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IT 223 -- 2/28/11

Review Questions

1. Suppose that S = x₁ + ... + x_n, where the x₁ are independent, all with expected value E(x) and standard deviation σ_x.
    
   
   1. What is E(S)?

      Ans: np.

   2. What is σ_S?

   Ans: sqrt(np(1 - p))

2. You flip a coin ten times and obtain 3 heads. How many ways are there of obtaining 3 heads out of 10?

   Ans: 10C3 = 10! / (3! (10-3)! = (10*9*8) / (3*2*1) = 5*3*8 = 120.

3. What is Pascal's Triangle? How can you use it to solve Problem 2?

   Ans: See the 2/24 Notes.

4. A major league batter has a batting average of 0.360.
   1. What are his chances of getting exactly 6 out of 12 hits in a game?

   2. What are his chances of getting 8 or more hits out of 12 in a game?

5. What does the Law of Large (LLN) numbers say?

   Ans: That S / n will get closer and closer to E(S)/n with high probability as n gets large.

6. Who first proved the LLN?

   Ans: Bernoulli in Switzerlans.

7. What does it mean to say that LLN works by swamping?

   Ans: That even if S/n is not very close to E(S)/n, if enough extra outcomes are obtained, they will swamp the outcomes previously obtained, so S/n will then be close to E(S)/n, with high probability.

8. What does the Central Limit Theorem (CLT) say?

   Ans: That even if x is not normally distributed, S will be approximately normally distributed if n is large.

9. Who first stated the CLT?

   Ans: DeMoivre from France.

10. Who first proved the CLT in its modern form?

    Ans: Lyapunov, from Russia.

11. Fifty four percent of 1000 voters surveyed in an exit survey say that they voted for Candidate A. Assuming that the voters that participated in the survey are telling the truth, and assuming that the sample is representative of all voters, give a 95% confidence interval for the true probability that a randomly chosen voter will vote for Candidate A.

    By the CLT, the sum S of independent random variables is approximately normally distributed, even if the original random variables are not. Therefore, if z = (S - E(S)) / σ_S is standard normal and a 95% confidence interval for z is [-2, 2]. (A more accurate confidence interval is [-1.96, 1.96].)

    Ans: n = 1000, S = 540, p^ = S / n = SE(S) = sqrt(1000 * p^ * (1-p^)) = sqrt(1000 * 0.54 * (1-0.54)) = 15.76. Then the confidence interval for the true value of p is obtained by solving for p in the following:

    -2 ≤ (S - np) / SE(S) ≤ 2
    -2 ≤ (540 - 1000p) / 15.76 ≤ 2
    -32.52 ≤ 540 - 1000p ≤ 31.52
    -571.52 ≤ -1000p ≤ -508.48
    0.57152 ≥ p ≥ 0.50848

    The confidence interval is (0.50848, 0.57152) or (50.8%, 57.2%).

Tests of Hypotheses

• A test of hypothesis can be used to determine if a coin or die is fair.

• For testing if a coin is fair, we test the null hypothesis H₀: p = 0.5 against the alternative hypothesis H₁: p ≠ 0.5.

• Here are the four steps for testing if the probability of success p for a Bernoulli random variable is equal to p₀:
  1. Write down the null and alternative hypothesis:
     H₀: p = p₀
     H₁: p ≠ p₀.

  2. Write down the test statistic, assuming the null hypothesis:
     z = (S - E(S)) / SE_S
  3. Write down a 95% confidence interval for z:
     I = [-1.96, 1.96]

  4. If z ∉ I, reject H₀; if z ∈ I, accept H₁.

• Example:   You flip a coin 20 times and obtain 8 heads. Is the coin fair?
  1. H₀: p = 0.5; H₁: p ≠ 0.5

  2. z = (S - E(S)) / SE_S = z = (S - np) / √p(1 - p) = (8 - 12*0.5) / √0.5 (1 - 0.5) = -0.894.

  3. A 95% confidence interval for z is [-2, 2].

  4. -0.894 ∈ [-2, 2], so accept the null hypothesis. Not enough evidence to conclude that the coin is not fair.

• Example:   Use SPSS to simulate 500 flips of a fair coin. Perform a test of hypotheses to see if the random number generator is fair. In addition to using SPSS to generate S with Rv.Binom, also use it to compute SE for S and z as well.

Project 4

• Finish discussing Project 4.

The Central Limit Theorem

• The Central Limit Theorem (CLT) was first postulated by Abraham de Moivre in 1733 to estimate probabilities of the number of heads resulting from many tosses of a fair coin.

• In 1901, the Russian mathematician Alexandr Lyapunov stated and proved the CLT in its modern form:
  If x₁, ... , x_n are independent observations from the same random variable x of any distribution and n is large enough, then the sum
  S = x₁ + ... + x_n

  is approximately normally distributed with expected value E(S) = nE(x) and standard deviation SE_S = σ_x√n.

• For our purposes, "n is large enough" if n ≥ 30.

Practice Problems

1. Use the CLT to estimate the following probabilities for the number of heads obtained from a fair coin:
   1. Obtaining 13 to 16 heads out of 25 tosses. (Because the normal table is continuous, use 12.5 to 16.5 tosses.)

      Ans: SE_S = √np(1-p) = √25(0.5)(1-0.5) = 2.5. Then z₁ = (S - np) / SE_S = (12.5 - 25(0.5)) / 2.5 = 0, z₂ = (S - np) / SE_S = (16.5 - 25(0.5)) / 2.5 = 1.6; the area under the normal curve for the bin [0,1.6] is 0.9452 - 0.5000 - 0.4452 = 44.5%.

   2. Obtaining between 60 to 75 heads out of 100 tosses. (Use 59.5 to 75.5 tosses.)

      Ans: SE_S = √np(1-p) = √100(0.5)(1-0.5) = 5. Then z₁ = (S - np) / SE_S = (59.5 - 100(0.5)) / 5 = 1.9, 1.9, 1.9, 1.9, 1.9, 1.9, 1.9, 1.9, 1.9, z₂ = (S - np) / SE_S = (75.5 - 100(0.5)) / 5 = 5.1; the area under the normal curve for the bin [1.9,5.1] is 1 - 0.9713 = 0.0287

   3. Obtaining exactly 30 heads out of 60 tosses. (Use 29.5 to 30.5 tosses.)

      Ans: SE_S = √np(1-p) = √60(0.5)(1-0.5) = 3.87. Then z₁ = (S - np) / SE_S = (29.5 - 60(0.5)) / 3.87 = -0.129, z₂ = (S - np) / SE_S = (30.5 - 60(0.5)) / 3.87 = 0.129; the area under the normal curve for the bin [-0.129, 0.129] is 0.5517 - 0.4880 = 0.0637 = 6%