1. 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
Use the fact (Fact 4) that statistic
has approximately N(0, 1) distribution under Ho (when p=p0) for large n.

2. TEST FOR PROPORTION: PROCEDURE
Let significance level =α.
STEP 1. Ho: p = po vs Ha: p ≠ po or Ha: p > po or Ha: p < po
STEP 2. Compute the test statistic:
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.

3. 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.

4. 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= Ha: p > 0.5
STEP 2. Test statistic
STEP 3. Critical value? z0.05 =1.645.
STEP 4. DECISION: z = 2 > = z reject Ho.
STEP 5. The coin favors heads.

5. EXAMPLE contd. P-value approach
STEP 3. To get p-value, we need the distribution of
By the Normal approximation to Binomial, under Ho, has an
approximately Normal distribution with mean po =0.5 and
st. deviation
P-value =
STEP 4. DECISION: P-value = < α= reject Ho.
STEP 5. The coin favors Heads.