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7/28/2019 Section2.2classnotm es

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Section 2.2:

What do samples tell us?


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Should gambling be legal?

What were the major ideas from this activity?

Finding a proportion for the TRUE

population (for everyone or everything)is almost impossible to do.

The more samples you take, the more

reliable (less variable) your data is.

The more samples you take, the better it

predicts the true population proportion!


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Several statisticians will use a fact about a sample toestimate the truth about the whole population.

However, in order to do this, its important to

understand whether a number describes a sample or

a population. The analogy below is used by many to

remember this vocabulary:

Parameter is to populationas statistic is to sample.


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1. Parameter:p

This is a number that describes thepopulation.

The parameter is a fixed number (inpractice we dont know its value, but wetry to estimate it using the outcomes ofour samples).

This is theproportion of yourpopulation.


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2. Statistic: p-hat

This is a number that describes the sample.

The value of a statistic is known when we

have taken a sample, but it can changefrom sample to sample.

It is the outcome(s) of taking samples. It istheproportion of your samples.

We use a statistic to estimate an unknownparameter.


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Example #1

A random sample of 1000 people whosigned a card saying they intended toquit smoking on November 20, 1995

(the day of the Great American Smoke-out), were contacted in June 1996. Itturned out that 210 (21%) of thesampled individuals had not smokedover the past six months. Specify the

population of interest, the parameter ofinterest, the sample, and the samplestatistics in this problem.


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Example #2On Tuesday, the bottles of tomato ketchup filled in a

plant were supposed to contain an average of 14ounces of ketchup. Quality control inspectors

samples 50 bottles at random from the days

production. These bottles contained an average of

13.8 ounces of ketchup. State the value of theparameter and the value of the statistic.


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Example #4

A recent report in the journal Nature examined whetherducks keep an eye out for predators while they sleep. Theresearchers, from Indiana State University, put four ducksin each of the four plastic boxes, which were arranged in a

row. Ducks in the two end boxes slept with one eye open31.8% of the time, compared to only 12.4% of the time forthe ducks in the two center boxes.

(a) State the values of the parameter and state the value of thestatistic.

(a) Is this an example of an observational study or a comparativeexperiment? Explain briefly.


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As you noticed with the gambling activity

(whether gambling should be legalized), the valueof the statistic,p-hat, will vary from sample tosample. Random samples eliminate bias from theact of choosing a sample, but they can still

estimate a population proportion badly becauseof the variability(how scattered the data is) thatresults when we choose at random.

If the variation when we take repeated samplesfrom the same population is too great, we cant

trust the results of any one sample!


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A couple of things to keep in

mindback to the gambling

discussion! Larger random samples have less variability (not as

scattered) than smaller samples. This is why you collected10 samples for the gambling activity. What would have

happened if you only found one sample and the proportionwas 10%?

While you were collecting samples, the proportion

p-hatwas sometimes higher than 0.6, but sometimes lowerthan 0.6. Due to the fact that thep-hatvalues you foundwere not always too high or too low,p-hat(the statistic)has no bias as an estimator ofp(the populationsparameter). This is true for both large and small samples.


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Two types of error in estimation:

Bias vs. Variability


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1. Bias

Bias occurs when there is consistent,repeated deviation of the samplestatistics from the population parameter

in the same direction when many samplesare taken (allp-hatvalues are understatedor overstated).

In order to reduce bias: use random

sampling (some statistics will be higherand some lower), so no repeateddeviations occur.


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2. Variability

Variability describes how spread out thevalues of the sample statistics are whenmany samples are taken (large variability

means that the result of sampling is notrepeatableits not good!).

In order to reduce variability: use a larger

sample (or many samples and take theaverage). Also, check that the instrumentused is valid!


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What you want

SMALL BIAS!

SMALL VARIABILITY!


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Visually representing bias and

variability using targets.

Bias means the archer systematically misses in the same direction.

Variabilitymeans that the arrows are scattered.


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Create your own data set of

6-values that holds true foreach set of conditions:

Low Variability

and Low Bias

High Variability

and Low Bias

Low Variability

and High Bias

High Variability

and High Bias

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