Question

sweet cravings bakery was interested in seeing if there was an association between cake flavor and frosting flavor. given are the results of the survey in a relative frequency table.

	vanilla buttercream	cream cheese icing	cookie butter buttercream	total
chocolate		0.15	0.10
total		0.34		1.00

which category has the lowest joint relative frequency?
chocolate cupcakes with cookie butter buttercream
vanilla cupcakes with cookie butter buttercream
cupcakes with cookie butter buttercream
cupcakes that are vanilla

Explanation:

Step1: Find missing values

First, for the "Cream Cheese Icing" column, the total is 0.34. We know the value for Vanilla is 0 (wait, no, the Vanilla row for Cream Cheese Icing: wait, the table is:

Rows: Vanilla, Chocolate, Total

Columns: Vanilla Buttercream, Cream Cheese Icing, Cookie Butter Buttercream, Total

Wait, let's re-express the table:

- Vanilla row:
- Vanilla Buttercream: 0.12
- Cream Cheese Icing:?
- Cookie Butter Buttercream:?
- Total: 0.40

- Chocolate row:
- Vanilla Buttercream:?
- Cream Cheese Icing: 0.15
- Cookie Butter Buttercream: 0.10
- Total:?

- Total row:
- Vanilla Buttercream:?
- Cream Cheese Icing: 0.34
- Cookie Butter Buttercream:?
- Total: 1.00

First, for Cream Cheese Icing column: Vanilla row value + Chocolate row value (0.15) = 0.34. So Vanilla row, Cream Cheese Icing: \( 0.34 - 0.15 = 0.19 \)? Wait, no, wait the Vanilla row total is 0.40. So Vanilla row: 0.12 (Vanilla Buttercream) + Cream Cheese Icing (let's call it \( x \)) + Cookie Butter Buttercream (let's call it \( y \)) = 0.40.

Chocolate row total: Vanilla Buttercream (let's call it \( z \)) + 0.15 (Cream Cheese Icing) + 0.10 (Cookie Butter Buttercream) = \( z + 0.25 \).

Total row:

- Vanilla Buttercream: 0.12 + \( z \)
- Cream Cheese Icing: \( x + 0.15 = 0.34 \) ⇒ \( x = 0.34 - 0.15 = 0.19 \)
- Cookie Butter Buttercream: \( y + 0.10 \)
- Total: 0.40 + ( \( z + 0.25 \)) = 1.00 ⇒ \( z + 0.25 = 0.60 \) ⇒ \( z = 0.35 \)

Now, Vanilla row: 0.12 + 0.19 + \( y \) = 0.40 ⇒ \( y = 0.40 - 0.31 = 0.09 \)

Now, let's list all joint relative frequencies (the cells where row and column intersect, not totals):

- Vanilla - Vanilla Buttercream: 0.12
- Vanilla - Cream Cheese Icing: 0.19
- Vanilla - Cookie Butter Buttercream: 0.09
- Chocolate - Vanilla Buttercream: 0.35
- Chocolate - Cream Cheese Icing: 0.15
- Chocolate - Cookie Butter Buttercream: 0.10

Now, let's check the options (the categories are the joint ones: Vanilla with Vanilla Buttercream, Vanilla with Cream Cheese Icing, Vanilla with Cookie Butter Buttercream, Chocolate with Vanilla Buttercream, Chocolate with Cream Cheese Icing, Chocolate with Cookie Butter Buttercream). Wait, the left side has the categories: Chocolate cupcakes with cookie butter buttercream, Vanilla cupcakes with cookie butter buttercream, Cupcakes with cookie butter buttercream, Cupcakes that are vanilla. Wait, maybe the categories are the ones in the left:

1. Chocolate cupcakes with cookie butter buttercream: that's Chocolate - Cookie Butter Buttercream: 0.10
2. Vanilla cupcakes with cookie butter buttercream: Vanilla - Cookie Butter Buttercream: 0.09
3. Cupcakes with cookie butter buttercream: Total - Cookie Butter Buttercream: \( 0.09 + 0.10 = 0.19 \)
4. Cupcakes that are vanilla: Vanilla total: 0.40

Wait, no, the question is "Which category has the lowest joint relative frequency?" The joint relative frequencies are the cells (not totals). Let's list all joint relative frequencies:

- Vanilla - Vanilla Buttercream: 0.12
- Vanilla - Cream Cheese Icing: 0.19 (from earlier calculation: 0.34 - 0.15 = 0.19? Wait no, wait the Vanilla row total is 0.40, so 0.12 (Vanilla Buttercream) + 0.19 (Cream Cheese Icing) + 0.09 (Cookie Butter Buttercream) = 0.40. Correct.

- Chocolate - Vanilla Buttercream: 0.35 (since Total Vanilla Buttercream: 0.12 + 0.35 = 0.47? Wait no, Total row, Vanilla Buttercream: 0.12 (Vanilla) + \( z \) (Chocolate) = 0.12 + 0.35 = 0.47. Then Total row, Cookie Butter Buttercream: 0.09 (Vanilla) + 0.10 (Chocolate) = 0.19. Then Total row: 0.47 (Vanilla Buttercream) + 0.34 (Cream Cheese Icing) + 0.19 (Cookie Butter Buttercream) = 0.47 + 0.34 = 0.81 + 0.19 = 1.00. Correct.

Now the categories on the left:

- Chocolate cupcakes with cookie butter buttercream: Chocolate - Cookie Butter Buttercream: 0.10
- Vanilla cupcakes with cookie butter buttercream: Vanilla - Cookie Butter Buttercream: 0.09
- Cupcakes with cookie butter buttercream: 0.09 + 0.10 = 0.19 (marginal, not joint)
- Cupcakes that are vanilla: 0.40 (marginal, not joint)

Wait, joint relative frequencies are the ones that are in the cells (row and column intersection), so the categories are:

1. Vanilla cupcakes with vanilla buttercream: 0.12
2. Vanilla cupcakes with cream cheese icing: 0.19
3. Vanilla cupcakes with cookie butter buttercream: 0.09
4. Chocolate cupcakes with vanilla buttercream: 0.35
5. Chocolate cupcakes with cream cheese icing: 0.15
6. Chocolate cupcakes with cookie butter buttercream: 0.10

Now, among these joint relative frequencies: 0.09 (Vanilla cupcakes with cookie butter buttercream) is the lowest (0.09 < 0.10 < 0.12 < 0.15 < 0.19 < 0.35).

Wait, but let's check the left side categories:

- Chocolate cupcakes with cookie butter buttercream: 0.10
- Vanilla cupcakes with cookie butter buttercream: 0.09
- Cupcakes with cookie butter buttercream: 0.19 (marginal)
- Cupcakes that are vanilla: 0.40 (marginal)

So the joint relative frequencies (the ones in the cells) are:

Vanilla - Vanilla Buttercream: 0.12

Vanilla - Cream Cheese Icing: 0.19

Vanilla - Cookie Butter Buttercream: 0.09

Chocolate - Vanilla Buttercream: 0.35

Chocolate - Cream Cheese Icing: 0.15

Chocolate - Cookie Butter Buttercream: 0.10

So the lowest is 0.09, which is "Vanilla cupcakes with cookie butter buttercream".

Wait, but let's confirm the calculations again.

Vanilla row total: 0.40 = 0.12 (Vanilla Buttercream) + Cream Cheese Icing + Cookie Butter Buttercream.

Cream Cheese Icing column total: 0.34 = Vanilla row Cream Cheese Icing + Chocolate row Cream Cheese Icing (0.15). So Vanilla row Cream Cheese Icing = 0.34 - 0.15 = 0.19. Then Vanilla row Cookie Butter Buttercream = 0.40 - 0.12 - 0.19 = 0.09. Correct.

Chocolate row total: Vanilla Buttercream + 0.15 + 0.10 = Vanilla Buttercream + 0.25. Total row total is 1.00, so Vanilla row total (0.40) + Chocolate row total = 1.00 ⇒ Chocolate row total = 0.60. So Vanilla Buttercream in Chocolate row = 0.60 - 0.15 - 0.10 = 0.35. Correct.

Now, the joint relative frequencies:

- Vanilla, Vanilla Buttercream: 0.12

- Vanilla, Cream Cheese Icing: 0.19

- Vanilla, Cookie Butter Buttercream: 0.09

- Chocolate, Vanilla Buttercream: 0.35

- Chocolate, Cream Cheese Icing: 0.15

- Chocolate, Cookie Butter Buttercream: 0.10

So the lowest is 0.09, which is the category "Vanilla cupcakes with cookie butter buttercream".

Step2: Compare all joint relative frequencies

Now we have all joint relative frequencies: 0.12, 0.19, 0.09, 0.35, 0.15, 0.10. The smallest among these is 0.09, which corresponds to the category "Vanilla cupcakes with cookie butter buttercream".

Answer:

Vanilla cupcakes with cookie butter buttercream