Question

The following table gives a frequency distribution:

The mean and variance of the above data are \( \mu \) and 19 respectively, where \( \mu \) is an integer. Find the value of \( (\lambda + \mu) \):

• A. 19
• B. 20
• C. 13
• D. 17

Hint:

For grouped data, always use class midpoints while calculating mean and variance.

Answer 1

The Correct Option is A

Step 1: Find class marks.
The class midpoints are: \[ 6,\; 10,\; 14,\; 18 \]
Step 2: Compute total frequency.
\[ N = 3 + \lambda + 4 + 7 = \lambda + 14 \]
Step 3: Use the formula for mean.
Mean is given by \[ \mu = \frac{\sum f x}{\sum f} \] \[ \mu = \frac{3(6) + \lambda(10) + 4(14) + 7(18)}{\lambda + 14} \] \[ \mu = \frac{200 + 10\lambda}{\lambda + 14} \]
Step 4: Use the variance formula.
Variance is given as 19: \[ \frac{\sum f(x-\mu)^2}{\sum f} = 19 \] Substituting values and simplifying gives \[ \lambda = 6,\quad \mu = 13 \]
Step 5: Final Answer.
\[ \lambda + \mu = 6 + 13 = 19 \]