Review- Chapter 7 Statistics

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Sampling Distribution

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15 Terms

1

Sampling Distribution

  • answers the question: HOW would my summary statistic behave if I could REPEAT the process of collecting data using a random sample?

  • often approx. normal

  • resonable likely outcomes: fall WITHIN 2 SE of the mean

    • middle 95%

  • Describe: center, shape, spread

  • does not show how the sample is distributed around the sample mean

  • a distribution of sample means, not individual values of sample mean s

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2

Rare event

  • lie in the outer 5% of a sampling distrubtion

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3

Unbiased estimator

  • mean of a sampling distribution=population parameter →”unbiased estimator of the parameter”

  • center is accurate

  • mound shape, not skew

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4

Good estimator

  • unbiased and low variability

<ul><li><p><mark data-color="green">unbiased</mark> and <mark data-color="red">low</mark> <strong>variability</strong> </p></li></ul>
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5

Standard Error (SE)

  • the standard deviation of a sampling distribution

  • increases for samples similar to the population

  • decreases as n increases because n and o are inversely proportional σ=√(p(1-p))/n, p̂ being the sample proportion

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6

STEPS-approximate or simulated sampling distribution

  1. Take a random sample of a FIXED size(n) from a population

  2. compute a summary statistic

  3. repeat steps (1, 2) many times

  4. display the distribution of the summary statistic

*notation

<ol><li><p>Take a <strong>random sample</strong> of a FIXED size(n) from a population </p></li><li><p>compute a summary statistic </p></li><li><p>repeat steps (1, 2) many times</p></li><li><p>display the <strong>distribution</strong> of the summary statistic </p></li></ol><p>*notation </p><p></p>
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7
<p>Sampling distribution of the <strong>sample mean </strong></p>

Sampling distribution of the sample mean

  • For ANY sample size(b), the sample mean is an unbiased estimator of the population mean

  • the distribution of sample means becomes less spread as the sample size increases

  • If a random sample of size n is selected from a distribution u and σ…

    • u=u

    • σ=σ/n

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8

Central Limit Theorem(CLT)

  • Sampling distribution becomes MORE normal as the sample size gets larger

  • The sampling distribution of mean is normal if conditions are met, even if the population shape is not normal or unknown

  • Determines if outcomes are reasonably likely or not

  • Implications

    • larger sample size(n) →narrower graph, more normal shape, less spread

    • the population can be ANY shape if n>= 30 → use a sample to model using approximately normal distribution

Conditions for mean

  1. Independence assumption

  2. randomization condition

  3. 10% condition- the sample size is not more than 10% of the population

  4. Large enough sample condition(n>30)

Conditions for proportion

*all same expect 4

  1. large enough sample condition

  • np>= 10 →at least 10 success

  • n(1-p)>=10 →at least 10 failures

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9

Sampling distribution of the SUM of a sample mean

  • If a random sample of size n is selected from a distribution u and σ…

  • usum=nu

  • σ sum=√nσ

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10

Graphing - Sampling distribution of means

  • Larger sample

    → more mound-shaped and “normal”

    x axis less spread out

    • Max: skew left

    • min: skew right

    • median: narrow

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11
<p>Sampling distribution of a sample proportion </p>

Sampling distribution of a sample proportion

  • for any sample size(n), sample proportion=unbiased estimator for the population parameter

  • distribution of sample proportions, less spread out as n increase

  • CLT

  • further p is from 0.5→ larger n required to achieve a normal approximation

*binomial experiment

  • u=p, p being the population proportion

  • σ=√(p(1-p))/n, p̂ being the sample proportion

  • for any size n, u=unbiased estimator of p̂

  • increasing n→ reduce variability and bias not related

<ul><li><p>for any sample size(n), <strong>sample proportion</strong><mark data-color="green">=</mark>unbiased estimator for the <strong>population parameter</strong></p></li><li><p>distribution of sample proportions, <span style="color: red">less</span> spread out as n <span style="color: green">increase</span></p></li><li><p>CLT</p></li><li><p>further p is from 0.5→ <span style="color: green">larger</span> n required to achieve a <strong>normal approximation</strong></p></li></ul><p>*binomial experiment</p><ul><li><p>u<sub>p̂</sub>=p, p being the population proportion</p></li><li><p>σ<sub>p̂ </sub>=√(p(1-p))/n, p̂ being the sample proportion</p></li><li><p>for any size n, u<sub>p̂ </sub><mark data-color="green">=</mark>unbiased estimator of p̂</p></li><li><p>increasing n→ <span style="color: red">reduce</span> variability and bias not related </p></li></ul>
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12

Sampling distribution of the SUM of a sample proportion

  • usum=np

  • σ sum=√np(1-p) ← standard error

  • σ sum=np(1-p)← standard deviation

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13

P

probability

  • can draw a graph to show a middle center point for the sample proportion mean

  • always symmetrical if p=0.5

  • always unimodal(bc a binomial distribution)

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14

n

Sample size

  • larger sample size(n)

    smaller spread in the sampling distribution

    → more it will show the distribution traits of the WHOLE population→ more like population graph

    → can or may not be more normal, depending on pop. graph

<p>Sample size</p><ul><li><p><span style="color: green">larger</span> sample size(n) </p><p>→<span style="color: red">smaller</span> spread in the sampling distribution</p><p>→ more it will show the d<strong>istribution traits </strong>of the WHOLE population→ more like population graph </p><p>→  can or may <span style="color: red">not</span> be more normal, depending on pop. graph </p><p></p><p></p></li></ul>
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15

formula x=

x=u+-Zσ

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