STAT 200 - Hypothesis testing basics (review sheet) Setup - Null hypothesis H0: the "nothing is happening" claim (for example, the mean score is 70). - Alternative hypothesis H1: what we suspect instead (the mean is not 70, or is greater than 70). - Test statistic: how far the sample result is from H0, measured in standard errors. - Significance level alpha: the cutoff we choose before looking at the data, commonly 0.05. The p-value - The p-value is the probability of getting a result at least as extreme as ours IF H0 is true. - Decision rule: if p <= alpha, reject H0. If p > alpha, fail to reject H0. - A small p-value is NOT the probability that H0 is true. - "Fail to reject" does not prove H0. It means we did not find enough evidence against it. Errors - Type I error: rejecting H0 when H0 is actually true. Its probability is alpha. - Type II error: failing to reject H0 when H0 is actually false. Its probability is beta. - Power = 1 - beta, the chance of correctly rejecting a false H0. Larger samples and larger true effects raise power. Worked example (z-test) - Claim: the average exam score is 70. The population standard deviation is sigma = 12 (known). - Sample: n = 36 students, sample mean = 74. - z = (74 - 70) / (12 / sqrt(36)) = 4 / 2 = 2.0 - The two-sided p-value for z = 2.0 is about 0.0455. - Since 0.0455 < 0.05, we reject H0 at the 5% level: the data are evidence that the true mean is not 70. Common mistakes - Choosing alpha after seeing the p-value. - Saying "we are 95% sure H0 is false". - Confusing statistical significance with practical importance.
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1. The notes
2. Notes StudyProofs made (6)
Hypothesis Testing Basics
Hypothesis testing involves a null hypothesis (H0) and an alternative hypothesis (H1). The null hypothesis claims that nothing is happening, while the alternative suggests a change or effect.
Source: “Null hypothesis H0: the "nothing is happening" claim (for example, the mean score is 70). - Alternative hypothesis H1: what we suspect instead (the mean is not 70, or is greater than 70”
Test Statistic and Significance Level
The test statistic measures how far the sample result is from H0 in standard errors. The significance level (alpha) is the predetermined cutoff for making decisions, commonly set at 0.05.
Source: “Test statistic: how far the sample result is from H0, measured in standard errors. - Significance level alpha: the cutoff we choose before looking at the data, commonly 0.05”
Understanding the p-value
The p-value indicates the probability of obtaining a result as extreme as the observed one, assuming H0 is true. A decision rule is applied: if p <= alpha, reject H0; if p > alpha, fail to reject H0.
Source: “The p-value is the probability of getting a result at least as extreme as ours IF H0 is true. - Decision rule: if p <= alpha, reject H0. If p > alpha, fail to reject H0”
Errors in Hypothesis Testing
Type I error occurs when H0 is rejected when it is true, with a probability equal to alpha. Type II error happens when H0 is not rejected when it is false, with a probability of beta. Power is defined as 1 - beta.
Source: “Type I error: rejecting H0 when H0 is actually true. Its probability is alpha. - Type II error: failing to reject H0 when H0 is actually false. Its probability is beta”
Worked Example: z-test
In a z-test example, the claim is that the average exam score is 70. With a sample mean of 74 from 36 students, the calculated z-value is 2.0, leading to a p-value of approximately 0.0455, which allows us to reject H0.
Source: “Claim: the average exam score is 70. The population standard deviation is sigma = 12 (known). - Sample: n = 36 students, sample mean = 74. - z = (74 - 70) / (12 / sqrt(36)) = 4 / 2 = 2.0”
Common Mistakes in Hypothesis Testing
Common mistakes include choosing alpha after seeing the p-value, misinterpreting confidence levels, and confusing statistical significance with practical importance.
Source: “Choosing alpha after seeing the p-value. - Saying "we are 95% sure H0 is false". - Confusing statistical significance with practical importance”
3. Flashcards (8)
Prompt
What is the null hypothesis (H0)?
Answer
The null hypothesis is the claim that nothing is happening, for example, the mean score is 70.
Source: “Null hypothesis H0: the "nothing is happening" claim (for example, the mean score is 70”
Prompt
What does a p-value represent?
Answer
The p-value represents the probability of getting a result at least as extreme as ours IF H0 is true.
Source: “The p-value is the probability of getting a result at least as extreme as ours IF H0 is true”
Prompt
What is a Type I error?
Answer
A Type I error occurs when we reject H0 when it is actually true, with a probability equal to alpha.
Source: “Type I error: rejecting H0 when H0 is actually true. Its probability is alpha”
Prompt
What is the significance level (alpha)?
Answer
The significance level (alpha) is the cutoff we choose before looking at the data, commonly set at 0.05.
Source: “Significance level alpha: the cutoff we choose before looking at the data, commonly 0.05”
Prompt
What is the power of a test?
Answer
Power is the chance of correctly rejecting a false H0, calculated as 1 - beta.
Source: “Power = 1 - beta, the chance of correctly rejecting a false H0”
Prompt
What does it mean to fail to reject H0?
Answer
Failing to reject H0 means we did not find enough evidence against it; it does not prove H0 is true.
Source: “Fail to reject" does not prove H0. It means we did not find enough evidence against it”
Prompt
What is the decision rule for hypothesis testing?
Answer
If p <= alpha, reject H0. If p > alpha, fail to reject H0.
Source: “Decision rule: if p <= alpha, reject H0. If p > alpha, fail to reject H0”
Prompt
What is a common mistake in hypothesis testing?
Answer
A common mistake is choosing alpha after seeing the p-value.
Source: “Choosing alpha after seeing the p-value”
4. Quiz (8)
1. What is the null hypothesis (H0)?
- A. The claim that nothing is happening
- B. The claim that something is happening
- C. The probability of making a Type I error
- D. The significance level of a test
Show the answer
A. The claim that nothing is happening
The null hypothesis is the claim that nothing is happening, for example, the mean score is 70.
Source: “Null hypothesis H0: the "nothing is happening" claim (for example, the mean score is 70”
2. What does a p-value less than alpha indicate?
- A. Reject H0
- B. Fail to reject H0
- C. Accept H0
- D. Increase alpha
Show the answer
A. Reject H0
A p-value less than alpha indicates that we reject the null hypothesis.
Source: “Decision rule: if p <= alpha, reject H0”
3. What is a Type II error?
- A. Rejecting H0 when it is true
- B. Failing to reject H0 when it is false
- C. Correctly rejecting H0
- D. None of the above
Show the answer
B. Failing to reject H0 when it is false
A Type II error occurs when we fail to reject H0 when it is actually false.
Source: “Type II error: failing to reject H0 when H0 is actually false”
4. What is the significance level commonly set at?
- A. 0.01
- B. 0.05
- C. 0.10
- D. 0.15
Show the answer
B. 0.05
The significance level is commonly set at 0.05 before conducting the test.
Source: “Significance level alpha: the cutoff we choose before looking at the data, commonly 0.05”
5. What does the power of a test represent?
- A. The probability of making a Type I error
- B. The probability of correctly rejecting a false H0
- C. The significance level of the test
- D. The p-value of the test
Show the answer
B. The probability of correctly rejecting a false H0
Power represents the probability of correctly rejecting a false null hypothesis.
Source: “Power = 1 - beta, the chance of correctly rejecting a false H0”
6. In a z-test example, what was the sample mean?
- A. 70
- B. 74
- C. 36
- D. 12
Show the answer
B. 74
The sample mean in the z-test example was 74.
Source: “Sample: n = 36 students, sample mean = 74”
7. What is a common mistake when conducting hypothesis tests?
- A. Choosing alpha before seeing the p-value
- B. Choosing alpha after seeing the p-value
- C. Calculating the p-value correctly
- D. None of the above
Show the answer
B. Choosing alpha after seeing the p-value
A common mistake is choosing alpha after seeing the p-value, which can bias the results.
Source: “Choosing alpha after seeing the p-value”
8. What does it mean to fail to reject H0?
- A. We are 95% sure H0 is false
- B. We did not find enough evidence against H0
- C. H0 is proven true
- D. None of the above
Show the answer
B. We did not find enough evidence against H0
Failing to reject H0 means we did not find enough evidence against it; it does not prove H0 is true.
Source: “Fail to reject" does not prove H0. It means we did not find enough evidence against it”
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