# PY 211 Lecture Notes - Lecture 9: Bias Of An Estimator, Central Limit Theorem, Variance

Chapter 7

Probability and Sampling Distributions

1. Standardized scores

a. Reminder

i. Transform scores from distributions with any mean and any SD

ii. Become z-scores

iii. Use unit normal table to look up probabilities

1. P(z > 1.96) = ? ļ .0250

2. P(z < -1.33) = ? ļ .0918

a. Swap that to p(z > 1.33) to look up on the table

iv. Empirical rule (68%, 95%)

1. One standard deviation to right and left captures about 70% of the

scores

2. Two standard deviations to the right and left captures about 95% of

scores

b. Importance of 95% (+ / - 2 SDs)

i. Most common responses

ii. What about remainder in the tails?

1. Split remaining 5% among tails

2. 5% 00> .05 probability

3. What is Ā½ of 0.05

Selecting Samples from Populations

ļ· Avoiding bias: random sampling

ļ· Random sampling

o Every item in pool has equal chance of selection

o The probability of remaining selections cannot change after selection (conditional

probability)

Selecting a Sample

ļ· In theory

o The order in which items are sampled matters

o Sampling with replacement

ļ· In practice

o The order in which items are sampled does not matter

o Sampling without replacement

Sampling Distribution: The Mean

ļ· Definition of sampling distribution

o A distribution of all possible sample means that could be obtainedā¦

o In sample of a given size,

o From the same population

ļ· The mean of the sampling distribution

o Symbol: ļm

o M ļ ļm = ļ

o M is an unbiased estimator of ļ

o On average, the sample mean will equal the value of ļ

o Central limit theorem

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## Document Summary

Random sampling: every item in pool has equal chance of selection, the probability of remaining selections cannot change after selection (conditional probability) In theory: the order in which items are sampled matters, sampling with replacement. In practice: the order in which items are sampled does not matter, sampling without replacement. Definition of sampling distribution: a distribution of all possible sample means that could be obtained , in sample of a given size, from the same population. The mean of the sampling distribution: symbol: m, m m = , m is an unbiased estimator of , on average, the sample mean will equal the value of , central limit theorem. The average of the sample variances: symbol: s^2, s^2 s^2 = ^2, s^2 is an unbiased estimator of ^2, on average, the sample variances will equal the value of ^2 . Divide population variance by sample size (n = 2)