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p-value : the probability that an event will happen purely by chance assuming the null hypothesis is true. The smaller thep-value, the stronger the evidence is against the null hypothesis. | https://openstax.org/books/introductory-statistics-2e/pages/9-key-terms |
Standard Deviation : a number that is equal to the square root of the variance and measures how far data values are from their mean; notation:sfor sample standard deviation andÏfor population standard deviation. | https://openstax.org/books/introductory-statistics-2e/pages/9-key-terms |
Student'st-Distribution : investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student. The major characteristics of the random variable (RV) are:It is continuous and assumes any real values.The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at... | https://openstax.org/books/introductory-statistics-2e/pages/9-key-terms |
Type I Error : The decision is to reject the null hypothesis when, in fact, the null hypothesis is true. | https://openstax.org/books/introductory-statistics-2e/pages/9-key-terms |
Type II Error : The decision is not to reject the null hypothesis when, in fact, the null hypothesis is false. | https://openstax.org/books/introductory-statistics-2e/pages/9-key-terms |
Two population means from independent samples where the population standard deviations are not known | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Random Variable:X¯1âX¯2X¯1âX¯2= the difference of the sampling means | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Distribution: Student'st-distribution with degrees of freedom (variances not pooled) | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
A hypothesis test of two population means from independent samples where the population standard deviations are known will have these characteristics: | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Random variable:X¯1âX¯2X¯1âX¯2= the difference of the means | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Distribution: normal distribution | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Test of two population proportions from independent samples. | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Random variable:p^Aâp^B=p^Aâp^B=difference between the two estimated proportions | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Distribution: normal distribution | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
A hypothesis test for matched or paired samples (t-test) has these characteristics: | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Test the differences by subtracting one measurement from the other measurement | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Random Variable:x¯dx¯d= mean of the differences | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Distribution: Studentâs-t distribution withnâ 1 degrees of freedom | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
If the number of differences is small (less than 30), the differences must follow a normal distribution. | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Two samples are drawn from the same set of objects. | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Samples are dependent. | https://openstax.org/books/introductory-statistics-2e/pages/10-chapter-review |
Standard error:SE=(s1)2n1+(s2)2n2(s1)2n1+(s2)2n2 | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
Test statistic (t-score):t=(x¯1âx¯2)â(μ1âμ2)(s1)2n1+(s2)2n2(x¯1âx¯2)â(μ1âμ2)(s1)2n1+(s2)2n2 | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
Degrees of freedom:df=((s1)2n1+(s2)2n2)2(1n1â1)((s1)2n1)2+(1n2â1)((s2)2n2)2df=((s1)2n1+(s2)2n2)2(1n1â1)((s1)2n1)2+(1n2â1)((s2)2n2)2 | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
where: | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
s1ands2are the sample standard deviations, andn1andn2are the sample sizes. | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
x¯1x¯1andx¯2x¯2are the sample means. | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
Cohenâsdis the measure of effect size: | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
d=x¯1âx¯2spooledd=x¯1âx¯2spooledwherespooled=(n1â1)s12+(n2â1)s22n1+n2â2spooled=(n1â1)s12+(n2â1)s22n1+n2â2 | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
Normal Distribution:X¯1âX¯2â¼N[μ1âμ2,(Ï1)2n1+(Ï2)2n2]X¯1âX¯2â¼N[μ1âμ2,(Ï1)2n1+(Ï2)2n2].Generallyµ1âµ2= 0. | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
Test Statistic (z-score):z=(x¯1âx¯2)â(μ1âμ2)(Ï1)2n1+(Ï2)2n2z=(x¯1âx¯2)â(μ1âμ2)(Ï1)2n1+(Ï2)2n2Generallyµ1-µ2= 0. | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
where:Ï1andÏ2are the known population standard deviations.n1andn2are the sample sizes.x¯1x¯1andx¯2x¯2are the sample means.μ1andμ2are the population means. | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
Pooled Proportion:pc=xF+xMnF+nMxF+xMnF+nM | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
Distribution for the differences:pâ²Aâpâ²Bâ¼N[0,pc(1âpc)(1nA+1nB)]pâ²Aâpâ²Bâ¼N[0,pc(1âpc)(1nA+1nB)] | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
where the null hypothesis isH0:pA=pBâorâH0:pAâpB= 0. | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
Test Statistic (z-score):z=(pâ²Aâpâ²B)pc(1âpc)(1nA+1nB)z=(pâ²Aâpâ²B)pc(1âpc)(1nA+1nB) | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
where the null hypothesis isH0:pA=pBâorâH0:pAâpB= 0. | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
where | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
pâ²Aandpâ²Bare the sample proportions,pAandpBare the population proportions, | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
Pcis the pooled proportion, andnAandnBare the sample sizes. | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
Test Statistic (t-score):t=x¯dâμd(sdn)x¯dâμd(sdn) | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
where: | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
x¯dx¯dis the mean of the sample differences.μdis the mean of the population differences.sdis the sample standard deviation of the differences.nis the sample size. | https://openstax.org/books/introductory-statistics-2e/pages/10-formula-review |
Degrees of Freedom (df) : the number of objects in a sample that are free to vary. | https://openstax.org/books/introductory-statistics-2e/pages/10-key-terms |
Pooled Proportion : estimate of the common value ofp1andp2. | https://openstax.org/books/introductory-statistics-2e/pages/10-key-terms |
Standard Deviation : A number that is equal to the square root of the variance and measures how far data values are from their mean; notation:sfor sample standard deviation andÏfor population standard deviation. | https://openstax.org/books/introductory-statistics-2e/pages/10-key-terms |
Variable (Random Variable) : a characteristic of interest in a population being studied. Common notation for variables are upper-case Latin lettersX,Y,Z,... Common notation for a specific value from the domain (set of all possible values of a variable) are lower-case Latin lettersx,y,z,.... For example, ifXis the numb... | https://openstax.org/books/introductory-statistics-2e/pages/10-key-terms |
The chi-square distribution is a useful tool for assessment in a series of problem categories. These problem categories include primarily (i) whether a data set fits a particular distribution, (ii) whether the distributions of two populations are the same, (iii) whether two events might be independent, and (iv) whether... | https://openstax.org/books/introductory-statistics-2e/pages/11-chapter-review |
An important parameter in a chi-square distribution is the degrees of freedomdfin a given problem. The random variable in the chi-square distribution is the sum of squares ofdfstandard normal variables, which must be independent. The key characteristics of the chi-square distribution also depend directly on the degrees... | https://openstax.org/books/introductory-statistics-2e/pages/11-chapter-review |
The chi-square distribution curve is skewed to the right, and its shape depends on the degrees of freedomdf. Fordf> 90, the curve approximates the normal distribution. Test statistics based on the chi-square distribution are always greater than or equal to zero. Such application tests are almost always right-tailed tes... | https://openstax.org/books/introductory-statistics-2e/pages/11-chapter-review |
To assess whether a data set fits a specific distribution, you can apply the goodness-of-fit hypothesis test that uses the chi-square distribution. The null hypothesis for this test states that the data come from the assumed distribution. The test compares observed values against the values you would expect to have if ... | https://openstax.org/books/introductory-statistics-2e/pages/11-chapter-review |
To assess whether two factors are independent or not, you can apply the test of independence that uses the chi-square distribution. The null hypothesis for this test states that the two factors are independent. The test compares observed values to expected values. The test is right-tailed. Each observation or cell cate... | https://openstax.org/books/introductory-statistics-2e/pages/11-chapter-review |
To assess whether two data sets are derived from the same distributionâwhich need not be known, you can apply the test for homogeneity that uses the chi-square distribution. The null hypothesis for this test states that the populations of the two data sets come from the same distribution. The test compares the observ... | https://openstax.org/books/introductory-statistics-2e/pages/11-chapter-review |
The goodness-of-fit test is typically used to determine if data fits a particular distribution. The test of independence makes use of a contingency table to determine the independence of two factors. The test for homogeneity determines whether two populations come from the same distribution, even if this distribution i... | https://openstax.org/books/introductory-statistics-2e/pages/11-chapter-review |
To test variability, use the chi-square test of a single variance. The test may be left-, right-, or two-tailed, and its hypotheses are always expressed in terms of the variance (or standard deviation). | https://openstax.org/books/introductory-statistics-2e/pages/11-chapter-review |
Ï2= (Z1)2+ (Z2)2+ ⦠(Zdf)2chi-square distribution random variable | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
μÏ2=dfchi-square distribution population mean | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
ÏÏ2=2(df)ÏÏ2=2(df)Chi-Square distribution population standard deviation | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
âk(OâE)2Eâk(OâE)2Egoodness-of-fit test statistic where:O: observed valuesE: expected valuesk: number of different data cells or categories | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
df=kâ 1 degrees of freedom | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
The number of degrees of freedom is equal to (number of columns - 1)(number of rows - 1). | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
The test statistic isΣ(iâj)(OâE)2EΣ(iâj)(OâE)2EwhereO= observed values,E= expected values,i= the number of rows in the
table, andj= the number of columns in
the table. | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
If the null hypothesis is true, the expected numberE=(row total)(column total)total surveyedE=(row total)(column total)total surveyed. | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
âiâj(OâE)2Eâiâj(OâE)2EHomogeneity test statistic where:O= observed valuesE= expected valuesi= number of rows in data contingency tablej= number of columns in data contingency table | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
df= (iâ1)(jâ1) Degrees of freedom | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
Ï2=Ï2=(nâ1)âs2Ï2(nâ1)âs2Ï2Test of a single variance statistic where:n: sample sizes: sample standard deviationÏ: population standard deviation | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
df=nâ 1 Degrees of freedom | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
Use the test to determine variation. | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
The degrees of freedom is the number of samples â 1. | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
The test statistic is(nâ1)âs2Ï2(nâ1)âs2Ï2, wheren= the total number of data,s2= sample variance, andÏ2= population variance. | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
The test may be left-, right-, or two-tailed. | https://openstax.org/books/introductory-statistics-2e/pages/11-formula-review |
Contingency Table : a table that displays sample values for two different factors that may be dependent or contingent on one another; it facilitates determining conditional probabilities. | https://openstax.org/books/introductory-statistics-2e/pages/11-key-terms |
The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or graphically from a plotted curve. (Lines are classified as straight curves.) Algebraically, a linear equation typically takes the... | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
Theslope of a lineis a value that describes the rate of change between the independent and dependent variables. Theslopetells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average. They-interceptis used to describe the dependent variable when the independent v... | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
Scatter plots are particularly helpful graphs when we want to see if there is a linear relationship among data points. They indicate both the direction of the relationship between thexvariables and theyvariables, and the strength of the relationship. We calculate the strength of the relationship between an independent ... | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
A regression line, or a line of best fit, can be drawn on a scatter plot and used to predict outcomes for thexandyvariables in a given data set or sample data. There are several ways to find a regression line, but usually the least-squares regression line is used because it creates a uniform line. Residuals, also calle... | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
The correlation coefficientrmeasures the strength of the linear association betweenxandy. The variablerhas to be between â1 and +1. Whenris positive, thexandywill tend to increase and decrease together. Whenris negative,xwill increase andywill decrease, or the opposite,xwill decrease andywill increase. The coefficien... | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
Linear regression is a procedure for fitting a straight line of the formÅ·=a+bxto data. The conditions for regression are: | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
LinearIn the population, there is a linear relationship that models the average value ofyfor different values ofx. | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
IndependentThe residuals are assumed to be independent. | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
NormalTheyvalues are distributed normally for any value ofx. | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
Equal varianceThe standard deviation of theyvalues is equal for eachxvalue. | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
RandomThe data are produced from a well-designed random sample or randomized experiment. | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
The slopeband interceptaof the least-squares line estimate the slopeβand interceptαof the population (true) regression line. To estimate the population standard deviation ofy,Ï, use the standard deviation of the residuals,s.s=SEEnâ2s=SEEnâ2. The variableÏ(rho) is the population correlation coefficient. To test ... | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
After determining the presence of a strong correlation coefficient and calculating the line of best fit, you can use the least squares regression line to make predictions about your data. | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
To determine if a point is an outlier, do one of the following: | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
Input the following equations into the TI 83, 83+,84, 84+:y1=a+bxy2=a+bx+2sy3=a+bxâ2sy1=a+bxy2=a+bx+2sy3=a+bxâ2swhere s is the standard deviation of the residualsIf any point is above y2or below y3then the point is considered to be an outlier. | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
y1=a+bxy2=a+bx+2sy3=a+bxâ2sy1=a+bxy2=a+bx+2sy3=a+bxâ2swhere s is the standard deviation of the residuals | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
If any point is above y2or below y3then the point is considered to be an outlier. | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
Use the residuals and compare their absolute values to 2swheresis the standard deviation of the residuals. If the absolute value of any residual is greater than or equal to 2s, then the corresponding point is an outlier. | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
Note: The calculator function LinRegTTest (STATS TESTS LinRegTTest) calculatess. | https://openstax.org/books/introductory-statistics-2e/pages/12-chapter-review |
y=a+bxwhereais they-intercept andbis the slope. The variablexis the independent variable andyis the dependent variable. | https://openstax.org/books/introductory-statistics-2e/pages/12-formula-review |
Least Squares Line or Line of Best Fit: | https://openstax.org/books/introductory-statistics-2e/pages/12-formula-review |
y^=a+bxy^=a+bx | https://openstax.org/books/introductory-statistics-2e/pages/12-formula-review |
where | https://openstax.org/books/introductory-statistics-2e/pages/12-formula-review |
a=y-intercept | https://openstax.org/books/introductory-statistics-2e/pages/12-formula-review |
b= slope | https://openstax.org/books/introductory-statistics-2e/pages/12-formula-review |
Standard deviation of the residuals: | https://openstax.org/books/introductory-statistics-2e/pages/12-formula-review |
s=SEEnâ2.s=SEEnâ2. | https://openstax.org/books/introductory-statistics-2e/pages/12-formula-review |
where | https://openstax.org/books/introductory-statistics-2e/pages/12-formula-review |
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