Question

a burrito restaurant recorded 200 customer orders. each order was categorized by whether it was takeout or dine-in and whether it was placed on a weekday or a weekend.

the results are shown below:

\

$$\begin{tabular}{|c|c|c|c|} \\hline order type & weekday & weekend & total \\\\ \\hline takeout & 66 & 34 & 100 \\\\ \\hline dine-in & 54 & 46 & 100 \\\\ \\hline total & 120 & 80 & 200 \\\\ \\hline \\end{tabular}$$

using this table, decide which statement about independence is true. which statement is correct?

• the events \takeout\ and \weekday\ are independent.
• the events \weekday\ and \weekend\ are independent.
• the events \order type\ (takeout or dine-in) and \day of the week\ (weekday or weekend) are not independent.
• the events \takeout\ and \dine-in\ are independent.

Explanation:

Calculate probabilities for independence test

Using the Independent Events and Two-Way Frequency Tables knowledge points

$$
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$$

Check for independence

Using the Independent Events knowledge point

$$
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$$

Since the joint probability does not equal the product of the marginal probabilities, the events "takeout" and "weekday" are not independent. This means the overall variables "order type" and "day of the week" are not independent.

Evaluate the given options

• Option 1: "takeout" and "weekday" are independent. (False, as shown above)
• Option 2: "weekday" and "weekend" are independent. (False, they are mutually exclusive complements)
• Option 3: "order type" and "day of the week" are not independent. (True, since the individual categories are dependent)
• Option 4: "takeout" and "dine-in" are independent. (False, they are mutually exclusive complements)

Answer:

• (A) The events "takeout" and "weekday" are independent.
• (B) The events "weekday" and "weekend" are independent.
• (C) The events "order type" (takeout or dine-in) and "day of the week" (weekday or weekend) are not independent. (Correct answer)
• (D) The events "takeout" and "dine-in" are independent.