Hypothesis Testing for One Proportion p (one sample)
1. State Hypotheses:
H0 : p = p 0
Ha : p > p0
Ha : p < p0
Ha : p 6= p0
2. Test Statistic:
z=s
p̂ − p0
p0 (1 − p0 )
n
We use p0 instead of p̂ in standard deviation (denominator) since we are assuming
H0 is true.
Note: Find p-values as test for µ when σ is known, using standard normal distribution
(Table A)
3. p-value: probability of getting more extreme test statistic than
z given H0 is true
Ha : p > p0
p-value = P (Z > z)
Ha : p < p0
p-value = P (Z < z)
Ha : p 6= p0
p-value = 2P (Z > |z|) = 2P (Z < −|z|)
4. Decision: Reject H0 if
p − value ≤ α
5. Conclusion: If reject H0 ⇒ statistically significant evidence to support the claim
made in Ha
If fail to reject H0 ⇒ no statistically significant evidence to support the claim made
in Ha
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Example: In a recent year, 73% of 1st-year college students responding to a national
survey identified “being very well-off financially” as an important personal goal. In
a random sample of 200 of its 1st-year students, a state university finds that 132
students say that this goal is important. Test the hypothesis that the proportion of
1st-year students at this university who think this goal is important differs from the
national value of 73% at the 0.10 significance level.
1. State Hypotheses:
2. Test statistic:
3. p-value:
4. Decision:
significant evidence to conclude that the
5. Conclusion: There is
proportion p of students at this university, who think being very well-off is important,
differs from the national value of 73% (i.e., p 6= 0.73).
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