BUSOBA 2320: W2: Sampling Distributions: (X-Bar, P-Hat) (Rec Notes)

REC NOTES

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Parameters

• describe populations (center, spread, skewness, etc.),

• are usually symbolized by letters from the Greek alphabet, like μ and σ

Statistics

• describe samples (center, spread, skewness, etc.),

• are usually symbolized by letters from the Roman alphabet, like 𝑋̅ , s, 𝑝

Samples should be selected using a

• probabilistic method to ensure independence between the sample

  observations and unbiased representation of the population.

  • We require (Simple Random Sample) SRS’s.

Simple Random Sample SRS

(each item in the population has an equal chance of selection and each combination of n items in the population has equal chance of selection).

n =

# of items in the sample, or sample size

𝜇𝑋̅ =

𝜇

𝜇𝑝̂ =

𝑝

𝜎𝑋̅ = 𝜎/√𝑛 and 𝜎𝑝̂ = √𝑝(1−𝑝)/𝑛 , when

sampling with replacement or sampling from a population of infinite size

Use correction (√𝑁−𝑛/𝑁−1), when

sampling more than 10% of a finite population without

replacement.

the correction (√𝑁−𝑛/𝑁−1) will be

close to 1 if the sample size is small relative to the population size.

we should always check to determine if

n ≤ 0.10N to make sure that the correction is not important.

N =

size of population

n = (2ND DEF)

size of sample

The sampling distribution (probability distribution) for 𝑋̅ depends on

the population that samples are being drawn from

if the population is Normal,

𝑋̅ ~ Normal, regardless of the sample size (even n = 1)

if the population is not Normal but n is sufficiently large,

𝑋̅ ~ approximately Normal by the Central Limit Theorem (CLT)

if the population is not Normal and n is small,

the distribution of 𝑋̅ will depend on the population and the sampling scheme

As n approaches infinity,

sampling distribution of x-bar converges to normal distribution. (Central Limit Theorem)

The sampling distribution, i.e., probability distribution, for 𝑝̂ depends on

n and p

𝑝̂ truly follows the

Binomial probability distribution

If np and n(1 – p) are both ≥ 10, the distribution of 𝑝̂ will be approximated well by

the Normal probability distribution.

The required expected number of successes (np) and expected number of failures (nq) is

• a rule of thumb and may be seen in other references as 5 rather than 10.

• The larger both values are, the better the Normal approximation will be. We will follow the rule of 10, because that is used in the Sharpe textbook.

More

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Infinite (Google):

limitless or endless 

Finite (Google):

having limits or bounds

Sampling Distribution (Khan):

shows every possible result a statistic can take in every possible sample from a population and how often each result happens - and can help us use samples to make predictions about the chance tht something will occur.

Probability Distribution (Prev Class):

The list of all possible outcomes and the probability of each outcome occurring.

Binomial Probability Distribution (Prev Class):

Results from a procedure that meets the following: 2 possible outcomes (success/failure), probability of success is a constant (p), trials are independent, and fixed number of trials.