  Question

The following table gives the distribution of the life time of 400 neon lamps: Life time (in hours)Number of lamps$$1500-2000$$$$2000-2500$$$$2500-3000$$$$3000-3500$$$$3500-4000$$$$4000-4500$$$$4500-5000$$$$14$$$$56$$$$60$$$$86$$$$74$$$$62$$$$48$$Find the median life time of a lamp.

Solution

Based on the given information, we can prepare the table shown above.Here, we have, $$n = 400$$$$\Rightarrow \dfrac n2 = 200$$The cumulative frequency just greater than $$\dfrac n2$$ is $$216$$ and the corresponding class is $$3000 – 3500$$.Thus, $$3000 – 3500$$ is the median class such that$$\dfrac n2 = 200, l = 3000, cf = 130, f = 86$$, and $$h = 500$$We know that,Median, $$M = l+\left(\dfrac{\dfrac n2 - cf}{f}\right)\times h$$Where, $$l\rightarrow$$ lower limit of the median class            $$n\rightarrow$$ total number of observations $$(\sum f)$$          $$cf\rightarrow$$ cumulative frequency of the class preceding the median class            $$f\rightarrow$$ frequency of the median class            $$h\rightarrow$$ class widthSubstituting the corresponding values in the formula, we get:$$M = 3000+\left(\dfrac{200-130}{86}\right)\times 500$$$$\Rightarrow M = 3000+\dfrac{70}{86}\times500 = 3000+406.98 = 3406.98$$       Hence, median life time of a lamp is $$3406.98$$ hours. MathematicsRS AgarwalStandard X

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