QTM 100 Test 2

when do you use pooled proportion

two proportion hypothesis testing

two proportion confidence interval

use CI formula for two proportions

two proportion hypothesis testing

use p-pooled and z score formula

Minimum sample size

formula with n

Rounding with minimum sample size formula

Round up

Basics for one proportion hypothesis test

• population parameter: p

• Point estimate: p-hat

• Conditions

Majority problems

Special kind of proportion test

Test for majority problems (more than half

No majority (even): null = 0.5

Majority =/ 0.5

Steps for hypothesis testing

1. Identify variables, hypothesis, claim, alpha

2. Check conditions

3. Find test statistic/zscorei, area, p-value, sketch

4. Compare p value to alpha, conclusion

Important note

Reject null hypothesis = significant difference between null and sample => STATE VALUE HIGHER OR LOWERRR (NOT JUST DIFFERENT)

how to interpret confidence interval

we are % confident that x%-x% of ____

Proportions are for

Categorical data

Proportions are written as

%, decimal, or fraction (x/n)

Sample proportion is called

P-hat

What is p-hat written as

%, decimal, Or fraction

P-hat is written as

^

P

P-hat without hat is

Population sample

Sample stats estimate

Population parameters

Sample stat is also known as

Point estimate

Sample stat should be ____ to pop parameter

very close

Sampling distribution

Average of all p-hats (point estimates)

Standard error

Variation or standard deviation in point estimates

SE vs SD

SE= categorical

SD= numerical

can a graph be curved for categorical data

No— not continuous data

Larger sample size = ___ SE

Smaller

Central Limit Theorem**

If observations are independent and sample size is sufficiently large, sample proportion will be nearly normally distributed (mean = p aka population proportion)

How to verify independence

Random sample less than 10% of pop

How to verify sufficiently large sample

Must meet success failure condition

(Np>=10) and n(1-p)>= 10

If p is not known, you can use

P-hat

Confidence interval

range of plausible values where we are likely to find population parameter

CI written as

(60%, 70%)

What do you need in confidence interval

Parenthesis!!

More commonly used confidence intervals

A- 90% = significance level, 1-0 confidence level

Leftover = alpha

95%

99%

Cutoff scores for critical values

90%+-1.65

95% +-1.96

98$ +-2.33

99%+2.58

(Given)

How to calculate confidence level

Point estimate +- ME

Or x* points estimate +- z* (SE)

what do you use to calculate uncertainty of point estimate

standard error

variables in SE formula

n= sample #

x= observed stat

p= point estimate/sample state

how to get point estimate from confidence interval problem

average it

how to get margin of error

distance from middle to endpoint

steps to solve confidence interval question

1. check conditions

2. find point estimate/sample stat (aka phat) and z score (on chart)

3. calculate with formula

4. put into words

parameter is also known as

population proportion

success failure condition

np>10 and (1-p)n>10

point estimate

sample value use to estimate population parameter

example of point estimates

sample mean, sample proportion, sample standard deviation

error

difference between observations and the parameter

when conditions are not met distribution is

discrete (not continuous)

skewed

why do we need a confidence interval?

cuz point estimates will likely not exactly hit population proportions

margin of error formula in confidence interval

z*+-SE

why do you need to check conditions

to make sure distribution will be near normal

What is statistical hypothesis testing?

Decision making process for evaluating claims mathematically

Distinguish between results that easily occur or are unlikely

What is a hypothesis

Claim or conjecture that may or may not be true

Two types of hypotheses in a test

Null and alternative

Null hypothesis

Always =, no difference, no change, status quo, written first (on top), H0

Alternative hypothesis

Inequality: not equal, difference or change written second, Ha

Example of null

Drug does not work

Example of alternative hypothesis

Drug works, does decrease level of depression!

do we like null or alt hypothesis in real world?

Alt because we want to see changes

Setup for proportions

H0: p=#

Ha: p=/ #

**number should be the same

what test is used for setup for proportions

Two tailed test

Two possible decisions

Reject the null hypothesis or fail to reject the null hypothesis

reject the null hypothesis

Not the same (significant difference)

fail to reject null hypothesis

Data is not convincing, no sig difference

Testing hypothesis using confidence intervals

Confidence interval: 95% sure real result will be captured in interval

So if result is within confidence interval, null hypothesis is TRUE (fail to reject)

Decision errors

Hypothesis tests are NOT flawless

Type 1 error

Reject null hypothesis when H0 is true

Type 2 error

Failing to reject null hypothesis when it is false

Alpha

probability of making a type 1 error (reject null when it is true)

If making type 1 error (null is really true) is dangerous, choose a

Small significance level (a=0.01) and be careful about rejecting null hypothesis aka demand strong evidence before rejecting null

If type 2 error (alternative is true) is dangerous, choose

Higher significance level (0.10) and be cautious about failing to rject H0 when null is actually false

Hypothesis testing using z score and p value

1. identify parameter, list hypothesis, identify significance level, identify p-hat and n

2. check conditions

3. calculatse z score and identify p value

4. conclusion (compare p value to alpha)

P value

Probability value is z-score area TIMES 2 because it is on both left and right side

If p value < a

Reject a

If p value > a

Fail to reject H0

Let alpha be ___ if not given

.05

why should we use p-value to test hypothesis instead of CI

CI is not always sustainable cuz confidence interval cannot always be constructed

interpretation for p-value

If the null hypothesis is true that [add context], then the probability

of getting our sample proportion of [context] or even more extreme

is

[p-value], which is [highly unlikely or plausible]

interpretation of inference

there is convining evidennce to reject the claim that there is no difference in the % of 1st and 2nd years that prefer JFK. Out data shows that more 2nd years like JFK

IF observations are independent,
sample size is

sufficiently large

parameter being estimated

proportion vs mean

standard error vs margin of error

margin of error is used to calculate margin of error, standard error measuresr variability

significance level vs confidence level

significance = alpha

confidence = xx% confident that blah blah

p-value is the

probability
of observing data at least as
extreme as the one found in
our sample data set, if the
null hypothesis is true

degrees of freedom in goodness of fit test

number of categories minus 1

Chi-square looks like

X²

Chi Squared is

• right skewed

• Values are positive because they are counts (how many people)

• Categorical data

• One parameter: degrees of freedom

Degrees of freedom are represented by

df

Goodness of Fit Test

Does sample fit population?

Does it represent population?

Do the observed counts equal the expected counts?

Conditions for chi square

1. Independent observations

2. E (expected) count is at least 5 in each category

3. Degrees of freedom at least 2

Setting the hypothesis for chi square s

NO SYMBOLS OR PARAMETERS => use sentences

General setting hypothesis format

H0: Observed counts follow same distribution as the expected counts (O=E)

HA: Observed count do not follow the same distribution as expected counts

Findings areas under the chi square curve

P-value = tail area under chi square distribution

Top row = p values, alpha at top

Two tests chi squared is used for

1. Goodness of fit: compares population with sample based on one characteristic

2. Independence (“related to“) : tests for relationship between 2 characteristics

How many variables are there in any given contingency table?

TWO (not 20 if 5 by 4 okok)

General format for hypothesis in chi square

H0: variables are independent (not related)

HA; variables are not independent (are related)

Observational study

Association or relationship does not mean CAUSATION (experimentation)