Stats Tri 1 Review



Stats Tri 1 Review

Chapter 1

o Statistics – the science and art of learning from data

o Data – numbers with a context

o Probability – the study of chance behavior; The regular pattern of something occurring that becomes clear after many trials; chance behavior is unpredictable in the short run but has regular and predictable patter in the long runl helps us make reasonable conclusions

o Statistical inference – produced answers to specific questions and statements of how confident we can be that the answer is correct.

o Available data – data that was produced in the past for some other purpose but may help answer a present question

o Data beats personal experience because data describes overall pattern not a few instances.

o Observational study – observes individuals and measure variables of interest but do not attempt to influence their responses.

▪ Surveys – popular way to gauge public opinion

o Experiment – deliberately do something to individuals in order to observe their responses.

▪ Cause and effect

▪ 3 Characteristics of a GOOD experiment: control group, randomization, and replication.

o Data Analysis – statistical practice of analyzing distributions of data through graphical displays and numerical summaries; organizing displaying and summarizing data

▪ Individuals – the objects described by a set of data; individuals may be people, animals, or things

▪ Variable – any characteristic of an individual; can take different values for different individuals

▪ Distribution – description of values a variable takes on and how often the variable takes on those values

▪ Begin by examining each variable by itself, and then study the relationship among variables. Begin with a graph or graphs, and then add numerical summaries of specific aspects of the data.

o Categorical Variables – places an individual into one of several groups or categories

▪ Distributions list the categories (X) by the count of each or the percent of individuals (Y)

▪ Bar graphs and pie charts

▪ Ex: Color, Gender, Grade, months, rock band

o Quantitative variable – takes numerical values for which arithmetic operations such as adding and averaging make sense.

▪ Distributions list the values (X) and number of times the variable takes on the value(Y)

▪ Dot plots, stem plots, histograms, box plots

▪ Ex: goals, score, time

o Dot plots – for quantitative data

▪ X= Goals per game Y= Number of times that occurred (frequency)

o Stem plots – Quick picture of the shape of the distribution while also including numerical values.

▪ Use for SMALL data sets!!

▪ You can double the number of stems therefore reducing the number of leaves.

▪ Back to back stem plot: when you are comparing 2 sets of data looks like: 4321|1|1234

▪ ALWAYS write the leaf unit!!

• Unit 1: 31=3|1 Unit 0.1: 3.1= 3|1 Unit 1000: 10,000 and 11,000 = 1|01

▪ Numbers on the far left of the graph show total values up to that point

• EX: 1 0|1

3 1|34

3 2|

7 3|6778

o Histogram – break the range of data values into classes and displays the count or percent of observations that fell into that class [frequency]. Quantitative data

▪ Shows shape of data

▪ Use with LOTS of data

▪ Divide the range of data into equal classes

▪ Count observations in each class = frequency

▪ BARS SHOULD TOUCH

▪ Calculator: Xlist = L1; Freq = L2; to change the number of bars do XCL=[(max-min)/classes]+1; Xmin and Xmax is max/min of data; Ymax is largest frequency of data

o Bar Graphs – for categorical data

▪ Side by side bar graph: compares 2 qualitative variables

o Pie Charts – for categorical data

▪ To show the part compared to the whole. Data equals 100%

o Summarize Data and graph: Center, Shape, Spread, Outliers

o Skew:

▪ Skewed right: tail to the right (Mean is larger than Median)

▪ Skewed left: Tail to the left (Mean is less than Median)

▪ Uniform: Square shape (Mean = Median)

▪ Approximately normal/symmetric/mound/unimodal: Upside-down U (Mean = Median)

▪ Bimodal: M shape

o Outlier – observations that lie outside the overall pattern of distribution

o Frequency graph - histogram that shows the total number of times (frequency) of each X value

o Cumulative frequency graph – adds up all the data from each previous bar. The last bar is the total of all the values.

o Relative frequency graph – histogram that shows percent of each X value. All bars should add up to 100%

o Relative Cumulative Frequency Graph/OJIVE – can be line graph or histogram what shows a cumulative percent of all of the data. Last bar should be 100%

o Time plot – drastically changes due to variations in the Y variable with time. Looks like ^v^^vv^v^

o Measures of center:

▪ Mean – Population mean= μ; Sample mean = Xbar ; very NONresistant to outliers; Average value; is the “balancing point” of a density curve if it were made of a solid material.

▪ Median – “Q2” or “M”; RESISTANT to outliers; Typical value; divides the area under a density curve in half

o Measures of spread:

▪ Range: High-low; Very nonresistant

▪ Quartiles: Inter-Quartile Range; Resistant to measures of spread

• {IQR=Q3-Q1}

• Q1 = Median of bottom 50%

• Q3 = Median of top 50%

• Outlier Test “1.5 IQR RULE”

o Q1-1.5IQR or Q3+1.5IQR

▪ Standard of Deviation: The average distance of all data points away from the mean

• Sample Standard of deviation = Sx

• Population Standard of deviation = ơ

• Equals zero when the data is one value

▪ Variance: Var; The squared standard of deviation; the larger the distance a value is from the mean in EITHER direction, the larger the variance.

• NOT on calculator so SQUARE Sx!! Sx2 or ơ2

o 5 Number Summary = Min-Q1-Med-Q3-Max; use 1-Var Stats

o Box Plot

MIN-Q1-MED-Q3-MAX

|-------- | ----------|

{25%}{ 50% }{25%}

Modified Box Plot

MIN-Q1-MED-Q3-MAX Outlier

|-------- | ----------| x

o Linear Transformations - when converting units, the measure of center and spread will change. HOWEVER linear transformations will only change center and spread but not shape.

▪ Multiplying each observation by b multiplies mean, median, IQR, and range

▪ Adding b to each observation adds b to mean, median, IQR but does not affect spread.

Chapter 2

o Standardized Value – one way to describe relative position in a data set is to tell how many Sx’s above or below the mean the observation is.

o Z-Score – How many standard of deviations away from the mean the value is.

▪ Z=(X-Xbar)/(ơ/√n)

▪ +Z is above mean; -Z is below mean

▪ The larger Z the larger the spread of data (roughly between -3 and 3)

o Percentiles – Measure of relative standing

▪ Pth percentile is value with P% of observations AT or BELOW is

▪ Ex: Jenny scored an 86% 22/25 scores are ≤ 86% Jenny is the 22/25 = 88th percentile

o Chebyshev’s Inequality –

o Density Curve – An idealized description of the overall pattern of a distribution. An area underneath =1 representing 100% of observations

▪ Area (Total) = Portion, Probability, Percentage of data

▪ Normal Density Curve = Mound/Bell; Mean=Median

▪ Mean = μ

▪ Uniform Distribution = Rectangle/Box shape

▪ Sx is at the inflection points

o 68-95-99.7 Rule

▪ 68% of data falls between -1 and 1 Z

▪ 95% falls between -2 and 2 Z

▪ 99.7% falls between -3 and 3 Z

o Standard Normal Distribution – mean of zero and a Sx of 1; N(0,1) or N(μ, ơ); when given the N() use the numbers to construct your graph

o Z-score table in front of book – given Z-Score of 2.85 look for 2.8 on left and .05 on top; find where they meet and that is the area to the LEFT of that z-score

▪ Any value in table that is off of the chart is assumed .0001 or .9999

o Solving Z-Score Problems!!

▪ State the problem in terms of X; DRAW A PICTURE and shade area of interest

▪ Standardized and Draw a picture

▪ Write X in terms of Z-Score and show calculations

▪ Write as Ex: P(Z>2.33) = 1-.9901 = .0099

▪ Write answer in context of the problem.

o Assessing Normality

▪ Construct a Histogram or Stem plot (Look at shape, check mean=median, compare to 68-95-99.7)

▪ Construct a Box plot and check for any skewness or outliers

▪ Construct a normal probability plot/Quartile Plot (must look like a straight line)

• Calculator: Graph info by selecting Statplot (Last graph option); Zoom 9

o Calculator instead of Z table: 2nd; VARS

▪ Normalcdf(Z1, Z2) Gives area between 2 Z-Scores

▪ Normalcdf(X1, X2, μ, ơ) Gives area between 2 X values given Mean and Sx

• If the X value is off the graph to the right or left use X1 = -1EE99 X2 = 1EE99

o EE = 2nd comma

▪ InvNorm(Area) = Z Score

▪ InvNorm(Area, μ, ơ) = X

Chapter 3

o Response Variable – Measures the outcome of the study; Dependant Variable; Y value

o Explanatory Variable – Helps explain or influence changes in the response variable; Independent Variable: X value

o Scatter plot – Shows the relationship between 2 QUANTATIVE* variables measured on the same individuals

*(so that you can calculate r)

▪ Interpreting:

• Look for overall pattern and any striking deviations

• Describe Direction (+/-), Form (linear for Ch. 3), Strength (Strong/Moderate/Weak)

• Look for outliers and effect on correlations

▪ Association: X^=Y^ Xˇ=Yˇ X^=Yˇ

o Insert sequence into L1 given an equation: 2nd List; OPS; Seq(equation, variable, start, # of numbers) Sto L1

o Correlation – measures strength and direction

▪ Correlation Coefficient: R; is between (-1,1)

• ONLY measures LINEAR

• VERY nonresistant to outliers

• Doesn’t account for X or Y; doesn’t matter which is which

• Doesn’t change if X and Y are multiplied by a constant

• No unit of measurement

▪ Coefficient of determination: R2; “what percent of the variation in Y is explained by the least squares regression line?”

• The bigger the R the better fit for the line

o Regression Line – line that describes how a response variable y changes as the explanatory variable X changes

▪ Interpolation – makes assumptions/predictions inside the range of the data

▪ Extrapolation – makes them outside of the data

▪ y=a+bx

o Least Squares Regression Line (LSRL) – y on x is the line that makes the sum of the squared vertical distances of the data points from the line as small as possible.

▪ Linear only!

▪ Point (Xbar, Ybar) is any point ON THE LINE! = centroid

▪ ŷ=a+bx a = y intercept = Ybar-b(Xbar) b = slope = r (Sy/Sx) *In your packet

▪ Calculator for equation: Stat; calc; #8; LinReg(a+bx) L1, L2, Y1

• Invalid dim? Check your lists for uneven lists

• R doesn’t appear? 2nd catalog diagnostics ON

o Influential Points

▪ Outlier becomes an influential point when it substantially changes the R of the regression line

C = Centroid

1. If the Y value of the centroid is changed, the slope of the LSRL stays the same but r becomes closer to zero because the line has shifted away from the bulk of the points. Y intercept changes.

2. If the influential point is way off of the data randomly, it makes r closer to zero

3. If the influential point is on the LSRL but further out, the r value significantly increases.

o Lurking Variables – there is some other variable that has a relationship with both variables compared that shows the two compared variables don’t have anything to do with each other.

▪ Ex: Ice cream sales increases; Drowning increase; Lurking Variable = Temperature increase/Summertime

o Residual – the difference between an observed variable of the response variable and the value predicted by the LSRL

▪ Residual = y- ŷ or Observed y – Predicted y

▪ Every point has a residual

▪ Tells how appropriate the model is for the data

▪ If there is no pattern then it’s a good model

▪ MUST do a new regression line calculation to update a residual graph after making data changes.

▪ Stat plot; 2nd graph option; make Ylist: RESID by doing 2nd Stat RESID

▪ Graph X value (X) by Resid (Y) and give title!!

Chapter 4

*** X is typically L1 Y is L2 Log X is L3 Log Y is L4

o Linear Equation – y=a+bx

▪ LinReg(X, Y, Y1)

▪ COMMON DIFFERENCE: y2-y1 ≈ y3-y2 etc

▪ r, r2 **ONLY linear!

o Exponential Equation – y=abx (Exponential Graph)

▪ Linearized Equation: LinReg(X, Logy, Y1) => log ŷ = a+bx (Makes a linear graph)

o Linear relationship between X and LogY, but the relationship between X and Y is exponential

▪ Inverse Transformations: ŷ = 10a(10b)x *OR* ExpReg(X, Y, Y1) = ŷ = abx

o Makes exponential graphs

o Notice: 10a ≈ a 10b ≈ b

▪ When you calculate LinReg or ExpReg, you get r and r2

▪ COMMON RATIO: y2/y1 ≈ y3/y2 etc

o Power Equation – y=axb (Power Graph)

▪ Linearized Equation: LinReg(LogX, LogY, Y1) => log ŷ = a+b(logx) (Makes a linear graph)

o Linear relationship between LogX and LogY, but the relationship between X and Y is power

▪ Inverse Transformations: ŷ = 10axb *OR* PwrReg(X, Y, Y1) = ŷ = axb

o Makes power graphs

▪ Gives you r and r2

▪ (0,0) usually makes sense in the context of the problem

o Do Transformations graphically… (Log) Y1= a+bx or (Log)Y1=a+b(Logx) Y2= 10^Y1

o Use Residual graphs to check for fit of the equation and r2

o Exponential or Power? Check for Common Ratio, (0,0), or the context of the question (what it asks for)

o If the explanatory variable is ‘years’ you may use ‘years since…’ so the numbers are smaller

o Relationship between two categorical Variables

▪ To analyze categorical data we use counts or percents of individuals that fall into various categories

▪ 2 way tables

▪ Bar graphs (side by side bar graph?)

▪ Marginal Distribution – totals of each row or column in the bottom and right side of the graph

▪ Conditional Distribution – comparing the percent of “infected and inoculated” with “Infected and not inoculated” and if there were more inoculated options, compare those too

▪ No graph (such as scatter plot) can show the relationship between two categorical variables. No single numerical measure (such as correlation) can summarize the strength of the association either.

o Simpsons Paradox – An association or comparison that holds for all of several groups can reverse direction when the data are combined to form a single group. Each group has positive association but when they are all together they form a negative association.

o CAUSATION:

▪ In the absence of experimental evidence, good evidence of causation requires:

o Strong association that appears consistently in many studies

o Clear explanation for the alleged casual link

o Careful examination of possible lurking variables

1. Causation – when X definitely causes Y to happen; the variables are correlated

2. Common Response – there is a lurking variable Z that relates X and Y; X and Y are correlated; Ice Cream and drownings example

3. Confounding – Either X or Z causes Y or a combination of the two; X and Y are correlated; Visiting the elderly with tea and improving mood

1) 1) 2) 3)

Chapter 5

o Observational Studies – Chapter 1

o Experiment – Chapter 1

o Population – entire group you desire data from

o Sample – Part of the population (CAN be the whole population too)

o Census – attempts to contact every individual in the entire population; a sample of the WHOLE population

o Sampling – studying a part of the population in order to gain information about it as a whole.

▪ Biased Sampling – systematically favors certain outcomes

• Voluntary response sampling – people who choose themselves by responding to general appeal. People who volunteer usually have a strong opinion, especially negative.

• Convenience Sample – choosing individuals who are easiest to reach

▪ Probability Sampling – sample chosen by chance

• Simple Random Sample (SRS) – each individual from the population has the same chance of being selected.

o Larger random samples gives us more accuracy than smaller samples

o Uses table of random digits

▪ Lable – assign numbers to individuals

▪ Table – select labels at random

▪ Stopping rule – indicate when to stop

▪ Identify Sample – Identify subjects of the selected labels

▪ Write out EVERYTHING!!

• Stratified Random Sample – divide population into groups (Strata) and choose an SRS from EACH stratum (You aren’t randomly selecting which stratum, just smaller samples from each stratum). Each stratum contains similar individuals; each stratum has different individuals than the other strata; every person within each stratum has the same chance as another person in the same stratum of being chosen. Like blocking in Experiments

• Cluster Sample – divides the population into groups or clusters. Some clusters are randomly selected. Then ALL of the individuals in the chosen cluster are selected to be in the sample. Each cluster has the same probability as the other clusters of being chosen.

• Systematic Random Sample – divide the population into the same number of groups you want. Randomly select a number from one group and that number individual from each group is chosen.

1 4 7 10 SRS 2

2 5 8 11

3 6 9 12

• Multistage Sample – SRS a country then SRS a state then SRS a city then SRS a subdivision…

o Bias (WE RUN)

▪ Wording Effect

▪ Response Bias – interviewer or interviewee may respond differently when being studied. Situation or lying

▪ Under coverage – individuals not represented by the sample

▪ Nonresponse – individuals who cannot be reached, who don’t respond, or who don’t cooperate

o Experiment

▪ Experimental Units/Subjects – Objects/People being experimented on

▪ Factors – Explanatory Variable (Ex. Color paints)

▪ Treatment – Experimental condition applied to the units (Ex: Red, Blue, Green, Yellow)

▪ Levels – (Ex: Amount of paint and Brand of paint) Each has its own box corresponding to another level from another factor: L1 L2 ----|----

▪ Placebo – dummy treatment to eliminate bias

▪ Control Group – overall effort to minimize variability in the way the experimental units are obtained and treated

▪ Randomization – the rule used to assign experimental units to treatments ***LABLE IN CHART!

▪ Principles of Experimental Design

• Control the effects of lurking variables on the response, most simply by comparing two or more treatments

• Replicate each treatment on many units to reduce chance variation on the results

• Randomize use impersonal chance to design experimental units to treatments

▪ Statistically Significant – an observed effect so large that it would rarely occur by chance.

▪ Block Design – Group of experimental units or subjects that are known before the experiment to be similar in some way that is expected to systematically affect the response to the treatment.

▪ Matched pairs design – compares two treatments on two similar subjects to eliminate differences or compare two treatments on one subject. Randomization occurs when giving the subjects which treatment or giving the one subject which treatment first or second.

o Random numbers on the calculator – Math Prob RandInt(first number, last number, # of numbers desired)

Chapter 6

o Relative Frequency – Number of times an event occurs when PERFORMING it (5 heads 6 tails)

o Probability Model – Theoretical Probability (50/50 Chance you get heads or tails)

o Simulation – not actually doing the act but simulating it in some way (On calculator)

o Probability of Finite samples – the probability of any event is the sum of the probabilities of the outcomes making up the event.

o OR = υ = Add

o AND = ̣π = Multiply

o Use Tree Diagrams and Venn Diagrams for probabilities!!

o At Least 1/One Or More = 1-NONE

o Probability that B occurs given A occurs…. Use P(B|A) Conditional Probablity

▪ P(A|B) = P(AπB)/P(B) **In Packet

o Mutually Exclusive – Two events have no outcomes in common

o Independence – The probability of one event doesn’t effect the probability of another.

o If something is mutually exclusive, it CANNOT be independent. Visa Versa applies

o P(AυB) = P(A)+P(B) (MUTUALLY EXCLUSIVE events)

o P(AυB) = P(A)+P(B)-P(AπB) (INDEPENDENT events)

o P(AπB) = P(A)+P(B) (INDEPENDENT events)

o P(AπB) = P(A|B)P(B) (Not independent – calculated from P(A|B) Equation)

Try to use a table to further understand conditional probability:

| |Event A |Event B |Totals |

|Event C |5 |10 |15 |

|Event D |15 |20 |35 |

|Totals |20 |30 |50 |

P(A|C) = 5/15

P(B|D) = 20/30

P(AπC) = 5/50

Stats Tri 2 Review

Chapter 7

o Discrete Random Variable – X has a countable (Integer 1, 2, 3) number of possible variables.

▪ Probability Distribution: lists the values and their probabilities and a histogram

▪ 0≤P(x)≤1

▪ Sum of all P(x) = 1

o Expected Value = Mean of discrete random variable

o Calculator: 1-Vars Stat L1, L2 (L1=X; L2=P(x))

o Continuous Random Variable – X has a measurable (all Real numbers +/- decimals etc) number of possible variables.

▪ Probability distribution: Density curve (doesn’t have to be normal)

▪ A fraction makes sense as an answer

▪ X≤A = Xn) = qn

o All cdfs have the same trend as binomial except it is: geocdf(P,r) geopdf(p,r)

o Calculator Strokes: (for binomial)

▪ P(X=r) binpdf(n,p,r)

▪ P(X≤r) bincdf(n,p,r)

▪ P(Xr)1-bincdf(n,p,r)

o Calculator Strokes: (for geometric)

▪ P(X=r) geopdf(p,r)

▪ P(X≤r) geocdf(p,r)

▪ P(Xr)1-geocdf(p,r)

Chapter 9

o Parameter – a number that describes a POPULATION. In stats the value of the parameter is unkown because we cannot examine the entire population

▪ μ, ơ, ơ2, N, P

o Statistic – a number that can be computed from a simple random sample

▪ Xbar, Sx, Sx2, n, phat

o Sampling Distribution – like meta analysis; the distribution of values taken by the statistic in all possible samples of the same size from the same population

o Unbiased – P= Phat or μphat = Phat

o As n increases, the distribution gets taller and narrower

▪ Sample n=100 (short wide mound) Sample n=500 (medium mound) Sample n=1000 (Tall thin mound)

o Short Answer Steps for Mean and Proportion Problems:

▪ Rule of Thumb 1: N>10n

▪ Rule of Thumb 2: np≥10 ; nq≥10 ; OR n≥30

▪ Problem Statement: P(Phat, Xbar ≥ #)

▪ Z-score

▪ Draw/shade curve

▪ Calculate

▪ Conclusion RESTATE QUESTION

o Cut variation (ơ) in half? 4n = .5 Variation ơ/√n …. ơ/√4n = ơ/2√n

o Mean Problem

▪ Given ơ

▪ Conditions:

• SRS

• Normality: Problem Says so OR Graph OR n≥30

• Independence: N>10n

▪ Statistic (Xbar, ơ/√n)

▪ Parameter (μ,ơ)

▪ Z= (Xbar- μ)/(ơ/√n)

▪ Ơxbar= ơ/√n

▪ P(Xbar≥≤#) when n>1

▪ P(X≥≤#) when n=1 “A”

o Proportion Problems

▪ P is like μ

▪ Phat is like Xbar

▪ Ơphat=√pq/n

▪ Conditions:

• SRS

• Normality: Problem Says so OR Graph OR np≥10 and nq≥10

• Independence: N>10n

▪ Phat = r/n (R= number of successes; n= Number in sample)

▪ Z= (Phat – P)/(√pq/n)

▪ P(Phat≥≤#) always

o Bias has to do with center not spread of a sampling distribution

o Parameters are fixed, while statistics vary depending on which sample is chosen

Chapter 10 – estimating where μ is with confidence

o Statistical Inference – provides methods for drawing conclusions about population from sample data

o Confidence interval – A level C confidence interval for a parameter has 2 parts

▪ A confidence interval calculated from the data usually in the form:

estimate ± margin of error = C

▪ A confidence level C which gives the probability that the interval will capture the true parameter (mean or proportion) value in repeated samples. Therefore the confidence level is the success rate for the method.

o Critical Values: z* and t* with the probability of p lying to its right under the standard normal curve called the upper p critical value

o Margin of Error – indicates how much error can be expected because of chance variation in randomized data production. It doesn’t account for other errors such as undercoverage or nonresponse. “Find it within” or “with a margin of error of”; always given; ROUND UP; length of the interval = 2(Margin of Error); decreases when…

▪ z* decreases/C decreases

▪ ơ decreases

▪ n increases

o Toolbox:

▪ Parameter: Identify the population of interest and the parameter you want to draw concludions about.

▪ Conditions:

• SRS

• Normality (n≥30; np>10, nq>10; graph, or it’s stated, EXPLAIN and draw graph)

• Independence (10n≤N)

▪ Calculations

▪ Interpretation: “ we are % confident that the true population mean of ___ lies within the interval (__,__)

o Mean Problems; [One Sample] Z Distributions:

▪ Known ơ

▪ z* Find using Invnorm(area below)

▪ n≥(z* ơ/m)2 m = Margin of error

▪ C = (Xbar ± z*( ơ/√n) Calculator: ZInterval

▪ Margin of Error = z*( ơ/√n)

o Mean Problems; [One Sample] T Distributions:

▪ UNKNOWN ơ

▪ t* find using t* table in packet and df (n-1); CRITICAL VALUE

▪ n≥(t* Sx/m)2

▪ C = (Xbar ± t*( Sx/√n) Calculator: TInterval

• With data use calculator to get Xbar and Sx

▪ Margin of Error = t*( Sx/√n)

▪ STANDARD ERROR = Sx/√n (The estimated SD from given data; we guess it because its not given)

▪ T Distributions always have a slightly larger spread than a normal distribution due to standard error (Estimated SD) and therefore more variation. As degrees of freedom increases, the density curve approaches the normal distribution.

▪ Paired T Procedure – used to compare the responses of two treatments in a matched pairs design or before-and-after measurements on the same subjects, apply one-sample T procedure to the observed difference

o Proportion Problems:

▪ Unknown ơ; You don’t need it

▪ Normality (of test): np≥10; nq≥10

▪ Normality (of confidence interval): nphat≥10; nqhat≥10

▪ C = phat +-z*(√phatqhat/n) Calculator: 1PropZInt( X = nphat Round to nearest whole #

▪ n≥(z*/m)2p*q* p* is given from a previous study otherwise its .5

▪ Margin of Error = z*(√phatqhat/n)

o Outliers strongly influence Xbar therefore can skew CInterval

o If Xbar is in the range of confidence it will ALWAYS contain μ

Chapter 11

o Null Hypothesis – (Ho) Statement being tested in a significance test; significance test is designed to assess the strength of the evidence against the null; μ=#

o Alternative Hypothesis – (Ha) Claim about the population that we are trying to find evidence for; μ(>, ................
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