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Start by computing the sample proportions and their difference, then calculate the confidence interval using both the unpooled normal approximation and the Wilson-Newcombe method. Compare whether the interval includes zero to determine statistical significance. Finally, apply Bonferroni correction for multiple comparisons and reassess the conclusion.
Pro tip: Always state the assumptions behind each method (e.g., normality, sample size) and note that the Wilson-Newcombe method is more robust for smaller samples or extreme proportions. This shows depth beyond just plugging numbers.
Calculate conversion rates for A and B: p_A = 410/5000 = 0.082, p_B = 470/5100 ≈ 0.09216. The observed difference is p_B - p_A ≈ 0.01016.
Compute standard error as sqrt(p_A(1-p_A)/n_A + p_B(1-p_B)/n_B). The 95% CI is difference ± 1.96*SE. Check if zero is inside the interval.
Compute Wilson intervals for each proportion separately, then combine them to get the Newcombe interval for the difference. This method adjusts for skewness and is more accurate for proportions.
For 3 comparisons against A, the family-wise error rate is 5%, so per-comparison alpha = 0.05/3 ≈ 0.0167. The corresponding confidence level is 1 - 0.0167 = 0.9833 or 98.33%. Recompute the CI using the adjusted z-value (e.g., z ≈ 2.128 for 98.33% two-sided).
Determine if the intervals include zero under each method and after Bonferroni adjustment. Discuss whether the conclusion about B changes.
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First, identify the sample size, mean, standard deviation, and confidence level. Then, use the t-distribution with 39 degrees of freedom to find the critical t-value, compute the standard error, and construct the confidence interval. Finally, state the margin of error clearly.
Pro tip: Always check whether the population standard deviation is known; if not, use the t-distribution. Also, mention that the t-distribution is robust to non-normality for sample sizes around 40, but verify assumptions if the data is heavily skewed.
Extract the sample size (n=40), sample mean (x̄=8.2), sample standard deviation (s=2.4), and confidence level (95%).
For a 95% confidence interval with 39 degrees of freedom, find the t-value (approximately 2.023). This can be obtained from a t-table or software.
Compute the standard error as s/√n = 2.4/√40 ≈ 0.3795.
Multiply the t-value by the standard error: 2.023 * 0.3795 ≈ 0.768 minutes.
The 95% confidence interval is x̄ ± margin of error: 8.2 ± 0.768, resulting in (7.432, 8.968) minutes.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
The formula is n = z squared times p times (1 minus p) divided by e squared.
Start by stating the formula for sample size for a proportion with conservative p=0.5, then plug in the values for 95% confidence (z=1.96) and margin of error 0.01. Compute the result and round up to the next integer, explaining that this is the minimum per-group sample size.
Pro tip: Mention that this is a conservative estimate because p=0.5 maximizes variance, and in practice you might use a pilot estimate to reduce sample size. Also, note that for A/B tests you need this sample size per variant, and consider any expected non-response or attrition.
Write the sample size formula for estimating a proportion: n = (z^2 * p * (1-p)) / E^2, where z is the z-score for the confidence level, p is the estimated proportion, and E is the margin of error.
For 95% confidence, z = 1.96. Use conservative p = 0.5. Margin of error E = 0.01.
Calculate n = (1.96^2 * 0.5 * 0.5) / 0.01^2 = (3.8416 * 0.25) / 0.0001 = 0.9604 / 0.0001 = 9604.
Since sample size must be an integer, round up to 9604. This means you need at least 9604 observations per group to achieve the desired precision.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.