A retail company wants to compare the proportion of customers who make a purchase after 

entering two of its stores. In Store A, out of 200 customers, 120 made a purchase. In Store B, out 

of 180 customers, 105 made a purchase. Test at the 5% significance level whether there is a 

significant difference in the proportion of customers who make a purchase between the two stores.

To test whether there is a significant difference in the proportion of customers who make a purchase between Store A and Store B, we can perform a **two-proportion z-test**. Here’s how we’ll do it step-by-step:

### Step 1: Define the null and alternative hypotheses

- **Null hypothesis (\(H_0\))**: There is no significant difference between the proportions of customers who make a purchase at Store A and Store B.
  \[
  H_0: p_A = p_B
  \]
  
- **Alternative hypothesis (\(H_A\))**: There is a significant difference between the proportions of customers who make a purchase at Store A and Store B.
  \[
  H_A: p_A \neq p_B
  \]

### Step 2: Calculate the sample proportions

For Store A:
\[
\hat{p}_A = \frac{120}{200} = 0.60
\]
For Store B:
\[
\hat{p}_B = \frac{105}{180} = 0.5833
\]

### Step 3: Calculate the pooled sample proportion

The pooled sample proportion (\(\hat{p}\)) is used because the null hypothesis assumes that the two population proportions are equal.
\[
\hat{p} = \frac{120 + 105}{200 + 180} = \frac{225}{380} = 0.5921
\]

### Step 4: Calculate the standard error of the pooled proportion

The standard error (SE) for the difference in proportions is given by:
\[
SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_A} + \frac{1}{n_B} \right)}
\]
where \(n_A = 200\) and \(n_B = 180\).

Let's calculate this next.The standard error (SE) of the pooled proportion is approximately 0.05049.

### Step 5: Calculate the z-score

The z-score for the difference in proportions is given by:
\[
z = \frac{\hat{p}_A - \hat{p}_B}{SE}
\]

Let's calculate the z-score next.The calculated z-score is approximately 0.33.

### Step 6: Determine the p-value

Since this is a two-tailed test, we compare the z-score to the critical value for a 5% significance level (α = 0.05). The critical z-value for a two-tailed test at α = 0.05 is approximately ±1.96.

### Step 7: Conclusion

- If \(|z|\) > 1.96, we reject the null hypothesis.
- If \(|z|\) ≤ 1.96, we fail to reject the null hypothesis.

In this case, since \(z = 0.33\), which is less than 1.96, we **fail to reject the null hypothesis**. This means that there is **no significant difference** in the proportions of customers who make a purchase between Store A and Store B at the 5% significance level.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the result change if we tested at a different significance level (e.g., 1%)?
2. What is the interpretation of the pooled proportion?
3. How does the sample size affect the z-test results?
4. Can we use a chi-square test for this data? If so, how would it compare?
5. What would be the impact of larger or smaller differences in proportions?

**Tip:** In hypothesis testing, always remember to interpret the results in context, not just rely on statistical significance.

A retail company wants to compare the proportion of customers who make a purchase after entering two of its stores. In Store A, out of 200 customers, 120 made a purchase. In Store B, out of 180 customers, 105 made a purchase. Test at the 5% significance level whether there is a significant difference in the proportion of customers who make a purchase between the two stores.

Learn how to conduct a two-proportion z-test to compare the proportion of customers making purchases in two retail stores. This step-by-step solution covers hypothesis testing, pooled proportions, and z-score calculations. Data: Store A (120/200), Store B (105/180). Suitable for Grades 11-12 and undergraduate students.

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