Question 7 of 21intermediate🔧 ApplyNumerical4 marks

To find out the concentration of SO₂ in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below:

Concentration of SO₂ (in ppm)Frequency
0.00 - 0.044
0.04 - 0.089
0.08 - 0.129
0.12 - 0.162
0.16 - 0.204
0.20 - 0.242

Find the mean concentration of SO₂ in the air.

Correct Answer

The mean concentration of SO₂ in the air is 0.099 ppm.

Exercise: EXERCISE 13.1 | Q: 7 | (Page 12)
For More Understanding

Explanation

This question requires finding the mean of grouped frequency data using the Direct Method. The class mark (midpoint) of each class interval is calculated first, then multiplied by the corresponding frequency. The sum of these products divided by the total frequency gives the mean. This is a standard method shown in Table 13.3 of the textbook context.

Solution Steps

  1. Step 1: Calculate the class marks (x_i) for each class interval using the formula: Class mark = (Upper limit + Lower limit) / 2

  2. Step 2: For 0.00-0.04: x_i = (0.00+0.04)/2 = 0.02; For 0.04-0.08: x_i = 0.06; For 0.08-0.12: x_i = 0.10; For 0.12-0.16: x_i = 0.14; For 0.16-0.20: x_i = 0.18; For 0.20-0.24: x_i = 0.22

  3. Step 3: Calculate f_i × x_i for each class: 4×0.024 \times 0.02=0.08, 9×0.069 \times 0.06=0.54, 9×0.109 \times 0.10=0.90, 2×0.142 \times 0.14=0.28, 4×0.184 \times 0.18=0.72, 2×0.222 \times 0.22=0.44

  4. Step 4: Find the sums: Σf_i = 4+9+9+2+4+2 = 30 and Σf_i x_i = 0.08+0.54+0.90+0.28+0.72+0.44 = 2.96

  5. Step 5: Apply the mean formula: x̄ = Σf_i x_i / Σf_i = 2.96/30 = 0.0987 ≈\approx 0.099 ppm

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