Ch 9/Ch 11: Testing Claims about Proportions/Means

significance test

significance test

 a formal procedure for using observed data to decide between two competing claims (called hypotheses) about given parameters

null hypothesis (Ho)

the claim that we weigh evidence against in a significance test

alternative hypothesis (Ha)

the claim that we are trying to find evidence for

one-sided alternative hypothesis

states that a parameter is greater than the null value or if it states that the parameter is less than the null value

two-sided alternative hypothesis

 states that the parameter is different from the null value (it could be either greater than or less than)

always state null and alternative hypotheses in terms of ______________

population parameters (NOT sample statistics)

if the p-value is less than or equal to the significance level (alpha)

reject H_o  and conclude there is convincing evidence for H_a (in context)

if the p-value is greater than the significance level (alpha)

fail to reject H_o  and conclude there is not convincing evidence for H_a (in context)

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

Type I error

 if we reject H_o  when H_o  is true —> the data give convincing evidence that H_a is true when it really isn’t

Type II error

occurs if we fail to reject H_o when H_a is true —> the data do not give us convincing evidence that H_a is true when it really is

probability of a type __ error is equal to the significance level

I

Ho true, reject Ho = _____ error

type I

Ha true, fail to reject Ho = _____ error

type II

Ho true/fail to reject Ho OR Ha true/reject Ho

correct conclusion

relationship between type I and type II errors

s the probability of a Type I error increases, the probability of a Type II error decreases

type I error is also known as a false ______

positive

type II error is also known as a false ______

negative

condition for significance test about a proportion - RANDOM

 the data come from a random sample from the population of interest

condition for significance test about a proportion - INDEPENDENT

10% condition: n < 0.10N

condition for significance test about a proportion - NORMAL

large counts condition → both np₀ and n(1-p₀) are at least 10.

p₀

 our hypothesized population proportion (aka the proportion we are assuming is true in our null hypothesis)

standardized test statistic

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

equation for standardized test statistic

(statistic-parameter)/standard deviation (error) of statistic

standardized test statistic if we KNOW the population standard deviation is ___

z

standardized test statistic if we DO NOT KNOW the population standard deviation is _____

t

power

the probability that the test will find convincing evidence for H_a when a specific alternative value of the parameter is true OR probability that we find convincing evidence that the alternative hypothesis is true, given that the alternative hypothesis really is true

equation for power

Power = 1 - P(Type II Error)

equation for probability of type II error

P(Type II Error) = 1 - Power

to increase power:
______ sample size (n)
______ significance level (α)

_______ distance between null and alternative parameter values

INCREASE

condition for performing a significance test about a difference between two proportions (RANDOM)

the data come from two independent random samples or from two groups in a randomized experiment

condition for performing a significance test about a difference between two proportions (INDEPENDENT)

10% condition: n1 < 0.10N AND n2 < 0.10N

condition for performing a significance test about a difference between two proportions (NORMAL)

 Large Counts Condition: expected numbers of successes and failures in each sample or group are all at least 10 (use p-hat cbmbined)

equation for p-hat combined

the Normal condition for a significance test about a mean can be satisfied in these 3 ways:

 1) told the population is Normal

2) n is large (greater than 30, per the CLT)

3) graph and sketch a dotplot/histogram and assess shape (look for massive skew or outliers)

formula for one-sample t-test for means

in what situation should you select “pooled” on your calculator?

NEVER

equation for degrees of freedom

df = n-1

inference procedure for paired data

one-sample t procedures for mean difference

inference procedure for quantitative data from independent random samples of two populations of interest/groups in a randomized experiment

twp-sample t-tet for mu1-mu2

if a condition is NOT met…

"although this condition has not been satisfied,

we will assume that it has been, and proceed

with caution."

characteristics of a paired t test

-one sample (data are paired)

-mean of differencces

-subtract, then average

characteristics of a two-sample t test

-two independent samples/groups

-difference of means mu1-mu2

-average, then subtract