Website Design Impact

Solution & Explanation

Understanding the Problem

The company is evaluating the impact of a new website design on conversion rates. They conducted an A/B test with two groups of 100 users each: one group saw the old design, and the other saw the new design. The conversion rates were 5% for the old design and 10% for the new design. We need to determine if this difference is statistically significant.

Statistical Test Selection

To assess the statistical significance of the difference in conversion rates, we use a two-proportion Z-test. This test is suitable for comparing proportions between two independent groups.

Hypotheses

• Null Hypothesis (H0): The conversion rate for the new design is equal to the conversion rate for the old design (p_new = p_old).
• Alternative Hypothesis (H1): The conversion rate for the new design is greater than the conversion rate for the old design (p_new > p_old).

Calculations

1. Sample Proportions:

   • New Design: $p_1 = \frac{10}{100} = 0.10$
   • Old Design: $p_2 = \frac{5}{100} = 0.05$

2. Pooled Proportion:

   • $p_{pooled} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{10 + 5}{100 + 100} = 0.075$

3. Standard Error (SE) of the Difference in Proportions:

   • $SE = \sqrt{p_{pooled} \times (1 - p_{pooled}) \times \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}$
   • $SE = \sqrt{0.075 \times 0.925 \times \left(\frac{1}{100} + \frac{1}{100}\right)} \approx 0.0373$

4. Z-Statistic:

   • $Z = \frac{p_1 - p_2}{SE} = \frac{0.10 - 0.05}{0.0373} \approx 1.34$

5. P-Value:

   • For a one-tailed test, we look up the Z-statistic in a standard normal distribution table.
   • $P(Z > 1.34) \approx 0.0901$
   • Since this is a one-tailed test, the p-value is approximately 0.0901.

Conclusion

The p-value (0.0901) is greater than the significance level of 0.05. Therefore, we fail to reject the null hypothesis. This means that the data does not provide sufficient evidence to conclude that the new design significantly improves conversion rates compared to the old design.

Considerations for Improvement

• Sample Size: The sample size of 100 users per group might be too small to detect a significant difference. A larger sample size could increase the power of the test.
• One-Tailed vs. Two-Tailed Test: If the interest is specifically in whether the new design improves conversion rates, a one-tailed test is appropriate. However, if any change (increase or decrease) is of interest, a two-tailed test should be considered.

The analysis indicates that, with the current sample size and results, the observed difference in conversion rates is not statistically significant.