**The following table shows the income distribution of 600 families. Find the minimum income of the riches 30% families. Also the limits of income of middle 50% of families, to the nearest rupees. Income Below 75 75- 150 150- 225 225- 300 300- 375 375- 400 400 & above No. of families 69 137 225 46 88 25 10 Ans.: the richest 30 % families earns Rs. 222 and above per week , the middle 50% families weekly income lies between 120 and 256.**

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Income 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

$a)$

To find the minimum income of the richest 30% families, we find the 70th percentile as follows.

$P_{70}=l+({70\times n\over100}-cf)\times{c\over f}$ where, $n=600$

$l$ 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 containing $P_{70}$

$P_{70}$ is in the $({70\times600\over100})^{th} =420^{th}$ position. Therefore, it lies in the class, 150-225

Thus,

$P_{70}=150+(420-206)\times{75\over225}=150+71.33=221.33\approx 222$ ( to the nearest rupees)

Therefore, the richest 30% families earns Rs. 222 and above per week

$b)$

The middle 50% families lies between the 25th and the 75th percentile. To find the range of their income, we find $P_{25}$ as the lower limit and $P_{75}$ as the upper limit.

Now,

$P_{25}=l+({25\times n\over100}-cf)\times{c\over f}$ where, $n=600$

$l$ is the lower class boundary of the class containing $P_{25}$

$f$ is the frequency of the class containing $P_{25}$

$c$ is the width of the class containing $P_{25}$

$cf$ is the cumulative frequency of the class preceding the class containing $P_{25}$

$P_{25}$ is in the $({25\times600\over100})^{th} =150^{th}$ position. Therefore, it lies in the class, 75-150

Thus,

$P_{25}=75+(150-69)\times{75\over137}=75+44.34=119.34\approx 120$ ( to the nearest rupees)

and,

$P_{75}=l+({75\times n\over100}-cf)\times{c\over f}$ where, $n=600$

$l$ is the lower class boundary of the class containing $P_{75}$

$f$ is the frequency of the class containing $P_{75}$

$c$ is the width of the class containing $P_{75}$

$cf$ is the cumulative frequency of the class preceding the class containing $P_{75}$

$P_{75}$ is in the $({75\times600\over100})^{th} =450^{th}$ position. Therefore, it lies in the class, 225-300

Thus,

$P_{75}=225+(450-431)\times{75\over46}=225+=255.98\approx 256$ ( to the nearest rupees)

Therefore, the middle 50% families weekly income lies between 120 and 256.