QTM Test 3

Central limit theorem for sample mean

When we collect sufficiently large sample of n independent observations from a population with mean and a standard deviation, the sampling distribution of will be nearly normal with mean and standard error

Two conditions for central limit theorem

1. Independent observations (random sample, <10% of pop)

2. Normality: if n< 30 and no clear outliers, we assume, the sample came from a nearly normal population

   If n>= 30 and there are no extreme outliers, then We assume WHATTT

To test for mean we use

T distribution

What is t distribution used for

• used for large and small samples because p-values are determined for each sample size using degrees of freedom

WHATT

Properties of t distribution

• bell shaped

• Symmetric so t is + or -

• Mean =0, s >1

• Less peaked

• Different curve for each degree of freedom

Hypothesis test: one mean t-test

H0: mean =#

HA: mean =/ #

Conditions for hypothesis test: one mean t-test

1. Independent observations

2. WHAT

Areas on t table get ____ as you move right

smaller

Paired data

Example of pairs data

Before/after data (SAT before and after)

To analyze paired data, look at the

Difference in outcomes of each pair of observations

Hypotheses for mean of differences

H0=average difference (before-after) = 0

Ha= average difference(before-after) =/0

Paired data aka

Difference of means

Conditions for paired data mean test

1. Independent observations

2. Normal distribution

What line on test 3 formula sheet is for paired data

3

If you can’t find degrees of freedom in paired mean problem

Go to nearest number that is SMALLER (ex: if df=199, you go to 150 NOT 200)

Interpretation of inference alt vs null claim

Alt: “support”

Null: “reject”

Type 1 vs Type 2 Error

If CI is + or -, mean difference is

Higher/lower