AP Statistics 3.8: Interpreting Errors

Interpret Type I and Type II errors in context, including which mistake matters more and how α or sample size affects the design.

The manager of a large company that sells pet supplies online wants to increase sales by encouraging repeat purchases. The manager believes that if past customers are offered $10 off their next purchase, more than 40 percent of them will place an order. To investigate the belief, 90 customers who placed an order in the past year are selected at random. Each of the selected customers is sent an e-mail with a coupon for $10 off the next purchase if the order is placed within 30 days. Of those who receive the coupon, 38 place an order.

Based on your conclusion from part (a), which of the two errors, Type I or Type II, could have been made? Interpret the consequence of the error in context.

Section 4

Because the null hypothesis was not rejected in part (a), a Type II error could have been made. A Type II error occurs when the null hypothesis is false and is not rejected. In this case, a Type II error is made by failing to reject the null hypothesis that 40 percent (or less) of all customers of the pet supply company would place an order within 30 days after receiving an e-mail with a coupon for $10 off the next purchase, when in fact, more than 40 percent would do so.

Consequently, the manager may decide not to use the coupon promotion when it actually would result in more than 40 percent of their customers making a purchase within 30 days.

Essentially correct (E) if the response satisfies the following two components:
1. States that a Type II error could have been
made
2. Provides a reasonable interpretation of the
consequence of the stated error AND uses
sufficient context, by including at minimum
"those who would place an order" or "coupon" or "sales" (e.g., indicating the
manager may not use the coupon promotion when it would actually lead to more than 40% of customers placing an order)
Partially correct (P) if the response satisfies only one of the two components.
Incorrect (I) if the response does not meet the
criteria for E or P.

Additional Notes:

- If the response to part (a) rejects the null hypothesis, then

Component 1 is satisfied if the response states that a Type I error could have been made.

Component 2 is satisfied if the response provides a reasonable interpretation of the consequence of a Type I error AND uses sufficient context by including at minimum "those who would place an order" or "coupon" or "sales."

- If the response states that a Type II error could have been made, followed by an incorrect description of a Type II error (e.g., "did not find convincing evidence that more than 40\% of customers will place an order when there actually was evidence of more than 40\%"), component 1 is not satisfied.

- The clarity and quality of the statement of the consequence may be used in a holistic approach to decide whether to score up or down (e.g., raising a score of 2.5 to 3 or reducing a score of 2.5 to 2).

Each essentially correct (E) part counts as 1 point, and each partially correct (P) part counts as $\frac{1}{2}$ point.}}

Score

Complete Response

Substantial Response

Developing Response

Minimal Response

If a response is between two scores (for example, $2 \frac{1}{2}$ points), use a holistic approach to decide whether to score up or down, depending on the strength of the response and quality of the communication.

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3.8.D—Interpret Type I and Type II errors question 1