A Statistics Summary-sheet - UBalt

A Statistics Summary-sheet

Sampling Conditions 2 is known X N (?, 2/n)

Confidence Interval

X

?

Z /2

n

Test Statistic Z = X - ?0

/ n

Yes

2 is unknown X N (?, 2/n)

X ? Z /2

s n

Z = X - ?0 s/ n

Is n is large, say over 30?

p ? Z / 2

p(1 - n

p)

Z = p - p0 p0 (1 - p0 ) n

No

X N (?, 2) and 2 is known X N (?, 2/n)

X

?

Z / 2

n

Z = X - ?0 / n

X N (?, 2) and 2 is unknown X tn-1 (?, 2/n)

X ? tn -1, / 2

s n

tn -1

=

X- s/

?0 n

If n is not large, say over 30 and X is not N (?, 2), cannot proceed with parametric statistics.

Formulas, Distributions, and Concepts

Counting and Probabilities

n Px

= n! (n - x)!

Permutations

n Cx

=

n! x!(n - x)!

Combinations

P(A | B) = P(A B) Conditional Probability P(B)

P(A B) = P(A | B)P(B) Probability of an Intersection

Discrete Probability Distributions

Px (x)

=

n! x!(n -

px (1 - x)!

p) n-x

Binomial Probability

Px (x)

=

e-? ? x x!

Poisson Probability

Continuous Probability Distributions

Random Variable Distribution (mean, variance)

Standard Normal

Z N(0,1)

Normal

X N(?,2)

Binomial Sample Mean By CLT, if n 30, Sample Proportion

X Binomial [np, np(1-p)]

X

N

?

,

2

n

X

N

?

,

2

n

p

p,

p(1 - n

p)

Confidence Intervals (Interval Estimation)

X ? z( / 2) n

If population is normal and population variance is known.

s X ? z( / 2) n If population variance is unknown and n 30.

s X ? t(n-1, / 2) n

If population is normal, population variance is unknown.

p ? z( / 2)

p(1- p) If n 30. n

( X - Y ) ? z( / 2)

2 1

+

2 2

n1 n2

variance for population variance.

If independent samples and either population variance known, or n 30 in which case, substitute sample

( X - Y ) ? t(nX +nY -2, / 2)

s

2

1 n1

+

1 n2

where

s2 = (n1 -1)s12 + (n2 -1)s22 If independent samples, population variances unknown, but statistically equal. n1 + n2 - 2

Estimating Sample Size

n

=

z2/2 E2

2

For estimation and CI for the population mean, normal population, 2 known, or estimated by a pilot run. E = absolute error.

Hypothesis Testing

1. Set up the appropriate null which must be in equality form, always and alternative hypotheses. 2. Define the rejection area. Take care as to whether the test is one-tailed or two-tailed. Look to the alternative hypothesis to determine this. 3. Calculate the test statistic. 4. State Decision. 5. Interpret your conclusion.

Hypothesis Testing (test statistics and their distributions under the null)

X

-

?0

z

When population variance known, or if n 30, substitute s for .

n

Xs

?0

tn-1,

When If population is normal, population variance unknown.

n

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