Statistics Formula Sheet Part 4

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. 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. Example of mutually exclusive (disjoint): A: roll an odd on a die B: roll an even on a die Examples of independent events: A: draw an ace B: draw a spade Odd and even share no outcomes P(odd and even) = 0 Therefore, they are mutually exclusive. P(Spade) =13/52 = 1/4 P(Spade | Ace) = 1/4 Knowing that the drawn card is an ace does not change the probability of drawing a spade 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) = 4/52 Therefore, they are not mutually exclusive. Mutually Exclusive events cannot be independent Examples that are dependent (not independent): A: roll a number greater than 3 B: roll an even P(even) = 3/6 = 1/2 P(even | greater than 3) = 2/3 Knowing the number is greater than three changes the probability of rolling an even number. Independent events cannot be Mutually Exclusive Mutually exclusive and dependent Independent and not mutually exclusive A: Roll an even B: Roll an odd A: draw a black card B: draw a king They share no outcomes and knowing that it is odd changes the probability of it being even. Knowing it is a black card does not change 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) 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 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 H0 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 H0 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" "Moderate or Good" "Strong" 0.05 < P-Value < 0.10 0.01 < P-Value < 0.05 P-Value < 0.01 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 H0 If the hypothesized parameter lies inside the C% confidence interval for the parameter I FAIL TO REJECT H0 Power of test: The probability that at a fixed level  test will reject the null hypothesis when and alternative value is true. Confidence Intervals One Sample Z Interval Use when estimating a single population mean and σ is known Conditions: -SRS -Normality: CLT, stated, or plots -Independence and N≥10n Interval: x z*  n One Sample t Interval Use when estimating a single mean and σ is NOT known [Also used in a matched paired design for the mean of the difference: PAIRED t PROCEDURE] Conditions: -SRS -Normality: CLT, stated, or plots -Independence and N≥10n Interval: x  t* s n CALCULATOR: 7: Z-Interval df = n-1 CALCULATOR: 8: T-Interval Two Sample Z Interval σ known (RARELY USED) Use when estimating the difference between two population means and σ is known Conditions: -SRS for both populations -Normality: CLT, stated, or plots for -both populations -Independence and N≥10n for both populations. Interval: Two Sample t Interval σ unknown Use when estimating the difference between two population means and σ is NOT known Conditions: -SRS for both populations -Normality: CLT, stated, or plots for both populations -Independence and N≥10n for both populations. Interval: * (x 1  x 2 )  z CALCULATOR: 9: 2-SampZInt 12 n1   22 n2 (x 1  x 2 )  t * s12 s 22  n1 n2 Use lower n for df (df = n-1) or use calculator CALCULATOR: 0: 2-SampTInt One Proportion Z Interval Use when estimating a single proportion Conditions: -SRS -Normality: nˆp ≥ 10, n(1  pˆ ) ≥ 10 -Independence and N≥10n Interval: ˆp  z* pˆ (1  pˆ ) n CALCULATOR: A: 1-PropZInt Two Proportion Z Interval Use when estimating the difference between two population proportions. Conditions: -SRS for both populations -Normality: -Independence and N≥10n for both populations. Interval: ˆp1 (1  pˆ1 ) ˆp2 (1  p ˆ2) *  ( pˆ1  pˆ2 )  z n1 n2 CALCULATOR: B: 2-PropZInt Confidence interval for Regression Slope: Use when estimating the slope of the true regression line Conditions: 1. Observations are independent 2. Linear Relationship (look at residual plot) 3. Standard deviation of y is the same(look at residual plot) 4. y varies normally (look at histogram of residuals) Interval: * b df = n - 2 CALCULATOR: LinRegTInt Use technology readout or calculator for this confidence interval. b  t SE Hypothesis Testing One Sample Z Test Use when testing a mean from a single sample and σ is known H o: µ = # Ha: µ ≠ # -or- µ < # -or- µ > # Conditions: -SRS -Normality: Stated, CLT, or plots -Independence and N≥10n Test statistic: z x  0  n 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] H o: µ = # Ha: µ ≠ # -or- µ < # -or- µ > # Conditions: -SRS -Normality: Stated, CLT, or plots -Independence and N≥10n Test statistic: df = n-1 t x  0 s n One Proportion Z test Use when testing a proportion from a single sample H o: p = # Ha: p ≠ # -or- p < # -or- p > # Conditions: -SRS -Normality: np0 ≥ 10, n(1-p0) ≥ 10 -Independence and N≥10n Test statistic: CALCULATOR: 1: Z-Test Two Sample Z test σ known (RARELY USED) Use when testing two means from two samples and σ is known H o : µ1 = µ 2 Ha: µ1 ≠ µ2 -or- µ1 < µ2 -or- µ1 > µ2 Conditions: -SRS for both populations -Normality: Stated, CLT, or plots for both populations -Independence and N≥10n for both populations. Test statistic: CALCULATOR: 2: T-Test CALCULATOR: 5: 1-PropZTest Two Sample t Test σ unknown Use when testing two means from two samples and σ is NOT known H o : µ1 = µ 2 Ha: µ1 ≠ µ2 -or- µ1 < µ2 -or- µ1 > µ2 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. Two Proportion Z test Use when testing two proportions from two samples CALCULATOR: 3: 2-SampZTest CALCULATOR: 4: 2-SampTTest Χ2 – Homogeneity of Populations r x c table Use for categorical variables to see if multiple distributions are similar to one another 2 Χ – Goodness of Fit L1 and L2 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 <5 Test statistic: df = n-1 H o : p 1 = p2 Ha: p1 ≠ p2 -or- p1 < p2 -or- p1 > p2 Conditions: -SRS for both populations -Normality: Independence and N≥10n for both populations.Test statistic: CALCULATOR: 6: 2-PropZTest 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 <5 Test statistic: df = (r-1)(c-1) Χ2 – 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 <5 Test statistic: df = (r-1)(c-1) CALCULATOR:D: Χ2GOF-Test(on CALCULATOR: C: Χ2-Test CALCULATOR: C: Χ2-Test 84) 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) 3. 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 t  b df = n-2 SEb Notation and Interpretations IQR µ s σ Inner Quartile Range Mean of a sample Mean of a population Standard deviation of a sample Standard deviation of a population Sample proportion p s2 σ2 M Σ Q1 Q3 Z z* t t* N(µ, σ) r r2 Population proportion Variance of a sample Variance of a population Median Summation First Quartile Third Quartile Standardized value – z test statistic Critical value for the standard normal distribution Test statistic for a t test Critical value for the t-distribution Notation for the normal distribution with mean and standard deviation Correlation coefficient – strength of linear relationship Coefficient of determination – measure of fit of the model to the data Equation for the Least Squares Regression Line a b ( y = axb y = abx SRS S P(A) Ac P(B|A) ∩ U X µX σX y-intercept of the LSRL Slope of the LSRL Point the LSRL passes through Power model Exponential model Simple Random Sample Sample Space The probability of event A A complement Probability of B given A Intersection (And) Union (Or) Random Variable Mean of a random variable 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 cdf n N CLT Probability distribution function Cumulative distribution function Sample size Population size Central Limit Theorem Mean of a sampling distribution Standard deviation of a sampling distribution df SE H0 Ha p-value α β χ2 Degrees of freedom Standard error Null hypothesis-statement of no change Alternative hypothesis- statement of change Probability (assuming H0 is true) of observing a result as large or larger than that observed Significance level of a test. P(Type I) or the y-intercept of the true LSRL P(Type II) or the true slope of the LSRL Chi-square test statistic z-score (z) The number of standard deviations an observation is above/below the mean slope (b) The change in predicted y for every unit increase on x y-intercept (a) Predicted y when x is zero r (correlation coefficient) strength of linear relationship. (Strong/moderate/weak) (Positive/Negative) linear relationship between y and x. r2 (coefficient of determination) percent of variation in y explained by the LSRL of y on x. variance (σ2 or s2) average squared deviation from the mean standard deviation (σ or s) measure of variation of the data points from the mean Confidence Interval (#,#) I am C% confident that the true parameter (mean µ or proportion p) lies between # and #. C % Confidence (Confidence level) Using my method, If I sampled repeatedly, C% of my intervals would contain the true parameter (mean µ or proportion p). p<α Since p < α I reject the null hypothesis H0 and I have sufficient/strong evidence to conclude the alternative hypothesis Ha p>α Since p > α I fail to reject the null hypothesis H0 and I have do not have sufficient evidence to support the alternative hypothesis Ha p-value The probability, assuming the null hypothesis is true, that an observed outcome would be as or more extreme than what was actually observed. Duality-Outside Interval Two sided test If the hypothesized parameter lies outside the (1 – α)% confidence interval for the parameter I can REJECT H0 for a two sided test. Duality-Inside Interval Two sided test If the hypothesized parameter lies inside the (1 – α)% confidence interval for the parameter I FAIL TO REJECT H0 for a two sided test. Power of the test The probability that a fixed level test will reject the null hypothesis when an alternative value is true standard error (SE) in general Estimates the variability in the sampling distribution of the sample statistic. standard deviation of the residuals (s from regression) A typical amount of variability of the vertical distances from the observed points to the LSRL standard error of the slope of the LSRL (SEb) This is the standard deviation of the estimated slope. This value estimates the variability in the sampling distribution of the estimated slope.