AP Statistics Unit 6 Inference for Proportions

Topic 6.7: Potential Errors When Performing Tests

Type I Error, Type II Error, and Power
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:

Key Vocabulary

Type I Error Rejecting H0 when H0 is actually true.
Type II Error Failing to reject H0 when H0 is actually false.
False Positive Another name for a Type I error: concluding there is convincing evidence for Ha when there really is not.
False Negative Another name for a Type II error: failing to find convincing evidence for Ha when Ha is actually true.
Power The probability that a test will correctly reject a false null hypothesis.

Video 1: Type I and Type II Errors

[0:00 – 6:15]
1. [0:31] If the p-value is α, we the null hypothesis and conclude there is convincing statistical evidence for Ha.
2. [0:54] A Type I error occurs when the null hypothesis is and is . This is sometimes called a .
3. [1:08] If the p-value is α, we H0. In this case, there is convincing statistical evidence for Ha.
4. [1:29] A Type II error occurs when the null hypothesis is , but it is . This is sometimes called a .
Decision Table: [1:45]
Decision H0 True Ha True
Reject H0 Type I Error Correct Decision
Fail to Reject H0 Correct Decision Type II Error
Context — "Does Green = More Natural?" [2:32] Two student researchers randomly selected 30 students from their school. Each student tasted a green cup and a white cup of identical lemonade and chose which tasted more natural. From previous videos: H0: p = 0.50, Ha: p > 0.50, where p is the proportion of all students at the school who would choose the green cup.
5. [3:20] Describe the errors in context:
(a) Type I: The researchers find convincing evidence that of all students at this school would choose the green cup, but the actual percentage is .
(b) Type II: The researchers do not find convincing evidence that of all students at this school would choose the green cup, but the actual percentage is .
6. [5:04] In this marketing example, the error is more consequential because it could lead to a .

Video 2: Power and Error Probabilities

[0:00 – 7:57]
7. [0:36] When the null hypothesis is true, the probability of a Type I error is equal to the significance level, .
8. [1:26] All other things being equal, decreasing the probability of a Type I error increases the probability of a error.
Key Power Relationships: [2:22]
Power = P(correctly reject a false H0)
P(Type I error) = α
P(Type II error) = 1 − power
9. [2:26] The power of a test is the probability that the test will a null hypothesis, so the probability of a Type II error is .
Context — Green Cup Example Revisited [3:49] Suppose the researchers used α = 0.05 and the power of the test against an alternative of p = 0.64 is 0.45. Interpret the power, then find the probability of a Type I and a Type II error.
10. [5:01] If the true proportion of students who would choose the green cup is 0.64, then there is a probability that we find convincing evidence that more than 50% of students would choose the green cup. The probability of a Type I error is , and the probability of a Type II error is .
Factors That Increase Power: [5:43]

Key Takeaways

[Summary]
  1. Type I error: reject H0 when H0 is actually true. This is a false positive.
  2. Type II error: fail to reject H0 when H0 is actually false. This is a false negative.
  3. The probability of a Type I error is α; the probability of a Type II error is 1 minus power.
  4. Power gets larger when sample size or α increases, standard error decreases, or the true parameter is farther from the null value.

Reflection Questions

[AI-Graded]
R1. A school cafeteria manager tests H0: p = 0.50 versus Ha: p > 0.50, where p is the proportion of students who would prefer a new lunch option. Describe a Type I error and a Type II error in context. Then explain which error would be more consequential and why.
R2. A significance test uses α = 0.05 and has power = 0.80 against the alternative that p = 0.64 in the green-cup study. Explain what this power means in context, find the probability of a Type II error, and name one change that would increase the power of the test.

Exit Ticket

A fitness app claims that 40% of high school students exercise at least 5 days per week. A researcher suspects the true proportion is higher. Let p be the proportion of all high school students who exercise at least 5 days per week. A significance test is performed with α = 0.01, and the power against a true value of p = 0.55 is 0.70.

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