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This document contains formulas for: 1) Calculating probabilities, means, variances, and standard deviations for probability distributions and sampling distributions. 2) Converting between random variables, standard scores, and z-scores. 3) Calculating population and sample characteristics like means, variances, standard deviations, and determining sample size for confidence intervals.

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57 views1 page

Reviewer

This document contains formulas for: 1) Calculating probabilities, means, variances, and standard deviations for probability distributions and sampling distributions. 2) Converting between random variables, standard scores, and z-scores. 3) Calculating population and sample characteristics like means, variances, standard deviations, and determining sample size for confidence intervals.

Uploaded by

chxrlslxrren
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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FORMULAS

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 Variance of the sampling distribution of the


𝑃(𝐸) =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 sample means (with replacement)

Mean of the Probability Distribution ̅ ) ∙ (𝑋


𝜎2𝑥̅ = ∑[ 𝑃(𝑋 ̅ − 𝜇)2 ]

𝜇 = 𝐸(𝑋) = ∑(𝑋 ∙ 𝑃(𝑋)) Mean of the sampling distribution of the sample


mean (without replacement)
Variance of the Probability Distribution
∑𝑋̅
𝜇𝑥̅ =
𝝈𝟐 = ∑[𝑿𝟐 ∙ 𝑷(𝑿)] − 𝝁𝟐 𝑛

Variance of the sampling distribution of the


Standard Deviation of the Probability Distribution
sample mean (without replacement)
𝝈 = √∑[𝑿𝟐 ∙ 𝑷(𝑿)] − 𝝁𝟐 ̅ − 𝜇𝑥̅ )2
∑(𝑋
𝜎𝑥̅2 =
𝑛
Converting a random variable x to a standard
normal variable z OR dealing with data from Standard Deviation of the sampling distribution
parameters of the sample mean (without replacement)
𝑥−𝜇
𝑧= ∑(𝑋̅ − 𝜇𝑥̅ )2
𝜎 𝜎𝑥̅ = √
𝑛
Converting standard normal variable z to
random variable x Convert the raw score to the standard score in
central limit OR dealing with data from sample
𝑥 = 𝑧𝜎 + 𝜇 mean
POPULATION MEAN 𝑥̅ − 𝜇
𝑧= 𝜎
∑𝑥
𝜇= √𝑛
𝑁
Lower limit : 𝐿𝐿 = 𝑋̅ − 𝐸
POPULATION VARIANCE
Upper limit: 𝑈𝐿 = 𝑋̅ + 𝐸
2
∑(𝑥 − 𝜇)2
𝜎 = 𝜎
𝑁 Margin of error: 𝐸 = 𝑍𝛼 ∙
2 √𝑛
POPULATION STANDARD DEVIATION
Length Of Confidence Interval:
∑(𝑥 − 𝜇)2 L = UL – LL 𝑜𝑟 𝐿 = 2𝐸
𝜎=√
𝑁
Determining Sample Size
SAMPLE MEAN
𝑧𝛼 𝜎 2
2
∑𝑥 𝑛=( )
𝑥̄ = 𝐸
𝑛

SAMPLE VARIANCE

∑(𝑥 − 𝑥̄ )2
𝑠2 =
𝑛−1

SAMPLE STANDARD DEVIATION

∑(𝑥 − 𝑥̄ )2
𝑠=√
𝑛−1

Combination of N (population size) objects


taken n (sample size) at a time.

𝑁!
𝐶
𝑁 𝑛
=
[𝑛! (𝑁 − 𝑛)!]

Mean of the sampling distribution of the sample


means. (with replacement)

𝜇𝑥̅ = ∑[𝑋̅ ∙ 𝑃(𝑋̅ )]

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