AP Stats Unit 6 & Confidence Interval & Significance Test

Confidence intervals and significance tests are for

Population

when estimating the proportion of success in a population use:

One sampl z interval

Conditions for one sample z intercal

independence in methods to collect the data

Conditions for one sample z interval for population proportion

1. The data are collected using a random sample from the population

2. . When sampling without replacement, the sample size is less than or equal to 10% of the population size

3. Both np(hat) >10 & 1-p(hat >10 at least ten
   
   

Confidence interval equation

Point estimate (statistic given) ± margin of error

margin or error equation

Critical value × standard error - (critical value is amount of confidence want in interval & standard error is an estimate of the standard devviation of the sampling distribution

If you don’t have a p(hat)

use 0.5

Interpret Confidence Interval

We are C% confident that the interval from ___ to ___ captures the (population parameter)

Justify a claim based on confident interva

Only if all intervals are consistent with the claim

Interpret Confidence Level

*In repeated random sampling with the same sample size (n=) (context), approximately C% of of C% confidence intervals will capture the population proportion.

EX: If we take many random samples of size 60 from the population of students at this highschool and use each sample to construct a 95% confidence intervals for the proportion of all students with a driver’s license, about 95% of those intervals would capture the population proportion.

Factors that effect margin of error (smaller)

1. margin of error gets smaller when sample size increases- inverse

2. The confidence is smaller- makes it more narrow

null hypothesis

A claim with “no difference” or “no change” -given proprotion is correct- we assume the null hypothesis is correct unless we have convincing evidence otherwise. H(o) = .5- always equals

Alternative Hypothesis

The claim we hope to support with evidence from data selected- so H A more than 50% of students would choose green cup- inequality greater or less than or not equal- Not equal is two-sided, greater or less than is one-sided- NEVER INCLUDE AP STATISTIC IN THE HYPOTHESIS

Conditions for one sample z test

1. The data is collected using a random sample

2. When sampling w/o replacement, sample size is less than or equal to 10% of the population size

3. Both np(null) and n(1-p null) are greater or equal than 10 Null hypothesis

Interpreting p- values- lemoande study

Assuming 50% of all students at this school would choose the green cup, there is a .1357 probability of getting a sample proportion of .60 or greater by chance alone in a random samples of 50 students from this school.

For differs, you say as extreme or more extreme than .29 in either directioction by chalnce along

Default significant level

0.05

If the p value is less than 0.05, then we

If the p value is greater than 0.05- we

1. Reject the null- convincing stat evidence

2. If p value is larger, we fail to reject- DON"‘T ACCEPT- not enough convincing stat evidence

Confidence interval acronym

PANIC

P- Parameter

A- assumption (radom sample or random assigenment, 10%, and large grous n times p and n(1-p)

N- Name what test?

I Interval (caclualte

C- conclsuion, interpret

When caculating stadnard error for ONE SAMPLE Z TEST- ALWAYS USE THE NULL HYPOTHESIS- NOT THE STATISITCS

Mean of p value in hypothesis test

The mean of the p-value in a hypothesis test represents the likelihood of observing the sample data, or something more extreme, assuming the null hypothesis is true.

what does the significant level of a null hypothesis mean

the probability of rejecting the null when the null hypothesis is true

Type 1 errors

Type 1 errors occur when the null hypothesis is incorrectly rejected, - you reject null but you are wrong

Type 2 errors

occur when the null hypothesis is not rejected when it is false. We fail to reject null but we are wrong-it needs to be rejected- mention convincg evidence

Probability of a type 1 error

the significance level (alpha)- so like 0.05 or. 10- so lower significance level is better- but makes a type 2 error- DECREASING PROBABILITY OF TYPE 1 ERROR INCREASES PROBABILITY OF TYPE 2 ERROR

power

the probability of avoiding a type 2 error- probability that a test will correctly reject a false null hypothesis- probability of finding convicing evidence for alternante when alternate is true

probability of a type 2 error

1- power

Factors that affect power - power of a test will be greater if:

sample size increases, significant level increaes, standard error decreases, the true paramter is farther from the null

Confidence intervals for the difference of two proportions conditions

1. random samples for both populations

2. 10% condition for both

3. Large counts condition for both - p hat (statistics)( both success and failure of ample greater than 10)

Interpret confidence interval for difference in population proportion- tree with 90% confidence

We are 90% confident that the interval from -0.029 to 0.079 captures the difference (High-low) in the proportion of all trees in the forests that have died from the disease

Justifying claim for difference

If two things are the same, the difference in the proportion would be zero- SO IF ZERO IS IN THE INTERVAL, THERE IS NOT CONVCING EVIDENCE IF THE DISEASE IS MORE LETHAL

interpreting confidence level

If all possible random samples of n from contexxt and a confidence interval was constructed from each pair of samples, then confidence level of all these intervals would succeed in capturing the differenece (day-night) in the proportion of all parts produced within speciifications by two shifts

Conditions for a two sample z test for population FOR DIFFERENCE IN PROPORTIONS 0 start with combined

For this test, , you need to combine the groups to find a p hat combined- add numberator and denoomintaro together and then fidning average

Actually conditinos

1. two indpendnet random samples from each populaigon

2. 10% sample without replaecement

3. the expected number of success and failures are at least 1o- USE P HAT COMBINED JUST FOR THIS

P VALUE

the probability of observing a test statistics as extreme or more extreme than the observed test statistics when the null hypothesis and probability model aer assumed to be true

Interpret p value for the difference of two population proprotion - two sample t test for proportion

Assuming the difference (A-B) in the trueproprotions of pink eye in pateitns like the ones in this experiement who would be cured is 0, thre is a 0.0122 probability of gettiing a different in proportion of .134 or greater, by chance along in the random assignement