1. In the following frequency distribution, if the arithmetic mean is 45.6, find out the missing frequency: Wages (₹): 10-20, 20-30, 30-40, 40-50, 50-60, 60-70 Number of Workers: 5... 1. In the following frequency distribution, if the arithmetic mean is 45.6, find out the missing frequency: Wages (₹): 10-20, 20-30, 30-40, 40-50, 50-60, 60-70 Number of Workers: 5, 6, 7, X, 4, 3 2. Calculate the weighted mean of the following data: Items: 96, 102, 104, 124, 148, 164 Weight: 5, 6, 3, 7, 9, 2 Read more

Steps to Solve

1. Find the midpoints of each wage interval

The midpoints are calculated as the average of the lower and upper limits of each interval.

$10-20: (10+20)/2 = 15$ $20-30: (20+30)/2 = 25$ $30-40: (30+40)/2 = 35$ $40-50: (40+50)/2 = 45$ $50-60: (50+60)/2 = 55$ $60-70: (60+70)/2 = 65$

2. Calculate the sum of the product of the midpoints and their corresponding frequencies

Let $x$ denote the missing frequency

Sum = $(15 \times 5) + (25 \times 6) + (35 \times 7) + (45 \times x) + (55 \times 4) + (65 \times 3)$ Sum = $75 + 150 + 245 + 45x + 220 + 195$ Sum = $885 + 45x$

3. Calculate the total frequency

Total frequency = $5 + 6 + 7 + x + 4 + 3 = 25 + x$

4. Use the formula for the arithmetic mean to find the missing frequency

Arithmetic mean = $\frac{\text{Sum of (midpoint $\times$ frequency)}}{\text{Total frequency}}$

We are given that the arithmetic mean is 45.6. Therefore,

$45.6 = \frac{885 + 45x}{25 + x}$

5. Solve for x

$45.6(25 + x) = 885 + 45x$ $1140 + 45.6x = 885 + 45x$ $45.6x - 45x = 885 - 1140$ $0.6x = -255$ $0.6x = -255$ $x = \frac{255}{0.6} $ $x = 425$

There appears to be a sign error, let's check the calculations carefully. $45.6(25 + x) = 885 + 45x$ $1140 + 45.6x = 885 + 45x$ $45.6x - 45x = 885 - 1140$ $0.6x = -255$ Something's amiss as frequency count can't be negative. Let's rework step 5, paying close attention to signs. However, it turns out that I made an error when subtracting. Should have been: $1140 - 885 = 45x - 45.6x$ $255 = -0.6x$ $x = -425$ Still doesn't make sense. I have found the correct numbers in steps 2 and 3, let me re-verify this. $15 \times 5 = 75$ $25 \times 6 = 150$ $35 \times 7 = 245$ $55 \times 4 = 220$ $65 \times 3 = 195$ $75 + 150 + 245 + 220 + 195 = 885$ ok $5 + 6 + 7 + 4 + 3 = 25$ ok But the equation $45.6 = \frac{885 + 45x}{25 + x}$ should be correct. Let's verify the arithmetic very carefully: $45.6 \times 25 = 1140$ $1140 + 45.6x = 885 + 45x$ $45.6x - 45x = 885 - 1140$ Hmm, error here ? Should be $1140 - 885$ $0.6x = -255$ Nope, should still be on $ RHS$ i.e. RHS - LHS

It's been pointed out that 45.6(25 + x) = 45.6x + 45.6 * 25 = 45.6x + 1140 so 45.6x + 1140 = 885 + 45x => 0.6x = -255 => x = -425 This can't be!

But 45 is indeed the central midpoint Wage is Rupees 10-20, 20-30, 30-40, 40-50, 50-60, 60-70 Frequency (Workers) is 5, 6, 7, x, 4, 3

Maybe there is an typo? Since this cannot be negative.

Let's re-arrange to $45.6 \times (25 + x) = (15 \times 5) + (25 \times 6) + (35 \times 7) + (45 \times x) + (55 \times 4) + (65 \times 3)$ Let's assume it's the case, what values would $x$ take? Assuming it's a positive integer? $45.6 \times (25 + x) = 75 + 150 + 245 + 45x + 220 + 195 $ $1140 + 45.6x = 885 + 45x$ $0.6x = -255 $ still same $x = -425 $ still same. Definitely mistake!

Okay, let's start problem 20

1. Weighted Mean Formula

The weighted mean is calculated as: $\text{Weighted Mean} = \frac{\sum (\text{Weight} \times \text{Value})}{\sum \text{Weight}}$

2. Apply the formula to the data Multiply each item's value by its weight and sum these products. $96 \times 5 = 480$ $102 \times 6 = 612$ $104 \times 3 = 312$ $124 \times 7 = 868$ $148 \times 1 = 148$ $164 \times 9 = 1476$ Sum of (Weight $\times$ Value) = $480 + 612 + 312 + 868 + 148 + 1476 = 3896$

3. Calculate the sum of the weights Sum of weights = $5 + 6 + 3 + 7 + 1 + 9 = 31$

4. Compute the weighted mean

$\text{Weighted Mean} = \frac{3896}{31} = 125.677 \approx 125.68$