Question

the table shows the yearly earnings, in thousands of dollars, over a 10 - year period for college graduates.
which statement is true about the distributions representing the yearly earnings?
the mean earnings of the self - employed are higher than the mean earnings of the wage earners.
the distribution of earnings for wage earners is more symmetric than the distribution of earnings for the self - employed.
the iqrs of the distributions are equal.
the standard deviations of the distributions are equal.

Explanation:

Step1: Analyze the mean

For self - employed: \((72 + 48+51 + 39+81+60+96+53+101+52)\div10=\frac{653}{10} = 65.3\)
For wage earners: \((65 + 51+58 + 44+84+62+81+64+89+68)\div10=\frac{666}{10}=66.6\)
So the mean of wage earners is higher than that of self - employed.

Step2: Analyze the symmetry

Self - employed data: \(39,48,51,52,53,60,72,81,96,101\)
The median is \(\frac{53 + 60}{2}=56.5\)
Wage earners data: \(44,51,58,62,64,65,68,81,84,89\)
The median is \(\frac{64 + 65}{2}=64.5\)
Self - employed data has a larger spread and is more skewed (with higher values like \(96,101\)), while wage earners' data is more symmetric.

Step3: Analyze the standard deviation and IQR

For self - employed:
First, find the mean \(\bar{x}=65.3\)
\((72 - 65.3)^2+(48 - 65.3)^2+(51 - 65.3)^2+(39 - 65.3)^2+(81 - 65.3)^2+(60 - 65.3)^2+(96 - 65.3)^2+(53 - 65.3)^2+(101 - 65.3)^2+(52 - 65.3)^2\)
\(=44.89+299.29+204.49+691.69+246.49+28.09+942.49+151.29+1274.49+176.89\)
\(=4060.1\)
Variance \(s^{2}=\frac{4060.1}{9}\approx451.12\), standard deviation \(s\approx21.24\)
For wage earners:
Mean \(\bar{y} = 66.6\)
\((65 - 66.6)^2+(51 - 66.6)^2+(58 - 66.6)^2+(44 - 66.6)^2+(84 - 66.6)^2+(62 - 66.6)^2+(81 - 66.6)^2+(64 - 66.6)^2+(89 - 66.6)^2+(68 - 66.6)^2\)
\(=2.56+243.36+73.96+510.76+302.76+21.16+207.36+6.76+501.76+1.96\)
\(=1872.4\)
Variance \(s^{2}=\frac{1872.4}{9}\approx208.04\), standard deviation \(s\approx14.42\)
For IQR:
Self - employed: \(Q_1=\frac{48 + 51}{2}=49.5\), \(Q_3=\frac{81+96}{2}=88.5\), \(IQR = 88.5-49.5 = 39\)
Wage earners: \(Q_1=\frac{51+58}{2}=54.5\), \(Q_3=\frac{81 + 84}{2}=82.5\), \(IQR=82.5 - 54.5=28\)

Answer:

The distribution of earnings for wage earners is more symmetric than the distribution of earnings for the self - employed.