1. Lesson 2: Section 9.1 (part 2)

2.  Interpret a Type I Error and a Type II Error in context, and give the consequences of each.  Understand the relationship between the significance level of a test, P(Type II error), and power.

3.  When we draw a conclusion from a significance test, we hope that it is correct – just like when a jury decides the verdict, they hope they are correct.  But sometimes it will be wrong.  There are two types of mistakes we can make.  Type I: We can reject the null hypothesis H 0 when it’s actually true  Type II: We can fail to reject a false null hypothesis H 0

4. H 0 is trueH 0 false (H a true) Reject H 0 Fail to reject H 0 Type I Error Correct Conclusion Type II Error

5.  A potato chip producer and main supplier agree that each shipment of potatoes must meet certain quality standards. If the producer determines that more than 8% of the potatoes in the shipment have “blemishes,” the truck will be sent away to get another load of potatoes from the supplier. Otherwise, the entire truckload will be used to make potato chips. To make the decision, a supervisor will inspect a random sample of potatoes from the shipment. The producer will then perform a significance test using the hypotheses:  H 0 : p = 0.08  H a : p > 0.08  Where p is the actual proportion of potatoes with blemishes in a given truckload.  Problem: Describe a Type I and Type II error in this setting, and explain the consequences of each.  Type 1 error: This would occur if the producer concludes that the proportion of potatoes with blemishes is greater than 0.08 when the actual proportion is 0.08 (or less). Consequence : The potato-chip producer sends the truckload of acceptable potatoes away, which may result in lost revenue for the supplier.

6.  A potato chip producer and main supplier agree that each shipment of potatoes must meet certain quality standards. If the producer determines that more than 8% of the potatoes in the shipment have “blemishes,” the truck will be sent away to get another load of potatoes from the supplier. Otherwise, the entire truckload will be used to make potato chips. To make the decision, a supervisor will inspect a random sample of potatoes from the shipment. The producer will then perform a significance test using the hypotheses:  H 0 : p = 0.08  H a : p > 0.08  Where p is the actual proportion of potatoes with blemished in a given truckload.  Problem: Describe a Type I and Type II error in this setting, and explain the consequences of each.  Type II error: This would occur if the producer does not send the truck away when more than 8% of the potatoes in the shipment have blemishes. Consequence: The producer uses the truckload of potatoes to make potato chips. More chips will be made with blemished potatoes, which may upset consumers.

7.  The manager of a fast-food restaurant wants to reduce the proportion of drive through customers who have to wait more than two minutes to receive their food once their orders is placed. Based on store records, the proportion of customers who had to wait at least two minutes was p=0.63.  To reduce this proportion, the manager assigns an additional employee to assist with drive-through orders. During the next month, the manager will collect a random sample of drive-through times and test the following hypotheses: H 0 : p = 0.63 H a : p < 0.63  Where p = the true proportion of drive-through customers who have to wait more than two minutes after their orders is placed to receive their food.  Problem: Describe a Type I and Type II error in this setting and explain the consequences of each.  Type I error: This would occur if the manager decides that the true proportion of drive-through customers who have to wait at least two minutes has been reduced when it really hasn’t been reduced. A consequence is that the manager will have to pay unnecessarily for an additional employee

8.  The manager of a fast-food restaurant wants to reduce the proportion of drive through customers who have to wait more than two minutes to receive their food once their orders is placed. Based on store records, the proportion of customers who had to wait at least two minutes was p=0.63.  To reduce this proportion, the manager assigns an additional employee to assist with drive-through orders. During the next month, the manager will collect a random sample of drive-through times and test the following hypotheses: H 0 : p = 0.63 H a : p < 0.63  Where p = the true proportion of drive-through customers who have to wait more than two minutes after their orders is placed to receive their food.  Problem: Describe a Type I and Type II error in this setting and explain the consequences of each.  Type II error: This would occur if the manager didn’t decide that the true proportion of customers who had to wait at least two minutes had been reduced when it really had been reduced. A consequence is that the restaurant would not have an additional employee helping with the drive-through, so it wouldn’t be providing faster service when it could.

9.  We can find the chances of making these mistakes.  Significance and Type I Error: The significance level (fixed) is the probability of a Type 1 error. (It is the probability that the test will reject H 0 when H 0 is in fact true.)

10.  Recall the potato chip problem where  H 0 : p = 0.08  H a : p > 0.08 Where p is the actual proportion of potatoes with blemishes.  Suppose that the potato-chip producer decides to carry out this test based on a random sample of 500 potatoes using a 5% significance level ( = 0.05)  (Remember: A type I error is to reject H 0 when H 0 is actually true.) If our sample results in a value of that is much larger than 0.08, we will reject H 0.  Assuming H 0 : p = 0.08 is true, calculate and draw the sampling distribution of and draw where we would make a Type I error.

11.  Recall the fast-food problem where  H 0 : p = 0.63  H a : p < 0.63 Where p is the true proportion of drive-through customers who have to wait more than two minutes after their order is placed to receive food.  Suppose that the manager decided to carry out this test based on a random sample of 250 orders and using a 10% significance level ( = 0.1)  (Remember: A type I error is to reject H 0 when H 0 is actually true.) If our sample results in a value of that is much smaller than 0.63, we will reject H 0.  Assuming H 0 : p = 0.63 is true, calculate and draw the sampling distribution of and draw where we would make a Type I error.

12.  We can also find the chances of failing to reject H 0 when H a is true (H 0 is not) – however, we are going to calculate the probability that the test WILL reject H 0 at a chosen significance level when the H a is true.  Power = The power of a test (against a specific stated H a ) is the probability that the test correctly WILL reject H 0 at a chosen significance level when the H a is true.  Power = 1 – Type II Error (aka 1 - )

13.  Recall the potato chip problem where  H 0 : p = 0.08  H a : p > 0.08 Where p is the actual proportion of potatoes with blemishes.  Suppose that the potato-chip producer wonders whether the significance test of H 0 : p = 0.08 versus H a : p > 0.08 based on a random sample of 500 potatoes has enough power to detect a shipment with say, 11% blemished potatoes.  He finds the power to be 75%, but what does this mean?  (When there is an actual 11% of blemished potatoes on the truck), he will reject the null hypothesis of p=0.08 (or 8%) and conclude that his truck as more than 8% blemished potatoes … but he will do this 75% of the time.  This means that he will have a Type II error 25% of the time (where he fails to reject the null, but the null really wasn’t true!)

14.  Recall the fast-food problem where  H 0 : p = 0.63  H a : p < 0.63 Where p is the true proportion of drive-through customers who have to wait more than two minutes after their order is placed to receive food.  Suppose that the manager wonders whether the significance test of H 0 : p = 0.63 versus H a : p < 0.63 based on a random sample of 250 orders has enough power to recognize when 45% of the customers had to wait more than 2 minutes for their order.  He finds the power to be 0.84 – interpret this.  The manager has an 84% chance that he will correctly reject the null hypothesis (of 63% of customers waiting longer than 2 minutes) and conclude that less than 63% of the customers wait 2 minutes or longer (when in fact, it is 45%).  This means that he will have a Type II error 16% of the time (where he fails to reject the null, but the null really wasn’t true!)

15. H 0 is trueH 0 false (H a true) Reject H 0 Fail to reject H 0 Type I Error Correct Conclusion Type II Error (Power or 1- ) ( ) ( 1 - )

16.  Assigned reading: p. 538-545  HW problems: p. 546 #15, 19, 20, 21, 23, 24, 25, 27-30  Check answers to odd problems.  STUDY FOR A QUIZ!