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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 width

Substituting 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.

498671_465497_ans.png

Mathematics
RS Agarwal
Standard X

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