Hypothesis Test
- evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. These two statements are called the null hypothesis and the alternative hypothesis.
- When you perform a hypothesis test, there are two types of errors related to drawing an incorrect conclusion:
- Type I Error / False Positive - the test rejects the null hypothesis that is actually true
- Type II Error / False Negative - the test fails to reject the null hypothesis that is actually false
- A test result is statistically significant when the sample statistic is unusual enough relative to the null hypothesis that you can reject the null hypothesis for the entire population. “Unusual enough” in a hypothesis test is defined by how unlikely the effect observed in your sample is if the null hypothesis is true
- A test result is not 100% accurate because they use a random sample to draw conclusions about the entire population
- If your sample data provide sufficient evidence, you can reject the null hypothesis for the entire population. Your data favor the alternative hypothesis
Statistical Hypothesis Test - Steps
Begin with a claim about the value of the population parameter (we will call the null hypothesis), then check whether or not the sample data provide evidence AGAINST this claim.
- Formulate 2 hypotheses (two mutually exclusive statements about population parameter 𝜃)
- Null Hypothesis (𝐻0) - the value of 𝜃 corresponding to “status quo”, “common belief”, “no change”, etc. Often, 𝐻0: 𝜃 = 𝜃0(a given value)
- Alternative Hypothesis (𝐻𝑎) - the claim the researcher is hoping to prove
- Compute the null distribution of 𝐻0
- Compute 𝑡𝑒𝑠𝑡-𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 of sample data using the null distribution
- Determine whether to reject the 𝐻0 in either 2 ways:
-
Critical Value Method
- Choose a significance level (𝛼) such as 0.05
- Compute 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙-𝑣𝑎𝑙𝑢𝑒 by using the null distribution and significance level
- Compare 𝑡𝑒𝑠𝑡-𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 with the 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙-𝑣𝑎𝑙𝑢𝑒(𝑠):
- 2-Sided Hypothesis Test - if 𝑡𝑒𝑠𝑡-𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 is outside the interval of [𝑙𝑜𝑤𝑒𝑟-𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙-𝑝𝑜𝑖𝑛𝑡, 𝑢𝑝𝑝𝑒𝑟-𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙-𝑝𝑜𝑖𝑛𝑡] the 𝐻0is rejected
- 1-Sided Upper Hypothesis Test - if 𝑡𝑒𝑠𝑡-𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 is GREATER than 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙-𝑣𝑎𝑙𝑢𝑒, then the 𝐻0is rejected
- 1-Sided Lower Hypothesis Test - if 𝑡𝑒𝑠𝑡-𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 is LESS than 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙-𝑣𝑎𝑙𝑢𝑒, then the 𝐻0is rejected
-
P-Value Method
- Compute the probability of 𝑡𝑒𝑠𝑡-𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 (i.e. probability of seeing sample data) under the assumption that 𝐻0is true
-
Info
- significance level (𝛼) - is the probability of REJECTING the 𝐻0 when it is true (i.e. the probability of Type I Error)
- critical value - is a point (or points) on the scale of the test statistic beyond which we reject the null hypothesis, and, is derived from the level of significance (𝛼) of the test
- p-value - is the probability of seeing sample data when null-hypothesis 𝐻0 is true
Statistical Hypothesis Test - Reject or Fail to Reject Null Hypothesis
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Either:
- reject 𝐻0
- fail to reject 𝐻0
We do not know the truth about the population. (If we knew, there was no point in collecting data)
𝐻0 is rejected only if there is strong evidence against it, otherwise, 𝐻0 is failed to be rejected
- if 𝐻0 is rejected, it doesn’t mean that 𝐻𝑎 is true. It simply means that the data strongly favors 𝐻𝑎
- if 𝐻0 is failed to be rejected, it doesn’t mean that 𝐻0 is true. It just means that there is not enough evidence in the data to reject it
The interesting part is to interpret our results correctly. Notice that conclusions like “My level 𝛼 test accepted the hypothesis. Therefore, the hypothesis is true with probability (1 − 𝛼)” are wrong! Statements 𝐻0 and 𝐻𝑎 are about a non-random population, and thus, the hypothesis can either be true with probability 1 or false with probability 1
- If the test REJECTS the hypothesis, all we can state is that the data provides sufficient evidence against 𝐻0 and in favor of 𝐻𝑎. This happens only because EITHER:
- 𝐻0 is not true
- 𝐻0 is true, and the sample is too extreme (this can only happen with probability 𝛼)
- If the test ACCEPTS the hypothesis, it only means that the evidence obtained from the data is not sufficient to reject it. In the absence of sufficient evidence, by default, we accept the null hypothesis
Statistical Hypothesis Test - Type I Error & Type II Error
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Link to original
- Type I Error occurs when we incorrectly REJECT the TRUE null hypothesis
- Type II Error occurs when we incorrectly ACCEPT the FALSE null hypothesis
Accept 𝐻0
Reject 𝐻0
𝐻0is true
correct
Type I Error
𝐻0is false
Type II Error
correct
probability of rejecting a TRUE null-hypothesis is the significance level (𝛼) of a test
- 𝛼 = 𝐏(reject 𝐻0 | 𝐻0 is true)
probability of rejecting a FALSE null-hypothesis is the power (𝑝) of the test
- 𝑝 = 𝐏(reject 𝐻0 | 𝐻0 is false)
Statistical Hypothesis Test - Types
Subpages
- Statistical Hypothesis Test - 1-Tailed & 2-Tailed
- Statistical Hypothesis Test - Effect (Statistics)
- Statistical Hypothesis Test - Null Distribution
- Statistical Hypothesis Test - Type I／II／One／Two Error
- Statistical Hypothesis Test - Types