AP Statistics Semester 1 Final Review

Study guide and note content from units 1 - 7

mean

mean

values in a data set added up and divided by the number of included values

median

middle value when data set is placed in ascending order

standard deviation

measure that is used to quantify the amount of set data values

variance

standard deviation squared

quantitative

numerical values - values that can be averaged

qualitative/categorical

values that are generally words, or grouped numbers - cannot “average”

range

the difference between the highest and lowest values in a data set

first quartile

middle value between the minimum and the median (the median of the bottom half)

third quartile

middle value between the median and the maximum (the median of the top half)

Interquartile range (IQR)

the difference between the first and third quartile

outliers

extreme values that are more than 1.5xIQR from the 1st/3rd quartile

resistant

resists the effects of outliers (ie: median, IQR)

nonresistant

influenced by the existence of outliers (ie: mean, standard deviation, range)

boxplot

displays general distribution of data

dotplot

each value represented by a dot - good for specific layout

histogram

displays data grouped into bins of the same width, but displaying the varying frequencies of values

bar chart

similar to histogram, but used for categorical variables - bars don’t touch and x-axis values are not continuous

stem-and-leaf plot

displays all but the last digit of each individual value as a stem, and last digit is the leaf - key must be included

statistical inference

method used to provide ways to answer specific questions from data with some guarantee of success

population

entire group of individuals to which the data is being generalized

sample

part of the group that is being studied

simple random sample

all samples size n have the same chance of being selected

probability sample

each member of a sample has a known chance greater zero of being selected

stratified random sample

dividing a population into groups of similar members and then choose a SRS within each smaller group to form the full sample

multistage sample design

process of selecting t counties, then x townships, y blocks in the township, and z households

cluster random sample

total population is divided into groups and a sample of the groups is selected

bias

contained in a study that systematically favors certain outcomes

voluntary response sample

sample that consists of people who choose themselves by responding to a general appeal

nonresponse

individual chosen for the sample can’t be contacted or refuses to cooperate

confounding variables

two variables whose effects on a response variable cannot be distinguished from each other

convenience sample

sample made from groups that are easiest to reach

response bias

when a responded lies about sensitive information or telescopes the timing of an event

observational study

data collector visually measures variables of interest, but does not attempt to influence the responses

statistically significant

an observed effect too large to attribute plausibly to chance

experiment

The most effective way to show a relationship between two or more variables

double-blind experiment

experiment where neither the person nor the data collector know the variable being applied to the person

matched pairs

special case of randomized block design used when the experiment has only two treatment conditions

blocking

grouping similar units to allow one to draw more specific, separate conclusions

experimental units

members on which an experiment is done

subjects

members of a group that are human beings

treatment

condition applied to a member or group

factor

different explanatory variables in an experiment

level

specific value of a factor

placebo

dummy treatment that can have effect

control group

group of people receiving a sham treatment

randomization

use of chance to divide experimental units into groups

principles of experimental design

1. control - basis comparison

2. Randomization - fair choice of experimental units/subjects

3. Replication - need to ensure that results continue to tell the same story

hidden bias

occurs when the experimenter does not treat all the subjects the exact same way

Median formula

(N + 1 )/2

Standardized Score (z-score)

the number of standard deviations a value is from the mean of its respective data set

normal distribution

bell-shaped curve centered at the mean of a data and distributed approximately as outlined below

types of distribution problems

1. raw value →formula → z-score → normalCDF →percentile

2. percentile → invNorm → z-score → algebra → raw value

symmetric, normal shape

bell-shaped as outlined on other side as well

symmetric, but not normal

mean and median are the same, mode may be different

skewed left

values drag out to the left (smaller numbers)

skewed right

values drag out to the right (larger numbers)

Best descriptive statistics when distribution is symmetric

mean and standard deviation

Best descriptive statistics when distribution is skewed

median and IQR

statistic

value that describes a sample (ie: sample mean, sample standard deviation)

parameter:

value that describes a population (ie: population mean, population standard deviation)

sampling distribution

the sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population

steps to create a sampling distribution

1. Take a larger number of samples from the same population

2. Calculate the p-hat or x-bar for each sample

3. Make a histogram of these values

4. Examine the distribution displayed in the histogram for overall pattern (shape), center, and spread

Bias versus Unbiased

If the sample is collected randomly, the mean of your sample should approach the mean of your population -- this is considered unbiased

variability

as you take the many samples of a sampling distribution, the bigger the sample size of each sample, the closer each sample mean will be to the population mean (bigger sample = less variability)

Central Limit Theorem

The sampling distribution of the means from any population whatsoever (regardless of shape) will be normal provided the sample size of the individual samples is large enough (generally 30+)

Sample means

The mean of the x-bars (sample means) is an unbiased estimator of the population mean

Sampling distribution requirements

1. SRS

2. n is greater than or equal to 30 (sample size of each individual sample is n)

Sample proportions

mean of the sampling distribution of p-hat is p (therefore p-hat is an unbiased estimator of p)

Sample proportion requirements

1. SRS

2. np & n(1-p) is greater than or equal 10

1 proportion z-test

testing a hypothesis regarding the proportion of a single population -- looking for evidence to reject Ho and statistically support Ha

2 prop. z-test

testing a hypothesis regarding the equivalence of the proportions of two populations -- determining if the evidence shows statistically a difference of higher/lower value between the two proportions

1 & 2 prop z-test: step 1

hypothesis; null and alternative hypothesis, and defining the parameter(s)

1 prop. z-test: step 2

type and conditions

A) one-proportion z-test

B) conditions (1. SRS, 2. success and failures greater than or equal 10)

2 prop z-test: step 2

type and conditions

A) two-proportion z-test

B) conditions (1. SRS, 2. success and failures greater than or equal to 5, 3. fair to believe the two populations are independent of each other)

1 & 2 prop. z-test: step 3

calculations; z-score, p-value

1 & 2 prop. z-test: step 4

conclusion; “based on our evidence [p-value compared to significance level], we [reject/fail to reject] the null hypothesis, so there [is/isn’t] significant evidence to support the alternative hypothesis [in context].”

1 prop. z-interval

using sample proportion to estimate a range of values that are likely to contain the population proportion

2 prop. z-interval

using our sample proportions to estimate a range of values that are likely to contain the difference in population proportions

1 & 2 prop. z-interval: step 1

defining in a sentence the population value/ difference in proportions that we are hoping to estimate (ie: “estimate the true proportion”)

1 prop. z-interval: step 2

type and conditions

A) one proportion z-interval

B) conditions (1. SRS, 2. success and failures greater than or equal to 10)

2 prop. z-interval: step 2

type and conditions

A) one proportion z-interval

B) conditions (1. SRS, 2. success and failures greater than or equal to 5, 3. fair to believe the two populations are independent of each other)

1 & 2 prop. z-interval: step 3

calculation; (calculator or formula)

1 & 2 prop. z-interval: step 4

interpretation; “we are % confident that our interval (,_) contains the true proportion/difference in proportions of [parameter of interest]

Type I Error

Rejecting Ho when Ho is true

Type II Error

Rejecting Ha when Ha is true

Power

The probability of accurately determining Ha as true

How can you increase power?

• Increase n (the best option)

• Increase a

• Move Ho and Ha further apart

• Decrease <del>o</del>

Calculator function: x → z → %

normalcdf

Calculator function: % → z → x

invNorm

1 sample t-test

Testing a hypothesis regarding the mean of a single population -- looking for evidence to reject Ho and statistically support Ha

2 sample t-test

Testing a hypothesis regarding the equivalence of the means of two populations -- determining if the evidence shows statistically a difference or higher/lower value between the two means

1/2 sample t-test: step 1

null and alternative hypothesis, define the parameter

1 sample t-test: step 2

Types and Conditions:

A) 1-sample t-test

B) Conditions (1. SRS, 2. Normality)

2 sample t-test: step 2

Types and conditions:

A) 2-sample t-test

B) Conditions (1. SRS’s, 2. Normality, 3. Independence b/w)

1/2 sample t-test: step 3

Calculations:

• test statistic

• degree of freedom

• p-value

1/2 sample t-test: step 4

conclusion; “based on our evidence [p-value compared to significance level], we [reject/fail to reject] the null hypothesis, so there [is/isn’t] significant evidence to support the alternative hypothesis [in context].”

1 sample t-interval

Using our sample mean to estimate a range of values that are likely to contain the population mean

2 sample t-interval

using our sample means to estimate a range of values that are likely to contain the difference in population means

1/2 sample t-interval: step 1

Defining the parameter we are estimating (“estimate the true mean/difference”)

1 sample t-interval: step 2

Type and conditions:

A) 1-sample t-interval

B) Conditions (1. SRS, 2. Normality)