Question

help me to solve this question..​

Answer 1

Question:

The following table gives the distribution of the life time of 400 neon lamps. Find the mode of the table.

$\begin{array}{|c|c|}\cline{1-2}\sf\:Life\:time\:(\:in\:hours\:) & \sf\:Number\:of\:Lamps\\\cline{1-2}\sf\:1500 - 2000 & \sf\:14\\\cline{1-2}\sf\:2000 - 2500 & \sf\:56\\\cline{1-2}\sf\:2500 - 3000 & \sf\:60\\\cline{1-2}\sf\:3000 - 3500 & \sf\:86\\\cline{1-2}\sf\:3500 - 4000 & \sf\:74\\\cline{1-2}\sf\:4000 - 4500 & \sf\:62\\\cline{1-2}\sf\:4500 - 5000 & \sf\:48\\\cline{1-2}\end{array}$

Answer:

The mode of the given data is 3342.11 hours (approx.).

Step-by-step-explanation:

We have given a frequency distribution table of 400 neon lamps.

$\begin{array}{|c|c|}\cline{1-2}\sf\:Life\:time\:(\:in\:hours\:) & \sf\:Number\:of\:Lamps\\\cline{1-2}\sf\:1500 - 2000 & \sf\:14\\\cline{1-2}\sf\:2000 - 2500 & \sf\:56\\\cline{1-2}\sf\:2500 - 3000 & \sf\:60\:\rightarrow\:f_0\\\cline{1-2}\boxed{\sf\:3000 - 3500} & \boxed{\sf\:86\:\rightarrow\:f_1}\\\cline{1-2}\sf\:3500 - 4000 & \sf\:74\:\rightarrow\:f_2\\\cline{1-2}\sf\:4000 - 4500 & \sf\:62\\\cline{1-2}\sf\:4500 - 5000 & \sf\:48\\\cline{1-2}\end{array}$

Now, here, the maximum frequency is 86.

Hence, the modal class is 3000 - 3500.

$\bullet\sf\:Lower\:class\:limit\:of\:modal\:class\:(\:L\:)\:=\:3000\\\\\\\bullet\sf\:Frequency\:of\:modal\:class\:(\:f_1\:)\:=\:86\\\\\\\bullet\sf\:Frequency\:of\:the\:class\:preceiding\:the\:modal\:class\:(\:f_0\:)\:=\:60\\\\\\\bullet\sf\:Frequency\:of\:the\:class\:succeeding\:the\:modal\:class\:(\:f_2\:)\:=\:74\\\\\\\bullet\sf\:Class\:interval\:of\:the\:modal\:class\:(\:h\:)\:=\:500$

Now, we know that,

$\pink{\sf\:Mode\:=\:L\:+\:\bigg[\:\dfrac{f_1\:-\:f_0}{2f_1\:-\:f_0\:-\:f_2}\:\bigg]\:\times\:h}\sf\:\:\:-\:-\:[\:Formula\:]\\\\\\\implies\sf\:Mode\:=\:3000\:+\:\bigg[\:\dfrac{86\:-\:60}{2\:\times\:86\:-\:60\:-\:74}\:\bigg]\:\times\:500\\\\\\\implies\sf\:Mode\:=\:3000\:+\:\bigg[\:\dfrac{26}{172\:-\:134}\:\bigg]\:\times\:500\\\\\\\implies\sf\:Mode\:=\:3000\:+\:\dfrac{26}{\cancel{38}}\:\times\:500\\\\\\\implies\sf\:Mode\:=\:3000\:+\:\dfrac{26}{19}\:\times\:250\\\\\\\implies\sf\:Mode\:=\:3000\:+\:\dfrac{6500}{19}\\\\\\\implies\sf\:Mode\:=\:\dfrac{3000\:\times\:19\:+\:6500}{19}\\\\\\\implies\sf\:Mode\:=\:\dfrac{57000\:+\:6500}{19}\\\\\\\implies\sf\:Mode\:=\:\cancel{\dfrac{63500}{19}}\\\\\\\implies\sf\:Mode\:=\:3342.105\\\\\\\implies\boxed{\red{\sf\:Mode\:\approx\:3342.11\:hours}}$

The mode of the given data is 3342.11 hours (approx.).

Answer 2

Life time (in hours) Number of lamps Cumulative Frequency(cf)

1500-2000 14 14

2000-2500 56 14+56 = 70

2500-3000 60 70+60=130

3000-3500 86 130+86=216

3500-4000 74 216+74=290

4000-4500 62 290+62=352

4500-5000 48 352+48=400

∑fi = 400

Mode= l+\frac{\frac{n}{2}-cf }{f}*hMode=l+

f

2

n

−cf

∗h

Where, l =lower limit of mode class=3000

h=class interval=5000-4500=500

n =∑fi

f=frequency of the mode class=86

cf = cumulative frequency before mode class=130

Mode= 3000+\frac{\frac{400}{2}-130 }{86}*500Mode=3000+

86

2

400

−130

∗500

Mode =3406.98