To determine which category has the lowest joint relative frequency, we need to analyze the given relative frequency table and the provided values.
1. The total for Vanilla Buttercream is given as 0.12.
2. The total for Cream Cheese Icing is given for Chocolate cupcakes as 0.15, with a total sum of 0.34 for that row. Therefore, the relative frequency for Vanilla cupcakes with Cream Cheese Icing is:
[tex]\[
0.34 - 0.15 = 0.19
\][/tex]
3. For Cookie Butter Buttercream, we are given the total for Chocolate cupcakes as 0.10. The total for all Vanilla cupcakes is given as 0.40. Hence, for Vanilla cupcakes with Cookie Butter Buttercream, we first compute the total relative frequency for Cookie Butter Buttercream:
[tex]\[
0.40 - 0.12 = 0.28
\][/tex]
4. Summarizing the relative frequencies:
- Chocolate cupcakes with Cookie Butter Buttercream: [tex]\(0.10\)[/tex]
- Vanilla cupcakes with Cookie Butter Buttercream: [tex]\(0.28\)[/tex]
- Total cupcakes with Cookie Butter Buttercream (sum of both Vanilla and Chocolate): [tex]\(0.10 + 0.28 = 0.38\)[/tex]
- Total cupcakes that are Vanilla: [tex]\(0.40\)[/tex]
To identify the category with the lowest joint relative frequency:
- Chocolate cupcakes with Cookie Butter Buttercream: [tex]\(0.10\)[/tex]
- Vanilla cupcakes with Cookie Butter Buttercream: [tex]\(0.28\)[/tex]
- Cupcakes with Cookie Butter Buttercream: [tex]\(0.38\)[/tex]
- Cupcakes that are Vanilla: [tex]\(0.40\)[/tex]
Comparing these values, the lowest joint relative frequency is for Chocolate cupcakes with Cookie Butter Buttercream, which is [tex]\(0.10\)[/tex].
Therefore, the category with the lowest joint relative frequency is:
Chocolate cupcakes with Cookie Butter Buttercream.
