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? self - employed wage earners 52 66 101 89 53 64 96 81 60 62 81 84 38 44 51 58 46 51 72 65 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: Calculate the mean of self - employed

The mean $\bar{x}_1$ of self - employed earnings: $\bar{x}_1=\frac{52 + 101+53+96+60+81+38+51+46+72}{10}=\frac{650}{10} = 65$

Step2: Calculate the mean of wage earners

The mean $\bar{x}_2$ of wage earners earnings: $\bar{x}_2=\frac{66+89+64+81+62+84+44+58+51+65}{10}=\frac{684}{10}=68.4$. So the first option is false.

Step3: Analyze symmetry

To check symmetry, we can look at the spread of data. For self - employed: $38,46,51,52,53,60,72,81,96,101$. For wage earners: $44,51,58,62,64,65,66,81,84,89$. The data for wage earners is more evenly spread around the middle values, so the distribution of earnings for wage earners is more symmetric.

Step4: Calculate the IQR for self - employed

First, order the self - employed data: $38,46,51,52,53,60,72,81,96,101$. The median of the lower half ($Q_1$) is the median of $38,46,51,52,53$, which is $51$. The median of the upper half ($Q_3$) is the median of $60,72,81,96,101$, which is $81$. So $IQR_1=Q_3 - Q_1=81 - 51 = 30$.

Step5: Calculate the IQR for wage earners

Order the wage earners data: $44,51,58,62,64,65,66,81,84,89$. The median of the lower half ($Q_1$) is the median of $44,51,58,62,64$, which is $58$. The median of the upper half ($Q_3$) is the median of $65,66,81,84,89$, which is $81$. So $IQR_2=Q_3 - Q_1=81 - 58 = 23$. So the third option is false.

Step6: Calculate standard deviation (approximate comparison)

The standard deviation formula is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$. Just by looking at the spread of data, we can see that the self - employed data has more extreme values relative to its mean compared to wage earners, so the standard deviations are not equal. So the fourth option is false.

Answer:

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