Normal approximation to binomial Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Normal approximation to binomial questions. See exactly how to solve problems on normal-approximation, mean-variance, conditions, model-choice.

normal-approximationmean-varianceconditionsmodel-choicecontinuity-correctionprobability
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
The random variable X∼B(40, 0.5)X\sim B(40,\ 0.5) is approximated by a normal distribution. Find the mean of that normal distribution.

Worked solution

  1. Identify the binomial parameters

    n=40, p=0.5n=40,\ p=0.5

    Read off the number of trials and the success probability.

  2. Apply the binomial mean and variance

    μ=np=20, σ2=np(1−p)=10\mu=np=20,\ \sigma^2=np(1-p)=10

    The normal approximation matches these two moments.

  3. State the mean

    μ=20\mu=20

    This is the mean of the approximating normal distribution.

Answer
2020
Question 2
2 markseasy
The random variable X∼B(40, 0.5)X\sim B(40,\ 0.5) is approximated by a normal distribution. Find the variance of that normal distribution.

Worked solution

  1. Identify the binomial parameters

    n=40, p=0.5n=40,\ p=0.5

    Read off the number of trials and the success probability.

  2. Apply the binomial mean and variance

    μ=np=20, σ2=np(1−p)=10\mu=np=20,\ \sigma^2=np(1-p)=10

    The normal approximation matches these two moments.

  3. State the variance

    σ2=10\sigma^2=10

    This is the variance of the approximating normal distribution.

Answer
1010
Question 3
2 markseasy
The random variable X∼B(50, 0.5)X\sim B(50,\ 0.5) is approximated by a normal distribution. Find the mean of that normal distribution.

Worked solution

  1. Identify the binomial parameters

    n=50, p=0.5n=50,\ p=0.5

    Read off the number of trials and the success probability.

  2. Apply the binomial mean and variance

    μ=np=25, σ2=np(1−p)=12.5\mu=np=25,\ \sigma^2=np(1-p)=12.5

    The normal approximation matches these two moments.

  3. State the mean

    μ=25\mu=25

    This is the mean of the approximating normal distribution.

Answer
2525
Question 4
2 markseasy
The random variable X∼B(50, 0.4)X\sim B(50,\ 0.4) is approximated by a normal distribution. Find the variance of that normal distribution.

Worked solution

  1. Identify the binomial parameters

    n=50, p=0.4n=50,\ p=0.4

    Read off the number of trials and the success probability.

  2. Apply the binomial mean and variance

    μ=np=20, σ2=np(1−p)=12\mu=np=20,\ \sigma^2=np(1-p)=12

    The normal approximation matches these two moments.

  3. State the variance

    σ2=12\sigma^2=12

    This is the variance of the approximating normal distribution.

Answer
1212
Question 5
2 markseasy
The random variable X∼B(100, 0.5)X\sim B(100,\ 0.5) is approximated by a normal distribution. Find the mean of that normal distribution.

Worked solution

  1. Identify the binomial parameters

    n=100, p=0.5n=100,\ p=0.5

    Read off the number of trials and the success probability.

  2. Apply the binomial mean and variance

    μ=np=50, σ2=np(1−p)=25\mu=np=50,\ \sigma^2=np(1-p)=25

    The normal approximation matches these two moments.

  3. State the mean

    μ=50\mu=50

    This is the mean of the approximating normal distribution.

Answer
5050

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