3B.5.1 Applying the Central Limit Theorem Quiz: Pearson Edexcel Further Maths, Unit 14

The 20 questions

1. For a large sample of size n from a population with mean mu and variance sigma^2, what is the approximate distribution of the sample mean?

   • Approximately N(mu, sigma^2 n)
   • Exactly N(mu, sigma^2)
   • Exactly Po(mu)
   • Approximately N(mu, sigma^2/n)

2. For n independent observations each with mean mu and variance sigma^2, what is the approximate distribution of their sum?

   • N(mu/n, sigma^2/n)
   • N(n mu, sigma^2)
   • N(n mu, n sigma^2)
   • N(mu, sigma^2/n)

3. A sample of size 64 is taken from a population with standard deviation 8. What is the standard error of the sample mean?

   • 64
   • 8
   • 1
   • 0.125

4. A population has mean 50 and standard deviation 10. For n = 100, what is P(sample mean > 52) to 4 decimal places?

   • 0.0228
   • 0.1587
   • 0.0456
   • 0.9772

5. A fair die is rolled 36 times and the scores are summed. What is the expected total?

   • 126
   • 3.5
   • 105
   • 36

6. A fair die is rolled 36 times and the scores are summed. What is the variance of the total?

   • 3.5
   • 105
   • 126
   • 35

7. A Poisson distribution Po(100) is approximated by a normal distribution. Which is correct?

   • Mean 10 and variance 100
   • Mean 100 and variance 100
   • Mean 0 and variance 1
   • Mean 100 and variance 10

8. X ~ B(200, 0.5) is approximated by a normal distribution. What is P(X > 110), to 4 decimal places?

   • 0.1587
   • 0.0228
   • 0.0688
   • 0.5000

9. A sample of size 25 is taken from a population with mean 8 and variance 4. What is the variance of the sample mean?

   • 4
   • 1.6
   • 0.04
   • 0.16

10. Each of 100 independent values has mean 2 and variance 9. What is the distribution of their sum?

    • Approximately N(2, 900)
    • Approximately N(200, 90)
    • Approximately N(200, 900)
    • Approximately N(200, 9)

11. A population has mean 10 and variance 36. A sample of size 36 is taken. What is P(9 < sample mean < 11), to 3 decimal places?

    • 0.317
    • 0.683
    • 0.954
    • 0.341

12. Does the Central Limit Theorem require the population itself to be normal?

    • Yes, the population must be Poisson
    • No, a large sample size is enough
    • No, but the sample size must be less than 10
    • Yes, the population must be normal

13. 100 independent geometric values with p = 0.5 are summed. What is the approximate distribution of the sum?

    • Mean 200, variance 200
    • Mean 100, variance 100
    • Mean 200, variance 400
    • Mean 50, variance 200

14. Forty independent values each have mean 3. What is the expected sum?

    • 43
    • 13.3
    • 3
    • 120

15. A sample of 16 values is taken from a population with standard deviation 12. What is the standard error of the sample mean?

    • 12
    • 3
    • 0.75
    • 192

16. Why is the normal approximation useful in practice?

    • It gives probabilities for sums and means without the exact distribution
    • It removes the need to calculate a variance, since the normal curve supplies it
    • It turns the population into a discrete distribution that is easier to work with
    • It gives exact probabilities for any sample size, whatever n happens to be

17. For a Poisson distribution with large lambda, what is the approximate variance of the normal approximation?

    • 1/lambda
    • lambda^2
    • lambda
    • sqrt(lambda)

18. A population has mean 10 and variance 36. What is P(sample mean > mu + sigma/sqrt(n)) for n = 36, using the normal approximation, to 3 decimal places?

    • 0.159
    • 0.841
    • 0.317
    • 0.023

19. For n independent variables each with mean mu, what is the expected sum?

    • mu^n
    • n mu
    • n + mu
    • mu / n

20. For n independent variables each with variance sigma^2, what is the variance of their sum?

    • sigma^2
    • n sigma^2
    • sigma^2 / n
    • n^2 sigma^2

Related quizzes

• Proof by mathematical induction Quiz · 1.1 · 20 questions
• Quadratic equations and complex arithmetic Quiz · 2.1-2.2 · 20 questions
• Matrix arithmetic and inverses Quiz · 3.1-3.2 · 20 questions
• Expectation of discrete random variables Quiz · 3B.1.1 · 20 questions
• The Poisson distribution Quiz · 3B.2.1 · 20 questions
• Geometric and negative binomial models Quiz · 3B.3.1 · 20 questions
• Hypothesis tests for the Poisson distribution Quiz · 3B.4.1 · 20 questions
• Goodness of fit tests for discrete distributions Quiz · 3B.6.1 · 20 questions
• Definition and derivation of probability generating functions Quiz · 3B.7.1 · 20 questions
• Type I and Type II errors Quiz · 3B.8.1 · 20 questions