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LARGE SAMPLE TESTS ON PROPORTIONS Binomial experiment with n trials and unknown proportion of successes p. GOAL: Test Ho: p=p0, where p0 is a specified, known value. DATA: Observe x successes, get sample proportion of successes x pˆ = . n GOAL: Test Ho: p=p0, where p0 is a specified, known value. Use the fact (Fact 4) that statistic z= pˆ − p0 p0 (1 − p0 ) n has approximately N(0, 1) distribution under Ho (when p=p0) for large n. TEST FOR PROPORTION: PROCEDURE Let significance level =α. STEP 1. Ho: p = po vs Ha: p ≠ po or STEP 2. Compute the test statistic: z = Ha: p > po or Ha: p < po pˆ − p0 p0 (1 − p0 ) n STEP 3. Find the critical number in the Z table. Two sided alternative critical value = zα/2. One sided alternative critical value = zα. STEP 4. DECISION. Ha: p ≠ po Reject Ho if |z|> zα/2; Ha: p > po Reject Ho if z > zα; Ha: p < po Reject Ho if z < - zα. STEP 5. Answer the question in the problem. P-value approach STEP 3. Compute the approximate p-value. Two sided test p-value: Ha: p ≠ po, approximate P-value: 2P( Z > |z|) One sided tests p-values: Ha: p > po, approximate P-value: P( Z > z) Ha: p < po, approximate P-value: P( Z < z) STEP 4. DECISION Reject Ho if p-value < significance level α. STEP 5. Answer the question in the problem. EXAMPLE Example. When a coin is tossed 100 times, we get 60 Heads. Test if the coin is fair versus the alternative that it is loaded in favor of Heads, using significance level of 5%. Solution. X=number of H in 100 tosses of a coin; n= 100, x=60, α = 5%. p= true probability of H on the coin, X ~Bin(100, p). STEP1. Ho: p=0.5 STEP 2. Test statistic Ha: p > 0.5 pˆ − po z= = po (1 − po ) n 0.6 − 0.5 = 2. 0.5(1 − 0.5) 100 STEP 3. Critical value? z0.05 =1.645. STEP 4. DECISION: z = 2 > 1.645 = z0.05 STEP 5. The coin favors heads. reject Ho. EXAMPLE contd. P-value approach STEP 3. To get p-value, we need the distribution of pˆ . By the Normal approximation to Binomial, under Ho, p̂ has an approximately Normal distribution with mean po =0.5 and st. deviation 0.5(1 − 0.5) = 0.05. 100 P-value = P ( pˆ > 0.6) ≈ P ( Z > 2) = 0.0228. STEP 4. DECISION: P-value = 0.0228 < α=0.05 STEP 5. The coin favors Heads. reject Ho.