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Normal: IOE marks of 5000 students

Short answer

Standardize: z=(x−50)/4. For marks<40, z=−2.5, probability 0.0062, so about 31 students. For marks between 35 and 60, z=−3.75 and 2.5, probability 0.9937, so about 4969 students.

The marks obtained by IOE students in statistics are 50 on average with variance 16. If 5000 students have given the exam, find the following: a) The number of students securing marks less than 40? b) The number of students securing marks between 35 to 60?

Marks X ~ N(μ = 50, σ² = 16), so σ = 4. Number of students N = 5000.

a) Marks less than 40

z = (40 − 50)/4 = −2.50

P(X < 40) = P(Z < −2.50) = 0.5 − 0.4938 = 0.0062

Number = 5000 × 0.0062 = 31 students.

b) Marks between 35 and 60

  • z₁ = (35 − 50)/4 = −3.75
  • z₂ = (60 − 50)/4 = 2.50

P(35 < X < 60) = P(−3.75 < Z < 2.50) = P(0 < Z < 3.75) + P(0 < Z < 2.50) = 0.4999 + 0.4938 = 0.9937

Number = 5000 × 0.9937 = 4969 students.

Answer: a) about 31 students b) about 4969 students.

✓ Answer checked by Claude Sonnet 5.5

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