Ch 6 Inference for Proportion

when a variable is categorical, use…

proportion

when a variable is quantitative, use…

mean

steps for inference

1. formulate the research question → use hypothesis testing

2. data collection

3. check conditions

4. calculate the test statistic

5. find the p-value

6. decision

7. conclusion & interpretation

null hypothesis (H_o)

always equal to population parameter

H_o : p = x

alternative hypothesis (H_a)

what the researcher claims

ex. H_a: p > x

p < x

left-tail test

p > x

right-tail test

p ≠ x

two-tail test

n

sample size

p̂

sample proportion

normality/large sample size condition

np ≥ 10

n(1-p) ≥ 10

both must be met

z (test statistic) formula

p̂ - p / SE

SE formula

√p(1-p)/n

z (test statistic)

measures the accuracy of a statistic

z interpretation(?)

z standard error below/above the H_o value

p-value

probability that null hypothesis is true for the given sample

p-value in R for left-tail test

pnorm(z)

p-value in R for right-tail test

1-pnorm(z)

p-value in R for two-tail test

2*pnorm(-x)

always use negative side

α (alpha)

used for level of significance

ex. 5% level of significance - α = 0.05

(decision rule) if p-value < α …

reject H_o

(decision rule) if p-value > α

do not reject H_o

if reject H_o…

“we have enough/sufficient evidence to conclude the H_a.”

if do not reject H_o…

“we do not have enough/sufficient evidence to conclude the H_a.”

high z-score =

more likely to reject (lower p-value)

if inc. sample size (n), what happens to SE?

it will decrease, and z (test statistic) will get larger.

with a larger test statistic (z), what will happen to the p-value?

it will dec. (think abt the proportion on a normal dist.)

what are the 2 types of error?

Type I error

Type II error

type I error

reject H_o when it’s actually true

type II error

do not reject H_o when it’s actually false

ex. H_o : Connor’s car is safe to drive

what is a type I error?

say Connor’s car isn’t safe to drive when it is

ex. H_o : Connor’s car is safe to drive

what is a type II error?

say Connor’s car is safe to drive when it’s not.

if you wanted to reduce the chance of a Type II error, would you use a larger or smaller alpha level?

larger

if you wanted to reduce the chance of a Type I error, would you use a larger or smaller alpha level?

smaller

confidence interval for proportion

when p is unknown; use p̂ (and n)

likely to capture population proportion p with high confidence

large sample size condition (when you dk p)

np̂ ≥ 10

n(1-p̂) ≥ 10

confidence interval calculation

p̂ ± z* (SE)

critical value (z*) is associated with…

confidence level

what part of the curve is being measured when it’s a 95% confidence interval?

the middle 95%

(UB = 0.975, LB = 0.025)

how do you calculate z* in R?

qnorm(UB) or qnorm(LB)

ex. qnorm(0.95) / qnorm(0.05) for 90% confidence interval

confidence interval interpretation (ex. 95% conf.)

“we have 95% confidence that the true/population proportion is btwn (LB)% and (UB)%.”

explain what confidence interval means (ex. 98%)

if you collect many samples, 98% of the confidence intervals produced will capture the true proportion.

margin of error (ME)

z* (SE)

how do you find p̂ & MOE given 95% confidence and an interval of (0.25, 0.85)

find the avg. of the UB & LB to get p̂.

find MOE by UB - p̂ or p̂ + LB

if n inc., what happens to SE, ME, & the interval?

SE dec, ME dec, the interval becomes narrower

if CL inc., what happens to z*, ME, & the interval?

z* inc, ME inc., wider interval

relationship between α and CL.

add up to 100

ex. α = 0.05, CL = 95%

ex. α = 0.01, CL = 99%

if you reject H_o, what does that tell you about the CI and p? ex. 95% CI, p = 0.7

the 95% CI does NOT capture p (true proportion) since you rejected H_o, and p = 0.7 is outside the interval.

if you do not reject H_o, what does that tell you about the CI and p? ex. 90% CI, p = 0.85

the 90% CI DOES capture p= 0.85 since you did not reject H_o. 

what does margin of error cover?

ONLY differences in sampling. does NOT count for human error/bias

if you reject H_o, do you have evidence for H_a or not?

have evidence

within 3% / ME 3% means…

ME is 0.03

formula to find n (given ME)

n = (z*/ME)²p(1-p)

(using p since you dk p̂)

what do you use if population proportion isn’t given?

p = 0.5

when solving for n and you get a decimal, what should you do?

round up! n should always be a whole number.