16=80\). \(s_{A}=\sqrt{\frac{80}{7}}\approx3.38\)

Step4: Calculate the standard deviation of Team B

\((5 - 6.875)^{2}+(9 - 6.875)^{2}+(11 - 6.875)^{2}+(4 - 6.875)^{2}+(6 - 6.875)^{2}+(11 - 6.875)^{2}+(2 - 6.875)^{2}+(7 - 6.875)^{2}\)
\(=(-1.875)^{2}+2.125^{2}+4.125^{2}+(-2.875)^{2}+(-0.875)^{2}+4.125^{2}+(-4.875)^{2}+0.125^{2}\)
\(=3.515625+4.515625 + 17.015625+8.265625+0.765625+17.015625+23.765625+0.015625 = 74.875\). \(s_{B}=\sqrt{\frac{74.875}{7}}\approx3.27\)

Answer:

C. mean \(A<\) mean \(B\), standard deviation \(A>\) standard deviation \(B\)