U5: Hypothesis Testing

Hypothesis Testing

● A hypothesis test is used to see whether an assumption is statistically plausible by using sample data

● The basic formula for a hypothesis test is: Statistic - Parameter/Standard Deviation of Statistic

● The higher the Z or t score, the lower the p value, and the more evidence there is to reject the null hypothesis

Five Steps

1. Hypothesis

2. Conditions/Assumptions

3. Formula

4. P Value

5. Conclusion

Step 1: Hypothesis

● “Where [µ, p, µ1, µ2, p1, p2] is [context of problem]”

○ Define ALL parameters in the context of the problem

● Whether Ha is <, >, or ≠ depends on the problem

Step 2: Conditions/Assumptions

● Random Sample

○ “The stem of the problem states that [sample] was chosen at random”

○ “The stem of the problem states that [participants] were randomly assigned to the groups”

● Approximate Normal Distribution

○ “The stem of the problem states the distribution is approximately normal”

○ “Since n = _ ≥ 30, by the Central Limit Theorem, we can assume the distribution is approximately normal”

■ For two samples, both n1 and n2 must be ≥ 30

○ “Since np = _ ≥ 10 and nq = _ ≥ 10, we can assume the distribution is approximately normal”

■ For two sample, n1p̂1, n1q̂1, n2p̂2, and n2q̂2 must all be ≥ 10

○ “Since the [graphical display] shows no outliers or strong skewness, we can assume the distribution is approximately normal”

■ You must provide a graphical display (preferably a box plot) if normality cannot be assumed by the other three ways

Step 3: Formula

● List the formula, your substitution, degrees of freedom (if using t) and your unrounded answer

○ One-Sample Means

○ Two-Sample Means

○ Proportions

Step 4: P Value

● The probability of obtaining a test statistic (Z or t) that is this much or more extreme

● If Ha is <

○ P (Z < _)

● If Ha is >

○ P (Z > _)

● If Ha is ≠

○ 2P (Z > _) if Z > 0

○ 2P (Z < _) if Z < 0

Step 5: Conclusion

● “Assuming H0 is true, since the p value ([p value]) is [greater/less] than α = _, we [fail to reject/reject] H0”

○ Α α will usually be given in the problem. If it is not, use α = .05

● “We [do not/do] have sufficient evidence to suggest Ha, that [context of problem]”

Type I and II Errors

● Type I Error: You reject H0 when you should not have

○ P (Type I Error) = α

● Type II Error: You fail to reject H0 when you should have

○ P (Type II Error) = β

○ Power of the test = 1 - β

■ P (Rejecting H0 when you should have)

■ Increases as α increases

● Which one is worse depends on the scenario

Match Paired t-Testing

● Most often used in a “before and after” scenario (e.g. dexterity before and after the subjects undergo a program) and comparing two things (e.g. amount of active ingredient in a name brand and generic brand drug).

● Hypothesis

○ H0: μd = 0

○ Ha: μd > or < or ≠ 0

○ *Where μd is the average difference (After - Before)

● Assuming Normality

○ Draw a box plot of A - B

● Compute the Match Paired t-test as if it were a normal hypothesis test (5 steps!)

Z vs t Distribution

● Means

○ Use a Z distribution when you have σ

○ Use a t distribution when you do not have σ (i.e. you have S)

● Proportions

○ ALWAYS use a Z distribution

Degrees of Freedom

● Only applies to t-distributions

○ The t-distribution varies with degrees of freedom

● df = n - 1

● For a Z-distribution, df = ∞

Calculator (TI-84 Plus)

● Stat → Test

○ Means

■ 1 = One-Sample Z-Test

■ 2 = One-Sample t-Test

■ 3 = Two-Sample Z-Test

■ 4 = Two-Sample t-Test

○ Proportions

■ 5 = One-Sample

■ 6 = Two-Sample