Models

Statistical Significance

  • our data is almost always collected from a sample

  • If we observe something unusual in the data

    • The observed sample relationship may have occurred because of a real relationship in the population - statistically significant

    • The observed sample relationship may have resulted due to chance

H0: The Null Hypothesis:

  • The status quo: an established claim

  • presumed true unless strong evidence to the contrary is demonstrated

HA: The alternative hypothesis

  • a rival claim

  • what the person doing the test wants to prove

  • must be “proved” true

conclusions

One hypothesis must be true and the other must be false

alpha = P(type 1 error) = P(reject H0 | H0 true)

confidence level = 1-alpha = P(accept H0 | H0 true)

beta = P(type 2 error) = P(accept H0 | H0 false)

Power = 1 - beta

Test Statistic

  • the statistic that we use to decide between the two hypothesis

  • what values f the statistic will support the alternative hypothesis?

  • reject the null if test statistics are sufficient

H0: Coin is fair → P(H) = p = .5

HA: Coin is not fair → P(H) = p ≠ .5

Test statistic: sample proportion of heads = phat

Decision rule: reject H0 is phat is unusually large or is unusually small

P-values

A ,measure of the strength of the evidence in favor of HA. The smaller the p-value, the stronger the evidence in favor of HA

Sampling distribution of Phat

Phat = sample proportion

original population has Population proportion = p

Sampling distribution of Phat for samples of size n will have…

  • mean = p

  • Standard dev = sqrt(p(1-p)/n)

  • for “large samples",” will be approximately normal

NULL hypothesis: population mean = u

  • test statistic: xBar

  • Standardized: z = (xBar-u)/(SampleStdDev/sqrt(n))

is population proportion = p

  • test statistic sample proportion = pHat

  • Standardized: z = (pHat - p)/sqrt(p(1-p)/n)

Is a coin biased in favor of heads?

H0: coin is fair → P(H) = p = .5

HA: Coin is biased towards heads → P(H) = p > .5

p-value = area to the right of z = 1-pnorm(z)

  • reject H0 if p-value <alpha

  • accept H0 is p-value >= alpha

Is a coin fair?

H0: coin is fair → P(H) = p = .5

HA: noic is not fai → P(H) = p ≠ .5

p-value = 2(pnorm(-abs(z))) = 2(1-pnorm(abs(z)))

  • reject H0 is p-value < alpha

  • accept H0 if p-value >= alpha