When we do a binomial test and obtain a P-value smaller than 0.05, then the 95% confidence interval for the proportion will almost certainly not include the proportion we hypothesized.

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A P-value smaller than 0.05 indicates that the observed data are unlikely under the null hypothesis, suggesting a significant difference from the hypothesized proportion. Consequently, the 95% confidence interval for the proportion is likely to exclude the hypothesized value, reflecting this significant difference.

When We Do a Binomial Test and Obtain a P-Value