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Testing for a Proportion

Averages of Random Variables

• The expected value of the average of n outcomes of a random variable:
  E(x) = E[(x₁ + ... + x_n) / n] = E[x₁ / n + ... + x_n / n]
        = E(x₁ / n) + ... + E(x_n / n) = E(x₁) / n + ... + E(x_n) / n
        = nE(x₁) / n = E(x₁)
  

  E(x) = E(x1)

• The variance of the average of n independent outcomes of a random variable:
  Var(x) = Var[(x₁ + ... + x_n) / n] = Var[x₁ / n + ... + x_n / n]
      = Var(x₁ / n) + ... + Var(x_n / n) = Var(x₁) / n² + ... + Var(x_n) / n²
      = = nVar(x) / n² = Var(x₁) / n
  

  Var(x) = Var(x1) / n

• The standard deviation of the average of n independent outcomes of a random variable:
  σ_x = √[Var(x)] = √[Var(x₁) / n] = σ_x / √n
  

  σx = σx / √n

• Recall that E(x) = E(x₁). How does σ_x compare to σ_x?
• For a Bernoulli random variable x, the average of n outcomes x = S / n of x is the proportion of successes. Recall that S is the number of successes and n is the number of trials.

Tests of Hypotheses for Proportions

• 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 = (p^{^} - p) / σ_phat
  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 1:   Flip a coin 100 times and obtain 43 heads. Is the coin fair?
  
  1. H₀: p = 0.5; H₁: p ≠ 0.5
  2. phat = S / n = 43/100 = 0.43, and σ_phat = √[np(1-p)] = √[(0.5)(1-0.5)/100] = 0.05,
     so z = (phat - p) / σ_phat = (0.43 - 0.5) / 0.05 = -1.4
  3. A 95% confidence interval for z is [-1.96, 1.96].
  4. -1.4 ∈ [-1.96, 1.96], so accept the null hypothesis. Not enough evidence to conclude that the coin is not fair.
• Example 2: To test if a four-faced die (shape of a tetrahedron) is fair, such a die is rolled 1600 times. 421 aces are obtained. Perform a test of hypothesis to test whether the die is fair.