Question 5

We need to find the median life time from the given data.

Step 1 — Calculate Cumulative Frequency

Let's add a cumulative frequency column. We sum frequencies to get cumulative frequency. The total number of lamps is $N$.

| Life time (in hours) | Number of lamps (f) | Cumulative Frequency (cf) | | :------------------- | :------------------ | :------------------------ | | 1500 - 2000 | 14 | 14 | | 2000 - 2500 | 56 | $14 + 56 = \textbf{70}$ | | 2500 - 3000 | 60 | $70 + 60 = \textbf{130}$ | | 3000 - 3500 | 86 | $130 + 86 = \textbf{216}$ | | 3500 - 4000 | 74 | $216 + 74 = \textbf{290}$ | | 4000 - 4500 | 62 | $290 + 62 = \textbf{352}$ | | 4500 - 5000 | 48 | $352 + 48 = \textbf{400}$ |

The total number of lamps is $\textbf{400}$.

$N = 400$

$\boxed{N = 400}$

Step 2 — Find Median Class

We find $N/2$. This helps us locate the median class. The cumulative frequency just greater than $N/2$ gives the median class.

$N/2 = 400/2$

$= 200$

The cumulative frequency just greater than 200 is 216. This corresponds to the class interval 3000 - 3500.

$\boxed{\text{Median class} = 3000 - 3500}$

Step 3 — Calculate Median

We use the median formula for grouped data. The formula is: Median $= l + \left(\frac{\frac{N}{2} - cf}{f}\right) \times h$. Here, $l$ is the lower limit of the median class. $cf$ is the cumulative frequency of the class before the median class. $f$ is the frequency of the median class. $h$ is the class size.

From the median class 3000 - 3500: $l = \textbf{3000}$ $N/2 = \textbf{200}$ $cf = \textbf{130}$ (from the class 2500-3000) $f = \textbf{86}$ $h = 3500 - 3000 = \textbf{500}$

Now, let's substitute these values into the formula.

$\text{Median} = 3000 + \left(\frac{200 - 130}{86}\right) \times 500$

$= 3000 + \left(\frac{70}{86}\right) \times 500$

$= 3000 + \frac{35000}{86}$

$= 3000 + 406.9767...$

$= 3406.9767...$

Rounding to two decimal places.

$\boxed{3406.98 \text{ hours}}$

Answer

The median life time of a lamp is 3406.98 hours.