Gender-Based Order Value Differences

Solution & Explanation

To determine whether the difference in average order value (AOV) between men and women is statistically significant, we need to perform a hypothesis test. Given the data:

• Men:
  • Average Order Value (AOV) = $46.3
  • Total Purchases (n1) = 2500
• Women:
  • Average Order Value (AOV) = $50.2
  • Total Purchases (n2) = 3500

We will use a two-sample t-test, which is suitable for comparing the means of two independent groups when the population variances are unknown and sample sizes are large.

Steps to Perform the Hypothesis Test:

1. State the Hypotheses:

   • Null Hypothesis (H0): There is no significant difference in AOV between men and women. $\mu_1 - \mu_2 = 0$
   • Alternative Hypothesis (H1): There is a significant difference in AOV between men and women. $\mu_1 - \mu_2 \neq 0$

2. Choose the Significance Level ($\alpha$):

   • Typically, $\alpha$ is set to 0.05.

3. Calculate the Test Statistic:

   • The test statistic for a two-sample t-test is given by: $t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
   • $\bar{x}_1$ and $\bar{x}_2$ are the sample means.
   • $s_1^2$ and $s_2^2$ are the sample variances, which are not provided but can be estimated if needed.
   • $n_1$ and $n_2$ are the sample sizes.

4. Determine the Degrees of Freedom (df):

   • For unequal sample sizes and variances, use the Welch-Satterthwaite equation: $df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}}$

5. Find the Critical Value(s):

   • Use a t-distribution table or calculator to find the critical value(s) for the chosen $\alpha$ and calculated df.

6. Decision Rule:

   • If the absolute value of the test statistic is greater than the critical value, reject the null hypothesis.
   • Otherwise, fail to reject the null hypothesis.

Conclusion:

Based on the calculation provided in the comments:

• Calculated t-value = -1.77
• Degrees of freedom (df) = 2998
• Critical value for $\alpha = 0.05$ and df = 2998 is approximately ±1.96

Since the absolute value of the calculated t-statistic (-1.77) is less than the critical value (1.96), we fail to reject the null hypothesis. Therefore, there is no statistically significant difference in the average order value between men and women at a 5% significance level.