Question

The average monthly expense of students residing in university hostel is ₹ 2000 and its standard deviation is ₹ 500. If the monthly expense of a student follows normal distribution, then

(1) Find percentage of students having expense between ₹ 750 and ₹ 1250.

(2) Find percentage of students having expense more than ₹ 1800.

(3) Find percentage of students having expense less than ₹ 2400.

Answer 1

Here, µ = 2000; σ = 500; X = Average monthly expense (in ₹).

(1) The percentage of students having expense between ₹ 750 and ₹ 1250 :

P[750 ≤ X ≤ 1250] = ?
For x₁ = 750,

∴ P[750 ≤ X ≤ 1250]
= P[- 2.5 ≤ Z ≤ – 1.5]
= P[- 2.5 ≤ Z ≤ 0] – P[- 1.5 ≤ Z ≤ 0]
= P[0 ≤ Z ≤ 2.5] – P[0 ≤ Z ≤ 1.5]
= 0.4938 – 0.4332
= 0.0606

Hence, the percentage of students having expense between ₹ 750 and ₹ 1250 = 0.0606 × 100 = 6.06%.

(2) The percentage of students having expense more than ₹ 1800:

P[X ≥ 1800] = ?
For x₁ = 1800,

∴ P[X ≥ 1800] = P[Z ≥ – 0.4]
= P[- 0.4 ≤ Z ≤ 0] + P[0 ≤ Z < ∞]
= P[0 ≤ Z ≤ 0.4] + P[0 ≤ Z < ∞]
= 0.1554 + 0.5000
= 0.6554

Hence, the percentage of students having expense more than ₹ 1800 = 0.6554 × 100 = 65.54%.

(3) The percentage of students having expense less than ₹ 2400:

P[X ≤ 2400] = ?
For x₁ = 2400,

∴ P[X ≤ 2400] = P[Z ≤ 0.8]
= P[- ∞ < Z ≤ 0] + P[0 ≤ Z =≤ 0.8]
= P[0 ≤ Z < ∞] + P[0 ≤ Z ≤ 0.8]
= 0.5000 + 0.2881
= 0.7881

Hence, the percentage of students having expense less than ₹ 2400
= 0.7881 × 100 = 78.81%.