A marketing firm wants to know if there is a difference in the best marketing strategy for new customers compared to returning customers. In order to determine the relationship between marketing strategy and customer type, the following information on number of sales generated is obtained for each group. Is there sufficient evidence at the 0.10.1 level of significance to conclude that an association exists between marketing strategy and customer type? Observed Sample of CustomersPaid searchSocial mediaWeb retargetingTotalNew customer

161

141

163

465

Returning customer

140

167

141

448

Total

301

308

304

913 Compute the value of the test statistic. Round your answer to three decimal places, if necessary. (**Note:** If calculating by hand, be sure to use at least six decimal places in numbers for all intermediate calculations to ensure accuracy of the result.)

This problem involves determining whether there is an association between **marketing strategy** and **customer type**. To test this, we can perform a **chi-square test of independence**. The null hypothesis for this test is that there is no association between the two variables (marketing strategy and customer type), and the alternative hypothesis is that there is an association.

### Steps to perform the chi-square test of independence:

1. **Set up the observed dataThis problem involves determining whether there is an association between **marketing strategy** and **customer type**. To test this, we can perform a **chi-square test of independence**. The null hypothesis for this test is that there is no association between the two variables (marketing strategy and customer type), and the alternative hypothesis is that there is an association.

### Steps to perform the chi-square test of independence:

1. **Set up the observed **

   The table shows the observed values:

   \[
   \begin{array}{|c|c|c|c|}
   \hline
   & \text{Paid Search} & \text{Social Media} & \text{Web Retargeting} & \text{Total} \\
   \hline
   \text{New Customer} & 161 & 141 & 163 & 465 \\
   \text{Returning Customer} & 140 & 167 & 141 & 448 \\
   \hline
   \text{Total} & 301 & 308 & 304 & 913 \\
   \hline
   \end{array}
   \]

2. **Calculate the expected counts:**

   The formula for the expected value for each cell is:

   \[
   E_{ij} = \frac{\text{(row total)} \times \text{(column total)}}{\text{grand total}}
   \]

   Let's calculate the expected counts for each cell.

3. **Compute the chi-square statistic:**

   The formula for the chi-square statistic is:

   \[
   \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}
   \]

   Where:
   - \(O_{ij}\) are the observed values
   - \(E_{ij}\) are the expected values

Let me perform these calculations step by step.The value of the chi-square test statistic is approximately **4.937**.

Would you like further details on how this value was calculated or any other assistance on interpreting the result? Here are five related questions to explore the concept further:

1. What are the degrees of freedom for this chi-square test?
2. How do you determine the critical value for the chi-square test at the 0.10 significance level?
3. Can you explain how to interpret the p-value in a chi-square test?
4. What are the assumptions for performing a chi-square test of independence?
5. How would the result change if the number of customers in each group was altered?

**Tip:** Always make sure to check the assumptions of the chi-square test, such as having sufficiently large expected frequencies in each cell.

A marketing firm wants to know if there is a difference in the best marketing strategy for new customers compared to returning customers. In order to determine the relationship between marketing strategy and customer type, the following information on number of sales generated is obtained for each group. Is there sufficient evidence at the 0.1 level of significance to conclude that an association exists between marketing strategy and customer type? Observed Sample of Customers Paid Search Social Media Web Retargeting Total New customer: 161, 141, 163, 465. Returning customer: 140, 167, 141, 448. Total: 301, 308, 304, 913. Compute the value of the test statistic.

Analyze whether there is a significant association between marketing strategy and customer type using the chi-square test of independence. This problem provides detailed calculations for the observed data of new and returning customers across different marketing strategies. Suitable for college-level statistics students.

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