Test Whether a Billboard Campaign Conversion Rate Exceeds 60%
A billboard campaign sample contains N = 100 users, and 65 of them converted. You want to test whether the true conversion rate is greater than 60%.
Constraints & Assumptions
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Treat this as a one-sample proportion test.
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Assume the sample is independent and reasonably representative of the population being evaluated.
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Use a one-sided test at
alpha = 0.05
.
-
Show both the confidence-interval intuition and the formal hypothesis-test result.
Clarifying Questions to Ask
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What counts as a conversion, and over what time window?
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Was the sample randomly selected from exposed users or from all eligible users?
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Are repeated users or clustered observations present?
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Is the 60% threshold a business target, a historical baseline, or a contractual benchmark?
Part 1 - State the Hypotheses
State the null and alternative hypotheses for the one-sided test.
What This Part Should Cover
-
A null hypothesis representing no evidence that the true conversion rate exceeds 60%.
-
An alternative hypothesis that the true conversion rate is greater than 60%.
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Recognition that the test is one-sided.
Part 2 - Compute Standard Error and Confidence Interval
Compute the standard error, 95% margin of error, and 95% confidence interval for the observed conversion rate.
What This Part Should Cover
-
Use of
p_hat = 0.65
and
N = 100
for the Wald confidence interval calculation.
-
Standard error
sqrt(p_hat * (1 - p_hat) / N)
.
-
Margin of error
1.96 * SE
and the interval
p_hat +/- margin
.
-
Explanation that the two-sided CI is a useful descriptive check but the formal one-sided test uses the null standard error.
Part 3 - Make the Test Decision
Using a one-sided z test, decide whether to reject the null at alpha = 0.05 and interpret the result.
What This Part Should Cover
-
Test statistic based on
p0 = 0.60
:
(p_hat - p0) / sqrt(p0 * (1 - p0) / N)
.
-
One-sided p-value and comparison with
0.05
.
-
Conclusion that distinguishes "not enough evidence" from "the conversion rate is definitely not above 60%."
What a Strong Answer Covers
A strong answer states the hypotheses clearly, computes the interval correctly, uses the proper one-sided test statistic for the decision, and explains the practical meaning of failing or succeeding to reject the null.
Follow-up Questions
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How would the conclusion change if the sample size were 1,000 with the same observed rate?
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When would you prefer an exact binomial test over a normal approximation?
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What additional validity checks would you run before trusting the result?