7.4 Confidence Intervals

Question 1

Review

Answer 1

Understand proportions for both population and sample (p= mean also)

Question 2

Estimate of proportions of TOI

Answer 2

Guessing to see where the mean will be (successes guess/ sample size- average of TOI)

Question 3

Confidence interval for PARAMETER

Answer 3

!!! For UD- the ALL!!! If we ask ALL (P) for TOI, then this interval shows that we’re 95% confident that this parameter (TOI/ pop) will be in between the p^s that represent the TOI as well.

OR distance between p^ and p will be LESS than 1.96 times SEest (m)
OR p is *BETWEEN (not more than but less than 2 SEs/ 1.96) each more or less’s SE (can be negative SE but just not more than ((more negatively than)) 2 SEs)
* if you need clarification on this look at page 4 graph i drew

!!So just combine the notions -> p^ +_ 1.96 TIMES SE equation
- find SE
- multiply that by 1.96 !! Leave answer rounded to 4 places (or convert to percent for show)
- take THAT and +/- to the p^ (fill in p^ at beginning of equation)
- put subtracted answer as first entry of interval and added as second
!!!! Since its 95% sure, there is 5% chance I miss the parameter

Question 4

Margin error (m)

Answer 4

Equation: 1.96 times SEest
Or 2nd way to find m-> 2nd # of CI - p^ / 2
!!! UD sentence!!! From ALL, we’re 95% sure that p^ plus/minus m would TOI (m being all the equations before +_’s result

See if the PARAMETER is WITHIN the margin of error
Translation: the estimate (p^) gives a margin error because of the sample size (n) -> too small (remember we’re showing that bigger samples are bigger)
So !!!! the parameter (containing the perfect amount of sample- to its own raw standard) will fall within the m if the p^ has right amount of sample size TOO and accounts for the error

SECOND part/FORM of CI (p) and m : p^ +_ m !!!!! <- literally m is so imp for making the literal CI
- create interval with the two answers
- is p within this interval?

Question 5

Finding p^ from given CI

Answer 5

45% + 41% (CI #s)
—————— = p^
2
!!! Remember its p^ because we’re working with p^ to see where it falls around p in the CI equation (so when we break up the CI equation like this ^, it will be p^ and not p we’re looking for)

Question 6

Compare width of m to width of CI

Answer 6

Subtract the bigger CI # - the smaller one = compare this result with the other CI to see which is wider (bigger result)

Question 7

For finding success if p^ is given and SC and confidence LEVEL !!!

Answer 7

0.24 = x / 601 -> multiply 601 by both sides
Find CI on statcrunch:
Proportion stats - one sample- with summary
Put x in “# of succcess”
N in “# of observations”
Select CI - LEVEL 0.95 and standard -wald

Confidence level= success rate of the method of finding CIs

Question 8

Zooming out more (getting confident about CIs)

Answer 8

250 CIs (.95) = 23.75 CIs captures p

Out of every 250 CIs, 95% of them capture p
So 5% of CIs dont capture p !!!!!(250 times .95)

250 - 23.5 (answer to 95%)= 12.5 CIs dont capture p

! Wider CIs are more confident (bigger # than 1.96)
_ it can fit the max min values of any lower CI range within it (fits values of 95% CI and 90%)
!!!Just cuz it seems like wider= more room for error, its actually more room for successes

Each is centered at given, singular p^ tho

!!! “If company was to conduct 200 surveys of same n (200, each with a CI) , so more CIs, 95% of them capture p

Question 9

Getting more confident in SC (and increase/decrease stats !!!)

Answer 9

! Wider CIs are more confident (bigger # than 1.96)

So go 99% confidence -> allows ur self a 1% error (bigger conf. -> smaller margin of error)
Get as many sample size units as possible

Use SC to see how exactly 99% is WIDER than other intervals

Bigger n= smaller SE= smaller m = bigger conf (wider CI)
Bigger n -> thin spread on dotplot (low variability- so many people start to look like sheep)
Bigger SE (small n) -> makes it more spread (opposite) ‘
!!!! P stays same as n changes because n is SAMPLE size not POP sizes

Use SC to see this based on the different sample sizes (n)

Question 10

Requirements of Central limit theorem

Question 11

Plausible claims / EVIDENCE

Answer 11

If the p claim from researchers (language or number) is within the CI

EX: p= majority do TOI , so its plausible if the CI contains at least 50% and above so it will capture that majority

Its literal evidence since its 95% sure and the confidence is in values that are well above the 50% (majority) mark

If an old p^ is in the interval then the sample proportion from 2016 hasn’t changed much til now CI (2017)

Question 12

UD CI

Answer 12

The confidence interval for ALL
For the proportion of ALL people in the country (p/ parameter) that TOI
Where p will land, where the average of actual TOIing will MOST land

Question 13

Remember

Answer 13

.074 > .048
Its the bigger # in the lineup

Question 14

Future bella