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

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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) = ? .0250
2. P(z < -1.33) = ? .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 ½ of 0.05
Selecting Samples from Populations
Avoiding bias: random sampling
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
In theory
o The order in which items are sampled matters
o Sampling with replacement
In practice
o The order in which items are sampled does not matter
o Sampling without replacement
Sampling Distribution: The Mean
Definition of sampling distribution
o A distribution of all possible sample means that could be obtained…
o In sample of a given size,
o From the same population
The mean of the sampling distribution
o Symbol: m
o M m =
o M is an unbiased estimator of
o On average, the sample mean will equal the value of
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)

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