Download Statistics and Probability: A Comprehensive Guide with Exercises and Explanations and more Cheat Sheet Statistics in PDF only on Docsity!
Important Concepts not on the AP Statistics Formula Sheet
Part I: IQR = Q 3 – Q 1 Test for an outlier: 1.5(IQR) above Q 3 or below Q 1 The calculator will run the test for you as long as you choose the boxplot with the oulier on it in STATPLOT
Linear transformation: Addition : affects center NOT spread adds to , M, Q 1 , Q3, IQR
not σ
Multiplication: affects both center and spread multiplies , M, Q 1 , Q3, IQR, σ
When describing data: describe center, spread, and shape.
Give a 5 number summary or mean and standard deviation when necessary.
Histogram: fairly symmetrical unimodal
skewed right
Skewed left Ogive (cumulative frequency)
Boxplot (with an outlier)
Stem and leaf Normal Probability Plot
The 80th^ percentile means that 80% of the data is below that observation.
HOW MANY STANDARD
DEVIATIONS AN
OBSERVATION IS FROM
THE MEAN
68-95-99.7 Rule for Normality N(μ,σ) N(0,1) Standard Normal
r: correlation coefficient, The strength of the linear relationship of data. Close to 1 or -1 is very close to linear
r^2 : coefficient of determination. How well the model fits the data. Close to 1 is a good fit. “Percent of variation in y described by the LSRL on x”
residual =
residual = observed – predicted
y = a+bx Slope of LSRL(b): rate of change in y for every unit x
y-intercept of LSRL(a): y when x = 0
Exponential Model: y = abx^ take log of y
Power Model: y = axb^ take log of x and y
Explanatory variables explain changes in response variables. EV: x, independent RV: y, dependent
Lurking Variable: A variable that may influence the relationship bewteen two variables. LV is not among the EV’s
Confounding: two variables are confounded when the effects of an RV cannot be distinguished.
Regression in a Nutshell
Given a Set of Data:
Enter Data into L 1 and L 2 and run 8:Linreg(a+bx)
The regression equation is:
y-intercept: Predicted fat gain is 3.5051 kilograms when NEA is zero.
slope : Predicted fat gain decreases by .00344 for every unit increase in NEA.
r: correlation coefficient
r = - 0.
Moderate, negative correlation between NEA and fat gain.
r^2 : coefficient of determination
r^2 = 0.
60.6% of the variation in fat gained is explained by the Least Squares Regression line on NEA.
The linear model is a moderate/reasonable fit to the data. It is not strong.
The residual plot shows that the model is a reasonable fit; there is not a bend or curve,
There is approximately the same amount of points above and below the line. There is
No fan shape to the plot.
Predict the fat gain that corresponds to a NEA of 600.
Would you be willing to predict the fat gain of a person with NEA of 1000?
No, this is extrapolation, it is outside the range of our data set.
Residual:
observed y - predicted y
Find the residual for an NEA of 473
First find the predicted value of 473:
observed – predicted = 1.7 - 1.87798 = -0.
predicted fatgain 3. 5051 0. 00344 ( NEA )
predictedfatgain 1.
predictedfatgain 3. 5051 0. 00344 ( 600 )
predictedfatgain 1. 87798
predictedfatgain 3. 5051 0. 00344 ( 473 )
Part II: Designing Experiments and Collecting Data:
Sampling Methods:
The Bad: Voluntary sample. A voluntary sample is made up of people who decide for themselves to be in the survey. Example: Online poll Convenience sample. A convenience sample is made up of people who are easy to reach. Example: interview people at the mall, or in the cafeteria because it is an easy place to reach people.
The Good: Simple random sampling. Simple random sampling refers to a method in which all possible samples of n objects are equally likely to occur. Example: assign a number 1-100 to all members of a population of size 100. One number is selected at a time from a list of random digits or using a random number generator. The first 10 selected without repeats are the sample. Stratified sampling. With stratified sampling, the population is divided into groups, based on some characteristic. Then, within each group, a SRS is taken. In stratified sampling, the groups are called strata. Example: For a national survey we divide the population into groups or strata, based on geography - north, east, south, and west. Then, within each stratum, we might randomly select survey respondents. Cluster sampling. With cluster sampling, every member of the population is assigned to one, and only one, group. Each group is called a cluster. A sample of clusters is chosen using a SRS. Only individuals within sampled clusters are surveyed. Example: Randomly choose high schools in the country and only survey people in those schools. Difference between cluster sampling and stratified sampling. With stratified sampling, the sample includes subjects from each stratum. With cluster sampling the sample includes subjects only from sampled clusters. Multistage sampling. With multistage sampling, we select a sample by using combinations of different sampling methods. Example: Stage 1, use cluster sampling to choose clusters from a population. Then, in Stage 2, we use simple random sampling to select a subset of subjects from each chosen cluster for the final sample. Systematic random sampling. With systematic random sampling, we create a list of every member of the population. From the list, we randomly select the first sample element from the first k subjects on the population list. Thereafter, we select every kth subject on the list. Example: Select every 5th^ person on a list of the population.
Experimental Design: A well-designed experiment includes design features that allow researchers to eliminate extraneous variables as an explanation for the observed relationship between the independent variable(s) and the dependent variable. Experimental Unit or Subject: The individuals on which the experiment is done. If they are people then we call them subjects Factor: The explanatory variables in the study Level: The degree or value of each factor. Treatment: The condition applied to the subjects. When there is one factor, the treatments and the levels are the same. Control. Control refers to steps taken to reduce the effects of other variables (i.e., variables other than the independent variable and the dependent variable). These variables are called lurking variables. Control involves making the experiment as similar as possible for subjects in each treatment condition. Three control strategies are control groups, placebos, and blinding. Control group. A control group is a group that receives no treatment Placebo. A fake or dummy treatment. Blinding : Not telling subjects whether they receive the placebo or the treatment Double blinding: neither the researchers or the subjects know who gets the treatment or placebo Randomization. Randomization refers to the practice of using chance methods (random number tables, flipping a coin, etc.) to assign subjects to treatments. Replication. Replication refers to the practice of assigning each treatment to many experimental subjects. Bias: when a method systematically favors one outcome over another.
Types of design: Completely randomized design With this design, subjects are randomly assigned to treatments. Randomized block design , the experimenter divides subjects into subgroups called blocks. Then, subjects within each block are randomly assigned to treatment conditions. Because this design reduces variability and potential confounding, it produces a better estimate of treatment effects. Matched pairs design is a special case of the randomized block design. It is used when the experiment has only two treatment conditions; and subjects can be grouped into pairs, based on some blocking variable. Then, within each pair, subjects are randomly assigned to different treatments. In some cases you give two treatments to the same experimental unit. That unit is their own matched pair!
Part II in Pictures: Sampling Methods
Simple Random Sample: Every group of n objects has an equal chance of being selected. (Hat Method!)
Stratified Random Sampling: Cluster Sampling:
Break population into strata (groups) Randomly select clusters then take all
then take an SRS of each group. Members in the cluster as the sample.
Systematic Random Sampling:
Select a sample using a system, like selecting every
third subject.
Experimental Design:
Completely Randomized Design: Randomized Block Design:
Matched Pairs Design:
Sampling distribution: The distribution of all values of the statistic in all possible samples of the same size from the population.
Central Limit Theorem: As n becomes very large the sampling distribution for is approximately NORMAL
Use (n ≥ 30) for CLT Low Bias: Predicts the center well High Bias: Does not predict center well Low Variability: Not spread out High Variability: Is very spread out
High bias, High Variability Low Bias, Low Variability
Low Bias, High Variability High Bias, Low Variability
See other sheets for Part IV
ART is my BFF
Type I Error: Reject the null hypothesis when it is actually True
Type II Error: Fail to reject the null hypothesis when it is False.
ESTIMATE – DO A CONFIDENCE INTERVAL EVIDENCE - DO A TEST
Paired Procedures
Must be from a matched pairs design:
Sample from one population where each
subject receives two treatments, and the
observations are subtracted. OR
Subjects are matched in pairs because they
are similar in some way, each subject
receives one of two treatments and the
observations are subtracted
Two Sample Procedures
Two independent samples from two different
populations OR
Two groups from a randomized experiment
(each group would receive a different
treatment) Both groups may be from the
same population in this case but will
randomly receive a different treatment.
Major Concepts in Probability
For the expected value (mean,μX) and the X or X^2 of a probability distribution use the formula sheet
Binomial Probability Simple Probability (and, or, not):
Fixed Number of Trials Probability of success is the same for all trials Trials are independent
If X is B(n,p) then (ON FORMULA SHEET)
Mean X np
Standard Deviation X np ( 1 p )
For Binomial probability use or: or use:
Exactly: P(X = x) = binompdf(n , p, x)
At Most: P(X ≤ x) = binomcdf(n , p, x)
At least: P(X ≥ x) = 1 - binomcdf(n , p, x-1)
More than: P(X > x) = 1 - binomcdf(n , p, x)
Less Than: P(X < x) = binomcdf(n , p, x-1)
You may use the normal approximation of the binomial distribution when np ≥ 10 and n(1-p) ≥ 10. Use then mean and standard deviation of the binomial situation to find the Z score.
Finding the probability of multiple simple events. Addition Rule: P(A or B) = P(A) + P(B) – P(A and B) Multiplication Rule: P(A and B) = P(A)P(B|A)
Mutually Exclusive events CANNOT be independent A and B are independent if the outcome of one does not affect the other. A and B are disjoint or mutually exclusive if they have no events in common. Roll two die: DISJOINT rolling a 9 rolling doubles
Roll two die: NOT disjoint rolling a 4 rolling doubles
Independent: P(B) = P(B|A) Mutually Exclusive: P(A and B) = 0
Geometric Probability Conditional Probability
You are interested in the amount of trials it takes UNTIL you achieve a success. Probability of success is the same for each trial Trials are independent
Use simple probability rules for Geometric Probabilities.
P(X=n) = p(1-p)n-^1 P(X > n) = (1 – p)n^ = 1 - P(X ≤ n)
μX is the expected number of trails until the first success or
Finding the probability of an event given that another even has already occurred.
Conditional Probability:
P A
P A B
P B A
Use a two way table or a Tree Diagram for Conditional Problems. Events are Independent if P(B|A) = P(B)
Normal Probability
For a single observation from a normal population
x
PX x Pz ( ) ( )
x
PX x Pz
To find P ( x X y )Find two Z scores and subtract the
probabilities (upper – lower)
Use the table to find the probability or use normalcdf(min,max,0,1) after finding the z-score
For the mean of a random sample of size n from a population.
When n > 30 the sampling distribution of the sample mean x
is approximately Normal with:
n
X
X
If n < 30 then the population should be Normally distributed to begin with to use the z-distribution.
n
x
P X x P z
n
x
P X x Pz
To find P ( x X y )Find two Z scores and subtract the
probabilities (upper – lower) Use the table to find the probability or use normalcdf(min,max,0,1) after finding the z-score
Mutually Exclusive vs. Independence You just heard that Dan and Annie who have been a couple for three years broke up. This presents a problem, because you're having a big party at your house this Friday night and you have invited them both. Now you're afraid there might be an ugly scene if they both show up. When you see Annie, you talk to her about the issue, asking her if she remembers about your party. She assures you she's coming. You say that Dan is invited, too, and you wait for her reaction. If she says, "That jerk! If he shows up I'm not coming. I want nothing to do with him!", they're mutually exclusive. If she says, "Whatever. Let him come, or not. He's nothing to me now.", they're independent.
Mutually Exclusive and Independence are two very different ideas
Mutually Exclusive (disjoint): P(A and B) = 0 Events A and B are mutually exclusive if they have no outcomes in common. That is A and B cannot happen at the same time.
Example of mutually exclusive (disjoint) : A: roll an odd on a die B: roll an even on a die
Odd and even share no outcomes P(odd and even) = 0 Therefore, they are mutually exclusive.
Example of not mutually exclusive (joint): A: draw a king B: draw a face card
King and face card do share outcomes. All of the kings are face cards. P(king and face card) = 4/ Therefore, they are not mutually exclusive.
Independence: P(B) = P(B|A) Events A and B are independent if knowing one outcome does not change the probability of the other. That is knowing A does not change the probability of B.
Examples of independent events: A: draw an ace B: draw a spade
P(Spade) =13/52 = 1/ P(Spade | Ace) = 1/ Knowing that the drawn card is an ace does not change the probability of drawing a spade
Examples that are dependent (not independent): A: roll a number greater than 3 B: roll an even
P(even) = 3/6 = 1/ P(even | greater than 3) = 2/ Knowing the number is greater than three changes the probability of rolling an even number.
Mutually Exclusive events cannot be independent
Mutually exclusive and dependent
A: Roll an even B: Roll an odd
They share no outcomes and knowing that it is odd changes the probability of it being even.
Independent events cannot be Mutually Exclusive
Independent and not mutually exclusive
A: draw a black card B: draw a king
Knowing it is a black card does not change the probability of it being a king and they do share outcomes.
Dependent Events may or may not be mutually exclusive Dependent and mutually exclusive A: draw a queen B: draw a king Knowing it is a queen changes the probability of it being a king and they do not share outcomes.
Dependent and not mutually exclusive A: Face Card B: King Knowing it is a face card changes the probability of it being a king and they do share outcomes.
If events are mutually exclusive then: P(A or B) = P(A) + P(B) If events are not mutually exclusive use the general rule: P(A or B) = P(A) + P(B) – P(A and B)
If events are independent then: P(A and B) = P(A)P(B) If events are not independent then use the general rule: P(A and B) = P(A)P(B|A)
Interpretations from Inference
Interpretation for a Confidence Interval:
I am C% confident that the true parameter (mean or proportion p) lies between # and #. INTERPRET IN CONTEXT!!
Interpretation of C% Confident: Using my method, If I sampled over and over again, C% of my intervals would contain the true
parameter (mean or proportion p). NOT : The parameter lies in my interval C% of the time. It either does or does not!!
If p < I reject the null hypothesis H 0 and I have sufficient/strong evidence to support the alternative hypothesis Ha INTERPRET IN CONTEXT in terms of the alternative.
If p > I fail to reject the null hypothesis H 0 and I have insufficient/poor evidence to support the alternative hypothesis Ha INTERPRET IN CONTEXT in terms of the alternative.
Evidence Against Ho
P-Value
"Some" 0.05 < P-Value < 0.
"Moderate or Good" 0.01 < P-Value < 0.
"Strong" P-Value < 0.
Interpretation of a p-value:
The probability, assuming the null hypothesis is true, that an observed outcome would be as
extreme or more extreme than what was actually observed.
Duality: Confidence intervals and significance tests. If the hypothesized parameter lies outside the C% confidence interval for the parameter I can REJECT H 0
If the hypothesized parameter lies inside the C% confidence interval for the parameter I FAIL TO REJECT H 0
Power of test:
The probability that at a fixed level test will reject the null hypothesis when and alternative value is true.
Hypothesis Testing
One Sample Z Test Use when testing a mean from a single sample and σ is known Ho: μ = # Ha: μ ≠ # - or- μ < # - or- μ > # Conditions:
- SRS
- Normality: Stated, CLT, or plots
- Independence and N≥10n Test statistic:
n
x
z
0
CALCULATOR: 1: Z-Test
One Sample t Test Use when testing a mean from a single sample and σ is NOT known [Also used in a matched paired design for the mean of the difference: PAIRED t PROCEDURE ] Ho: μ = # Ha: μ ≠ # - or- μ < # - or- μ > # Conditions:
- SRS
- Normality: Stated, CLT, or plots
- Independence and N≥10n
Test statistic: df = n- 1
n
s
x t^0
CALCULATOR: 2: T-Test
One Proportion Z test Use when testing a proportion from a single sample Ho: p = # Ha: p ≠ # - or- p < # - or- p > # Conditions:
- SRS
- Normality: np 0 ≥ 10, n(1-p 0 ) ≥ 10
- Independence and N≥10n Test statistic:
CALCULATOR: 5: 1-PropZTest
Two Sample Z test σ known (RARELY USED) Use when testing two means from two samples and σ is known Ho: μ 1 = μ 2 Ha: μ 1 ≠ μ 2 - or- μ 1 < μ 2 - or- μ 1 > μ 2 Conditions:
- SRS for both populations
- Normality: Stated, CLT, or plots for both populations
- Independence and N≥10n for both populations. Test statistic:
CALCULATOR: 3: 2-SampZTest
Two Sample t Test σ unknown Use when testing two means from two samples and σ is NOT known Ho: μ 1 = μ 2 Ha: μ 1 ≠ μ 2 - or- μ 1 < μ 2 - or- μ 1 > μ 2 Conditions:
- SRS for both populations
- Normality: Stated, CLT, or plots for both populations
- Independence and N≥10n for both populations. Test statistic: use lower n for df (df = n-1) or use calculator.
CALCULATOR: 4: 2-SampTTest
Two Proportion Z test Use when testing two proportions from two samples Ho: p 1 = p 2 Ha: p 1 ≠ p 2 - or- p 1 < p 2 - or- p 1 > p 2 Conditions:
- SRS for both populations
Normality:
Independence and N≥10n for both populations.Test statistic:
CALCULATOR: 6: 2-PropZTest
Χ^2 – Goodness of Fit L1 and L Use for categorical variables to see if one distribution is similar to another
Ho: The distribution of two populations are not different Ha: The distribution of two populations are different Conditions:
- Independent Random Sample
- All expected counts ≥ 1
- No more than 20% of Expected Counts < Test statistic: df = n- 1
CALCULATOR: D: Χ^2 GOF-Test(on 84)
Χ^2 – Homogeneity of Populations r x c table Use for categorical variables to see if multiple distributions are similar to one another Ho: The distribution of multiple populations are not different Ha: The distribution of multiple populations are different Conditions:
- Independent Random Sample
- All expected counts ≥ 1
- No more than 20% of Expected Counts < Test statistic: df = (r-1)(c-1)
CALCULATOR: C: Χ^2 - Test
Χ^2 – Independence/Association r x c table Use for categorical variables to see if two variables are related Ho: Two variables have no association (independent) Ha: Two variables have an association (dependent) Conditions:
- Independent Random Sample
- All expected counts ≥ 1
- No more than 20% of Expected Counts < Test statistic: df = (r-1)(c-1)
CALCULATOR: C: Χ^2 - Test
Significance Test for Regression Slope: Ho: β = 0 Ha: β ≠ # - or- β < # - or- β > # Conditions: 1. Observations are independent 2. Linear Relationship (look at residual plot)
- Standard deviation of y is the same(look at residual plot) 4. y varies normally (look at histogram of residuals)
CALCULATOR: LinRegTTest Use technology readout or the calculator for this significance test SEb
b (^) t df = n- 2
Notation and Interpretations
IQR Inner Quartile Range Mean of a sample
μ Mean of a population s Standard deviation of a sample σ Standard deviation of a population Sample proportion
p (^) Population proportion s (^2) Variance of a sample σ (^2) Variance of a population M (^) Median Σ (^) Summation Q 1 First Quartile Q 3 Third Quartile Z Standardized value – z test statistic z^ Critical value for the standard normal distribution t Test statistic for a t test t^ Critical value for the t-distribution N(μ, σ) Notation for the normal distribution with mean and standard deviation r Correlation coefficient – strength of linear relationship r^2 Coefficient of determination – measure of fit of the model to the data Equation for the Least Squares Regression Line
a y-intercept of the LSRL b Slope of the LSRL ( Point the LSRL passes through
y = axb^ Power model y = abx^ Exponential model SRS Simple Random Sample S Sample Space P(A) The probability of event A Ac^ A complement P(B|A) Probability of B given A ∩ Intersection (And) U Union (Or) X Random Variable μX Mean of a random variable σ (^) X Standard deviation of a random variable Variance of a random variable
B(n,p) Binomial Distribution with observations and probability of success Combination n taking k
pdf Probability distribution function cdf Cumulative distribution function n Sample size N Population size CLT Central Limit Theorem Mean of a sampling distribution Standard deviation of a sampling distribution
df Degrees of freedom SE Standard error H 0 Null hypothesis-statement of no change Ha Alternative hypothesis- statement of change p-value Probability (assuming H 0 is true) of observing a result as large or larger than that observed α Significance level of a test. P(Type I) or the y-intercept of the true LSRL β P(Type II) or the true slope of the LSRL χ^2 Chi-square test statistic
