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Final

# FINAL CHEAT SHEET STATS 2320.docx

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York University

Administrative Studies

ADMS 2320

Michael Rochon

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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