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

   f n ≥ 30 and there are no extreme outliers, then we
   assume the sampling distribution is nearly normal

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 (aka based on sample size)

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. approximately normal

Areas on t table get ____ as you move right

smaller

Paired data

each observation in one set is related to or
corresponds with exactly one observation in
the second sample

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 t-test aka

mean of differences

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

Mean of differences

From dependent or matched pair samples (aka paired t-test)

difference of two means

From independent samples

Look at means of both samples

Mean 1= mean 2 (mean 1-mean 2=0)

Can difference of two means have different sample sizes? What about mean of differences?

Yes

No

Conditions for 2 means

1. Both sample approx normal

2. Independent observations within the samples

3. Independent observations BETWEN SAMPLES, meaning two samples are not related

Which degrees of freedom do you use for two means

Smaller N-1

How do you recognize two means

Two of everything: SD, mean, samples

ANOVA

Analysis of variance

Used to compare means from two or more groups by using variance

Variance

Measures spread of data

Between vs within in anova formula is

On the top vs on the bottom

Variability within groups in anova is also known as

Sampling error

Variability between samples is

variability within samples is

Due to different treatments

Due to regular sampling error

F distribution is for

ANOVA

F distribution is always

Right skewed

Large F statistic

Higher variability between each group

Repeated measures

Use the same group of subjects with each treatment

One way ANOVA

Samples are compared using one category/factor/characteristic

Conditions for ANOVA

1. Observations should be independent within groups

2. Observations should be independent between routes

3. The observations within each group should be nearly normal

4. Variances across groups should be about equal

Post hoc

After the fact

ANOVA table: Row 1

Variability between sections

ANOVA table: Row 2

Residuals (aka error aka WITHIN)

K in Anova

Number of categories

Degrees of freedom of sections for ANOVA

K-1

Degrees of freedom in residuals in Anova

N-K

Sum sq

Sum of squares

How to find sum of squares in ANOVA

Difference, square, add up (WHAT THOUGH)

Mean sq

Subtract every grade from mean square and add up

Get by divide sum sq by df

Pr(>+F)

Probability of getting f or more extreme aka P VALUE

Power of the test

Probability of correctly rejecting the null (1-beta)

To increase power of a test

Increase sample size

Increase alpha (but type 1 more likely)

Decrease standard deviation

sampling distribution would be

the mean and SD of several means

formal definition of sampling distribution

distribution of a statistic across an infinite number of samples

SE of sample means

SD/sq root of sample size

population vs sampling distribution

population: more spread out because larger SE

sampling distribution: less spread out

most frequently used test statistic

t

t is more accurate for

small samples

what parameter is needed for t distribution

standard deviation

as degrees of freedom increase, the t distribution

approaches standard normal (aka peak goes higher)

unique aspect of tails of t distributions

tails are thicker, meaning more observations are more likely to fall beyond 2 SD from mean

t score and z scores are similar in that

both measure spread on one

how to find estimated p value

look on df row
for your test statistic, t.s., then go
straight up to the top for p-value for interval like chi square

interpreting inference format

there is not enough/enough evidence to support (alt)/reject (null) the claim that _____________. Then compare different.

interpretation of p value

if the mean ___ is ___,
the probability of getting our sample mean of _____ or more extreme is less than [pvalue], which is highly unlikely/likely.

interpreting confidence interval

we are % confidence that the true population mean is between _____ and _______

type 1 vs type 2 errors

type 1: reject null but you should’ve failed to reject

type 2: fail to reject but you should’ve rejected

how to find sample size of the mean

first formula on formula sheet (always round up)

how to get two sets of data into one

subtract

Parameter for paired t test

Average difference
between the reading and writing scores of
all high school students.

Point estimate for paired t test

Average difference between
the reading and writing scores of sampled
high school students.

subscript for paired data

d

hypothesis for two means test

H0: Mean 1- Mean 2 =0

HA: Mean 1- Mean 2 =/ 0

example of interpreting beta value (beta =0.15)

if beta = 0.15, the probability of failing to reject the null hypothesis when you should have rejected it is 0.15

the power would then be 1-0.15=0.85 so probability of correctly rejecting the false null hypothesis is .85

rejecting null means you found a difference so

if you tested hypothesis 100
times, [power %] of the tests would
detect the difference

hypothesis test for ANOVA

H0: μ1 = μ2 = ... = μk
HA: At least one mean is different

how to get f in ANOVA

mean sq between/mean sq within

how to set up post hoc tests

find Bonferroni alpha (a*) => used as new alpha for comparison

steps for post hoc tests

1. get means and populations for each option

2. plug into formula with whichever two data you are comparing (have to do for all combinations!)

how to get sample error from anova table?

mean sq from residual

overall anova conclusion

some combinations will work and some may not so write that in conclusion and make inference based on data

what is linear regression?

Determines if a linear relationship exists
If one numerical variable can predict another

can linear regression show you causation

no

how to model relationship between two variables

linear regression line w/ points in (x,y) form

x vs y in linear regression

x= explanatory variable

y= response variable

points in linear regression model

usually do not fall exactly on line, instead cluster around the line

why is regression line called a regression line

Regress means to move
towards a previous or less
formed state, so the
points would all be moving
toward the line in a
perfect world

correlation coefficient

described with R

scale from -1 to 1, with -1 and 1 having the strongest correlation (line) and 0 being the weakest (no correlation)

different names for regression line

line of best fit, least squares line

residuals of error

how far the points are from the line

difference between observed data and predicted data

how to calculate residual

observed – predicted (formula with e)

variables in residual formula (y and y-hat)

y= observed data.
y-hat= predicted data.

Residual > 0

predicted data is an
underestimate.

Residual < 0

the
predicted value is an
overestimate.

The Least Squares Method

square all the residuals/errors, add them

purpose of least squares method

to find the best line of fit

why least square method?

• Most commonly used
  • Easier to compute
  • Highlights the errors

basis formula for line

y=mx+b (m is slope and b is y intercept)

how to find slope for regression line

formula with b1=

how to find intercept for regression line

formula with b0=

interpret the slope

For each additional % _____ in ______, we would
expect the % _______ to decrease/increase (based on positive or negative) on average by ___% (slope)

interpret the intercept

situation when explanatory variable (x) is 0, ___% (intercept) of situation

interpretations of the intercept tend to be

unbelievable