Lesson Worksheet: Approximating a Binomial Distribution Mathematics

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In this worksheet, we will practice approximating a binomial distribution with a normal distribution.

Q1:

Which of the following four binomially distributed random variables may be approximated by a normal distribution?

  1. 𝑋∼𝐵(98,0.42)
  2. 𝑋∼𝐵(7,0.55)
  3. 𝑋∼𝐵(700,0.75)
  4. 𝑋∼𝐵(200,0.59)
  • Aa and b
  • Ba and c
  • Cb and c
  • Db and d
  • Ea and d

Q2:

𝑋 is a binomially distributed random variable, and 𝑋∼𝐵(250,0.47). Write down a normal approximation of 𝑋∼𝑁𝜇,𝜎, stating the values of 𝜇 and 𝜎.

  • A𝑋∼𝑁(55.225,117.5)
  • B𝑋∼𝑁(250,62.275)
  • C𝑋∼𝑁(62.275,117.5)
  • D𝑋∼𝑁(117.5,62.275)
  • E𝑋∼𝑁(117.5,55.225)

Q3:

A discrete random variable 𝑋 is binomially distributed, and 𝑋∼𝐵(200,0.47). Using a normal approximation, estimate 𝑃(𝑋≤96).

Q4:

A discrete random variable 𝑋 is binomially distributed, and 𝑋∼𝐵(200,0.47). Using a normal approximation, estimate 𝑃(𝑋>80).

Q5:

A discrete random variable 𝑋 is binomially distributed, and 𝑋∼𝐵(175,0.56). Using a normal approximation, estimate 𝑃(100≤𝑋<110).

Q6:

A discrete random variable 𝑋 is binomially distributed, and 𝑋∼𝐵(175,0.56). Using a normal approximation, estimate 𝑃(𝑋=106).

Q7:

Matthew flips a fair coin 84 times. By using a suitable normal approximation, calculate the estimated probability that the coin lands on heads more than 50 times.

Q8:

A farm produces eggs. The farmer claims that 60% of the eggs weigh more than 64 grams. A random sample of 20 eggs is taken. Find the exact probability that at least 15 of the eggs weigh more than 64 grams.

A random sample of 1,000 eggs is taken. Use a normal approximation to estimate the probability that more than 550 and less than 600 eggs weigh more than 64 grams.

Q9:

Some seeds are planted. The probability that a particular seed will germinate after 4 days is 58%. Of a random sample of 10 seeds, find the exact probability that 7 of the seeds will germinate after 4 days.

A normal approximation predicts a probability of 0.65 that at least 𝑘 seeds will germinate from a random sample of 300 seeds. Find the value of 𝑘.

  • A171
  • B161
  • C181
  • D195
  • E105

Q10:

A population of frogs contains female and male frogs in the ratio 53∶47. A random sample of 100 frogs is taken. Find the percentage error when using a normal approximation to calculate the probability that exactly 60 of the frogs are female.

  • A0.34%
  • B0.56%
  • C0.23%
  • D0.17%
  • E0.68%

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