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## Here's the Solution to this Question

Class f cf

0-75 69 69

75-150 137 206

150-225 225 431

225-300 46 477

300-375 88 565

375-400 25 590

400+ 10 600

To find the minimum income of the riches 30% families, we determine the $70^{th}$ percentile given as,

$P_{70}=l+({70\times n\over 100}-cf)\times{c\over f}$ where, $n=600$ and

$l$ is the lower class boundary of the class containing $P_{70}$

$f$ is the frequency of the class containing $P_{70}$

$c$ is the width of the class containing $P_{70}$

$cf$ is the cumulative frequency of the class preceding the class with $P_{70}$.

The class containing $P_{70}$ is,

$({70\times n\over 100})^{th} class=({70\times 600\over 100})=420$. Therefore the class with $P_{70}$ is 150-225

Thus,

$P_{70}=150+(420-206)\times {75\over 225}=150+71.33=221.33\approx 222$

The limits of income of middle 50% of families is same as determining $Q_1$ and $Q_3$ where,

$Q_1$ is the lower limit and $Q_3$ is the upper limit.

Therefore,

$Q_1=l+({n\over4}-cf)\times {c\over f}$ where,

$l$ is the lower class boundary of the class containing $Q_1$

$f$ is the frequency of the class containing $Q_1$

$c$ is the width of the class containing $Q_1$

$cf$ is the cumulative frequency of the class preceding the class with $Q_1$.

The class containing $Q_1$ is,

$({n\over 4})^{th} class={600\over 4}=150$.Therefore, the class with $Q_1$ is 75-150

Thus,

$Q_1=75+(150-69)\times {75\over 137}=119.34\approx 120$

and,

$Q_3=l+({3n\over4}-cf)\times {c\over f}$ where,

$l$ is the lower class boundary of the class containing $Q_3$

$f$ is the frequency of the class containing $Q_3$

$c$ is the width of the class containing $Q_3$

$cf$ is the cumulative frequency of the class preceding the class with $Q_3$.

The class containing $Q_1$ is,

$({3n\over 4})^{th} class={1800\over 4}=450$.Therefore, the class with $Q_3$ is 150-225

Thus,

$Q_3=225+(450-431)\times {75\over 46}=255.98\approx 256$

Therefore, the limits of income of middle 50% of families is 120 and 256.