AP Stats Chapter 8

confidence intervals

- estimate the value of a parameter from a sample statistic

- calculate probs that would describe what would happen if we used the inference method many times

95% confidence level

looking at every sample w margin of error --> 95% of all samples will include the population parameter, the other 5% won't

point estimator

a stat that provides an estimate of a population parameter (ie. mean, median, min, IQR, etc)

point estimate

the value of that statistic from a sample --> best guess at the population parameter

confidence interval formula

point estimate +/- critical value • standard deviation of the stat

margin of error formula

critical value • sd of stat

critical value

basically the z-score, standardized score

confidence levels

- tells us how likely it is that the method that we are using will produce an interval (margin of error) that captures the pop parameter if we use it many times

- DOES NOT tell u that chance the population parameter is in there

wording for confidence levels

in ____% of all possible samples of the same size, the resulting confidence interval would capture the true (detail in context)

confidence interval wording

we are 95% confident that the interval from ______ to _______ captures the actual value of the (detail in context)

when does the margin of error get smaller

confidence level decreases

sample size n increases

wording for margin of error

if we were to repeat the sampling procedure many times, on average, the sample proportions would be within ____% points of the true proportion in 95% of samples

standard error

sd of proportion but using p hat

what is the process called for creating confidence intervals

1-sample z interval for p

What are the conditions to calculate a confidence interval

random, independent and normal

confidence interval calculations steps: state

p = true proportion

___% confidence level

confidence interval calculations steps: plan

method: 1-sample z interval for p

conditions

- random

- normal (large counts)

- independent (10%)

confidence interval calculations steps: calc (1 Sample Z interval for P)

show sd calc

draw pics

use your calc

to calculate confidence intervals on calc (1 Sample Z interval for P)

stat --> test

1-propZint

x is the numerator of the proportion

n is the denominator of the proportion (sample size)

when trying to find sample size, what do you pal in for the value of p

.5

What information does the margin of error provide?

The margin of error shows how close we believe our guess is based on the variability of the estimate in repeated SRSs.

Interpret a confidence level: "To say that we are 95% confident is shorthand for .....

If we take many samples of the same size from this population, about 95% of them will result in an interval that captures the actual parameter value.

Why do we want high confidence and a small margin of error?

High confidence is good because that means that our method almost always gives correct answers. A small margin of error says that we have pinned down the parameter quite precisely.

What is the z-score for the confidence level of 95%

1.96

What is the z-score for the confidence level of 90%

1.645

What is the z-score for the confidence level of 99%

2.58

What is the z-score for the confidence level of 80%

1.28

confidence interval formula for mean when you know the population standard deviation

x bar +/- z ( pop sd/√n)

conditions for finding a confidence interval of mean when you know the population standard deviation

same as pop proportion

- random

- normal --> n≥30

- independent --> 10%

formula for minimum sample size for mean when you know the population sd

z (pop sd / √n) ≤ ME

why do you have to use a t-distribtion if you do not know the pop sd

bc u can't treat it as normal

degrees of freedom formula (1 Sample T interval)

n-1

what happens when the degrees of freedom/sample size increases

the distribution looks more normal and the tails get smaller

what is the confidence interval formula for a t-interval

x bar +/- t (Sx / √n)

how do you find the t score if the actual degree of freedom does not appear on the chart

use the greatest df available that is less than your desired df

what is the method for finding a confidence interval of a t-distribution called

1-sample t-interval for μ

how do you find a t interval on calc

stat

test

t-interval

data (no mean and stuff)

list: L1

freq: 1

how do you see if the distribution is approximately normal in a t-distribution when n≥30 or the population is normal

it is normal bc passes CLT

how do you see if the distribution is approximately normal in a t-distribution when n<30

draw box plot to determine if skewed or outliers

if none, then normal

if STRONGLY skewed or outliers present then not normal

How can you arrange to have both high confidence and a small margin of error?

You can increase the sample size to decrease the margin of error while keeping a high confidence level

What happens to the margin of error when you increase the sample size? Why?

decrease --> The more data you collect, the more accurate your results are going to be

what happens to the margin of error when you increase the confidence level (95% to 99%)

increase --> If you are including almost all of the samples, there will be more variability (its less precise).

what happens to the margin of error when you decrease sd

decrease --> Margin of error is calculated using standard deviation so if the standard deviation decreases, the margin of error will decrease.

Similarities between a standard normal distribution (z) and a t distribution

The density curves of the t distribution are similar in shape to the standard Normal curve. They are symmetric about 0, single-peaked, and bell-shaped

Differences between a standard normal distribution (z) and a t distribution

The spread of the t distributions is a big greater than that of the standard Normal distribution.

what is the standard error of the sample mean (1 Samp t int)

SEx = Sx / √n

It describes how far x bar will typically be from pop mean in repeated SRSs of size n

when is the stated confidence level of a one-sample t interval for μ exactly correct

when the population distribution is exactly normal but no population of real data is exactly normal

Robust inference procedure

an inference procedure (the 1-sample x interval for p OR the 1-sample t interval for μ) is called robust if the provability calculation involved in the procedure remain fairly accurate when a condition for using the procedures is violated

when are t procedures robust? when are they not robust?

the t procedures are relatively robust when the population is non-normal, especially for larger sample sizes

the t procedures are not robust against outliers

the t procedures are quite robust against non-normality of the population except when outliers or strong skewness are present

what improves the accuracy of critical values from the t distributions when the population is nor normal

larger samples

standard error wording

in repeated sampling, the average distance between the sample means and the population mean will be about (# got for SE)