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York University (9,812)
Final

FINAL CHEAT SHEET STATS 2320.docx

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School
York University
Department
Course
Professor
Michael Rochon
Semester
Fall

Description
  E(X) = μx= μ = ∑ xP(x) V(X ) =σ =σ /n σ =σ / n ν 2 n −1 2 N p ± z p(1− p x x  Larger values of s produce wider confidence intervals.   α / 2 n     Larger sample size narrow confidence interval. Chapter 9 Sample Distribution: X is normal▯X  is normal | If X is Non­normal   is appr. normal  distribution for n>30 | larger sample  ▯closer to normal. σ x ± z α / 2 μ μ n  Confidence interval estimator of : z-estimate of : 2  Sample Mean:         zα / 2 σ = (σ / n) ( (N − n)/(N −1) σ = ∑ x P(x)− μ2 μ n =  W   Finite Population: x                (N − n)/(N −1) Minimum sample size to estimate   Rule of thumb: Pop. Size > 20 Sample size  ▯omit correction factor (                 ) ( round up) : σ σ - An unbiased estimator of a population is an estimator whose expected value is equal to that Z = X −μ P (μ − α / 2 < X < μ + Zα / 2 ) = 1 − α parameter. It is said to be consistent if the difference between the estimator and the parameter  Z­test Statistic : σ / n               or n grows smaller as the sample size grows larger.  E.g. s = 2, m=12 x − μ10 −1 - If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to P(x ≤10) =Pσ < 2 = P(Z ≤ −1) = .5−3413 = .1587 be relatively efficient  Prob. of 1 cells = 10 or less:  - Narrower interval provides more meaningful info. Larger confidence level produces a wider P(x ≤10)= P − <10−1= P(Z ≤−2)=.5− 4771=.0228 confidence interval. Increasing the sample size decreases the width of confidence interval.  Prob. Mean of 4 cells = 10 or less: σ / n 2/ 4 Prob. of Total 4 cells = 40 or less:   [P(x ≤1])= (.1587) = .0006 X CH: 11 HYPOTHESIS TESTING  Prob. All 4 cells = 10 or less:  P = 2. test statistic: randomly sample the population and calculate the sample mean If the test ˆ ˆ 2 n statistic’s value is inconsistent with the null hypothesis, we reject the null hypothesis and infer that  Proportion: E(P) = μp= p V(P) =σ P p(1− p)/n σPˆ p(1− p)/n the alternative hypothesis is true? Z = P − p Rejection region: Z> Za (one tail, right) . Z< -Za ( one tail Left )  Z­Stat for Proportion:  p(1− pE.g. 80% agree(p), ask 350(n), P<75%=? z =x −μ Test statistic of z-test of : σ / n  P − p .75−.80  P(P α), we do not reject the null hypothesis. P > is 1 – probability. Relationship between Type I and Type II errors: deceasing the significance level (a), increases the value of (B) and vice versa. If the probability of Type II is too large (B), we CHAPTER 9 Central Limit Theorem – the sampling dist of the mean of a random sample drawn form any can reduce it by increasing (a) and/or increasing the sample size. Power of a test is defined as 1– population is approximately normal for a sufficiently large sample size. The larger the sample size, the represents the probability of rejecting the null hypothesis when it is false. CHAPTER 12: INFERENCES ABOUT ONE POPULATION more closely the sampling distribution of X will resemble a normal distribution σ N −n Inference about a population when σ is unknown: Describe a population, interval, central 4.σ x nd location. n N−1 SE finite population; 2 correction factor It is impossible to construct a confidence interval for a population mean, if there is a non normal Standard error of the proportion= p(1−p)/n population, with small sample and unknown variance Required conditions: the population is normally distributed or not extremely non-normal. Chi squared distribution approaches the shape σ μ = μ σ x of a normal distribution as D.O.F increase Expected value 1 smpl mean E(X) = x Stn error : n x − μ t = Z = X −μ ˆ μ = p μ s / n z score 1 smpl mean: σ/ n Expec. Val. sample pro E(P) = p - t-test for : σ = p(1− p) s Standard error |1 smpl proportion: p n x ±t α/2,n−1 n P − p Z = μ z score |1 sample proportion: p(1− p) n t-estimate of : v = n-1 Rejection region: t > t a, v Inference about a population Variance: (variability): Describe a population, interval, Variability Expected value |Difference between 2 sample means: μ = μ −μ Required conditions: sample population be normal, and is valid unless population is non normal E(X 1 X ) 2 x1−x 2 1 2 2 2 (n−1)s σ 1 σ 2 χ 2 2 χ = 2 σx1−x2 = + -Test ofσ : σ Stn. error |Difference b/w 2 sample means: n1 n2 (n−1)s 2 2 z score (standardized score) | Difference between 2 sample means: (n −1)s (X 1 X )2−(μ −1μ )2 2 Z = 2 2 χ 2 σ1 + σ 2 2 α/2,v−1 χ n1 n2 Estimate of σ : LCL = UCL = 1−α/2,v−1 CHAPTER 10: INTRODUCTION TO ESTIMATION χ 2 1 – a = level of confidence, a= significance level 2 α/2,v−1 2 2 1-α α/2 Zα/2 Rejection Region: χ > (greater) χ < χ 1−α/2,v−1 (less) α 0.9 Z0.05=1 p 0 .10 .05 .645 Inference about a Population Proportion: Describing a population, Nominal. ** =0.5 if 0.9 Z0.25=1 unknown. Required: np and n(1-p) >5 5 .05 .025 .96 p− p 0.9 Z.01=2. z = x 8 .02 .01 33
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