The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 resepectively. Each of these 10 observations is multiplied by p and then reduced by q, where p ≠ 0 and q ≠ 0. If the new mean and new s.d. become half of their original values, then q is equal to

Solution

Maths sir · 4 steps

The s.d. ignores q and only feels |p|, so fix p from the s.d. first, then the mean gives q.

1. New observations. Each xᵢ becomes yᵢ = p xᵢ - q. The mean goes through the same change, so ȳ = px̄ - q. Subtracting q does not change the spread, so σ_y = |p| σₓ.
2. Use the s.d. The new s.d. is half of 2, which is 1. So 2|p| = 1, giving |p| = 1/2. That means p = 1/2 or p = -1/2.

   20p - q = 10

3. Test both signs. The new mean is half of 20, which is 10. With p = 1/2: 10 - q = 10, so q = 0, which is not allowed. With p = -1/2: -10 - q = 10, so q = -20.
4. So. p = -1/2 and q = -20, which is option (b).

The idea to keep. Under y = px - q, the mean becomes px̄ - q but the s.d. becomes |p|σ. Shifts never touch the s.d., and the modulus means you must check both signs of p.

Answer from the NTA final answer key. Solution by PrepWiser.