AP Stats Ch 9 & 11

null hypothesis (H⌄0)

a claim that we weigh evidence against in a significance test

alternative hypothesis (H⌄a)

a claim that we are trying to find evidence for

one-sided alternative hypothesis

states that a parameter is either greater than or less than the null

two-sided alternative hypothesis

states that the parameter is different from the null value (greater than and less than)

P-value of a test

the probability of getting a p̂ or more extreme purely by chance

significance level (𝛼)

the value that we use as a boundary for deciding whether an observed result is unlikely to happen by chance alone when the null hypothesis is true 

• 𝛼 usually is 0.05 or less - statistically significant

• 𝛼 should be stated before the data is produced

standardized test statistic (z score)

measures how far a sample statistic is from what we would expect if the null hypothesis were true (in standard deviation units)

• standardized test statistic = (statistic - parameter) / standard deviation of statistic

significance test: the PHANTOMS process

• Parameter: P = _______

• Hypothesis: H⌄0: ____, H⌄a: ____, p̂ = ____, & 𝛼 = ____

• Assumptions: random, 10% condition, large counts condition

• Name: “All conditions are met to create a one-sample (proportion) Z test for P.”

• Test statistic: z = ____ 

  • draw N(0, 1) normal distribution for z

• Obtain P-value: (& interpret)

  • one-sided: P-value = P(z ≥/≤ ___)

  • two-sided: P-value = 2P(z ≥ ___)

  • state: “Assuming H⌄0 is true (P = ___), there is a <p-value> probability of getting a sample proportion of <p̂> or <more extreme / greater than / less than> purely by chance.”

• Make a decision about H⌄0:

  • P-value < 𝛼 → reject H⌄0

  • P-value > 𝛼 → fail to reject H⌄0

  • no naked numbers

• State your conclusion about H⌄a:

  • reject H⌄0 bc convincing evidence for <H⌄a in context> 

  • fail to reject H⌄0 bc, not convincing evidence for <H⌄a in context>

type I error

occurs if we reject H⌄0 when H⌄0 is true (false positive)

• P(type I error) = 𝛼

• “We found convincing evidence for <H⌄a> when in reality <H⌄0> is true.”

power of a test

the probability that the test will find convincing evidence for H⌄a when H⌄a is true

the power of a significance test will become larger when…

• the sample size (n) is larger

  • decreases variability

• the significance level (𝛼) is larger

• the null and alternative parameter values are farther apart

difference in proportions significance test: the PHANTOMS process

• Parameter: 

  • P⌄1 = _______

  • P⌄2 = _______

  • P⌄1 - P⌄2 = the true difference in proportions of _______________  _______ vs. _______ (<P1> - <P2>)

• Hypothesis: 

  • H⌄0: P⌄1 - P⌄2 = 0

  • H⌄a: ____

    • P⌄1 - P⌄2 > 0 (1 tail)

      • P⌄1 > P⌄2

    • P⌄1 - P⌄2 < 0 (1 tail)

      • P⌄1 < P⌄2

    • P⌄1 - P⌄2 ≠ 0 (2 tail)

  • p̂⌄c = (x1 + x2) / (n1 + n2) = ____

  • 𝛼 = ____

• Assumptions: 

  • Is the sample random?

  • Do the samples meet the 10% condition?

  • Do the samples meet the large counts condition?

    • n1(p̂⌄c) ≥ 10 & n1(1 - p̂⌄c) ≥ 10

    • n2(p̂⌄c) ≥ 10 & n2(1 - p̂⌄c) ≥ 10

• Name: 

  • “All conditions are met to create a two-proportion Z test for P⌄1 - P⌄2.”

• Test statistic: 

  • z = [(p̂⌄1 - p̂⌄2) - (P⌄1 - P⌄2)] / S⌄p̂⌄1 - p̂⌄2 

  • S⌄p̂⌄1 - p̂⌄2 = √[p̂C(1 - p̂C)/n1 + p̂C(1 - p̂C)/n2]

  • draw N(0, 1) normal distribution for z

• Obtain P-value: 

  • one-sided: P-value = P(z ≥/≤ ___) = ____

  • two-sided: P-value = 2P(z ≥ ___) = ____

• Make a decision about H⌄0:

  • P-value < 𝛼 → ________ ∴ reject H⌄0

  • P-value > 𝛼 → ________ ∴ fail to reject H⌄0

• State your conclusion about H⌄a:

  • reject H⌄0 bc convincing evidence for (H⌄a in context) 

  • fail to reject H⌄0 bc there is not convincing evidence for (H⌄a in context)

significance test for a population mean

μdiff

the mean difference in A versus B (A-B) (one-sample t test for μdiff)

significance test for a difference in means