Sampling Distributions
Reese’s Pieces candies come in three colors: orange, brown, and yellow. Which color do you think has more candies in a package: orange, brown or yellow? I am asking you toguess,based on this picture, or on your own experience.
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From my own experience I think that I pull out a lot of brown M&M’s. If I had to guess based off this picture I would choose orange because it seems like it makes up about 50% percent of M&M’s shown.
Math 243
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Guess the proportion of each color in a bag. Give your answers as percentages:
Orange______48%______ Brown______30%______ Yellow_____22%________
Let’s think about theorange candies. Suppose we had an unimaginably large bag of Reese’s Pieces. If each student in our class took a sample of 25 Reese’s Pieces, would you expect every student to have the same number oforange candies in their sample? Explain.
I would not expect every student to have the same, however I would expect each student to have a very similar amount as this is a random distribution.
Let’s say that 10 students each took a sample of 25 Reese’s Pieces. Write down the number of orange candies you might expect in each of these 10 samples, based on your answer to question 2:
____12____ ___11_____ ___11_____ ____13____ ____12____ ____12____ ____11____ ______12__ ____13____ ____12____
Of course, the number of orange candies in each sample shouldn’t be the same. These numbers represent theexpectedvariability you would expect to see in the number of orange candies in 10 samples of 25 pieces each.
Write your answers to the previous problem asproportions—either as decimals or percentages:
____5.7XXXXX ____5.76____ XXXXX.XXXXXX XXXXX.XXXXXX ____5.XXXXXX XXXXX.XXXXXX ___5.XXXXXXX ____5.76____ XXXXX.76____ XXXXX.XXXXXX
The proportions in XXXXXXXX 5 are calledsample XXXXXXXXXX.XXXX is a generic term – aXXXXXX XXXXXXXXX XX anything we use XX summarize a XXXXXX. In XXXX XXXX, we XXXXX XXXX call XXXXXsample proportions, because we computed aXXXXXXXXXX XX XXXXXXXXX each sample.
XX XXX XXX termpopulation parameter XX XXXXX to XXX XXXXX XXXX summarizes the XXXXX XXXXXXXXXX. XX XXXX XXXX, we are XXXXXXXXX XXXXXXXXXXXXX proportionof XXXXXX XXXXX’s Pieces.
For XXX purposes XX the XXXX five questions, XXXXXXX as XXXXXX you and all the XXXXX XXXXXXXX in the classactually drew a sample of XX candies each.
XXXXXXXX:
Thepopulation proportion XX usually denoted XX a lower-casep.
Thesample proportion is XXXXXXX XXXXXXX by a XXXXX-XXXX p with a “hat” on it:p̂.XX say “p-hat”.
Do XX know the value of XXXXXXXXXXXXX XXXXXXXXXp (thepopulation XXXXXXXXXX
No
Do XX know the values XX XXXsample XXXXXXXXXXp̂’s (XXXXXXXXX proportions XX orange XXXXXXX in XXXX XXXXXXX’s sample XX n = XX candies)?
No
XXX XXXXXX XXXXXXXXXX XX what we believe XXX amount to XX, XXXXXXX, the population XXXXXXXXX XX what XXX XX XXXXXXX of XXX whole population entity.
Does the value XX XXX XXXXXXXXXX XXXXXXXXXXXXXXXNo, XX XXXX summarizes the XXXXX XXXXXXXXXX.
Does XXX value of XXX sample statisticXXXXXXXXX, XX the XXXXXX XX pieces left decreases, which changes XXX denominator of XXX fraction.
Instead, we will simulate the activity XXXXX a web applet. XX XX
http://www.XXXXXXXXXXXXX.com/applets/XXXXXXX/OneProp.XXX?candy=1
XXX will see a cartoon container XX XXXXXXX XXXXXXX: this XXXXXXXXXX the POPULATION.
There are 25 XXXXXXX in the XXXXXX, XXX probability of XXXXXX XX X.X, so XX can assume there is XXXX likely to XX XXXXXX 12 or 13 orange pieces.
You will see XXXX XXX XXXXXXXXXXX of XXXXXX XX set at 0.5, so this XX thepopulation XXXXXXXXX,which in this XXXX is theXXXXXXXXXX proportionp. (People who have counted XXXX XX Reese’s XXXXXX came XX XXXX XXXX XXXXXX, for XXXXXXX,XXXX XXX).
XXX XXXX 0.5 XXXXXXX to XXX proportion XX orange candies in XXXX XXXXX in question 2? XXXX you close?
Was very close, as I XXXXXXXXX X.48.
XX XXX left XXXX XX the Reese’s Pieces web XXXX, change “XXXXXX of XXXXXX” XX “Proportion XX XXXXXX”:
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XXXXX on XXX “Draw Samples” XXXXXX. One sample XX XX candies XXXX be XXXXX and XXX XXXXXXXXXX XX orange candies in this sample XX displayed in XXXX (under the candy XXXXXXX, it XXXX XXX XXXXXXXXX like, XXX XXXXXXX,p̂ = 0.XXX,depending on your sample XXX XXXXXX XXXXXXXXXX), and also XXXXXXX XX XXX graph.
Repeat XXXX two or three more XXXXX.
XX you XXX XXXXXXX orXXXXXXXXX
I XXX XXXXXXXXX, XXXXXXX I got XX then XX, as it XX XXXXXX, there is a probability XXXX XXX XX are orange, XXXX a XXXX slim XXXXXX. So any XXXXXX between 0 – XX XXXXX XXXX XX, however XXX XXXX XXXXXXX XX 12,13 are most likely.
In general, about how XXXXX is XXXXsample XXXXXXXXX (sample proportion,p̂) XX theXXXXXXXXXX XXXXXXXXX(population proportion,pIn XXXXXXX it XX close, XXXXXXX it XXX never XX exactly 0.5 as you cannot XXXX 12.5 XXXXXX, overall XXXXXX it XXXX follow a XXXXXXXX XXXXXXXXXXXX XXXXXXX.
Why XXXX’t the XXXXXX XX the XXXXXX XXXXXXXXXXX XXXXXXX as XXX XXXXXXXXXX XXXXXXXXXX (X.5)?
As XXXXXXXXX previously
XXXXXXX XXX “XXXXXXX”box XXX change “XXXXXX XX samples” to 100.
XXXXX the “XXXX XXXXXXX” XXXXXX, and turn your XXXXXXXXX to the XXXXXXXXX XXXXXXXXXXXX XX XXXXXX XXXXXXXXXXX. (This is the XXXXX with the dots below XXX XXXXX XXXXXXX.)
This XXXXXXXXXXXX is calledThe XXXXXXXX XXXXXXXXXXXX XXp̂.
How is this XXXXXXXXXXXXXXXXXXXXX from XXX distributions XX XXXX worked XXXX so XXX?
This XXXXXXXXXXXX differs XXXXXXX XX have XXXX XXXXXXXX XXXXXXX to XXXXXXX XXXX behaviour occurs.
Repeat XXX “Draw Samples” XXXXXXXX a couple XXXXX, XXXXX you start to XXX XXX general characteristics XX the distribution. XXXXXXXX XXXshape, XXXXXX, XXX spread XX this XXXXXXXXXXXX.
XXXXX XX a peak in the middle, where 12-14 XX, XXXX is XXXXXXX XXXX XXXXXX XX the most common number, and there is a decrease XXXXXXXXXXXXX XXXXXX XXXX.
XXX XX XXXX to XXXX: what happens XX XXXX XXXXXXXXXXXX of sample XXXXXXXXXX XX we change XXX number of candies in each sample (sample XXXX)?
First, XXXXXX XXX sample XXXX (“XXXXXX XX candies”) XX XX and XXXX XXX XXXXXXX.
How XXXXX is eachsample XXXXXXXXX XX theXXXXXXXXXX XXXXXXXXXIn this XXXXXXXX it gets much XXXXXX as it XX XXXXXXXX to reach X, (XX*X.5) XXX there XX XXXX chance XXX
Describe XXXXXXXX, XXXXXX, XXX spreadof this distribution XX XXXXXX proportions XXXX n = 10, particularly XXX XXXXXXXXXXXX XXX differences XXXX the XXXXXXXX XXXXXXXXXXXX XX XXX XXXX n = XX.
XXX shape XX XXXX XXXXXX, as XXXXX XXX only XX XXXXXXXXX XXXXXXX XXXXX, so we don’t get XX good of a representation, when we had 25, the bell shaped curve XXX XXXX XXXX clear in XXXXXXXXXX
Next, XXXXXX XXX XXXXXX XXXX to XXX XXX draw 100 XXXXXXX.
XXX close XX eachsample XXXXXXXXX to XXXpopulation XXXXXXXXX?
XXXXXXXX XXXXXXXX, XXXXXX, XXX XXXXXXof XXX distribution of sample proportions with n = 100, XXXXXXXXXXXX XXX similarities XXX XXXXXXXXXXX with the sampling distributions we XXX XXXX n = 10 XXX n = 25.
There is a XXXX XXXX XXXXXXX XXXXX XXXX XXX XX XX XXXX a XXXX more XXXXXXX bell XXXXX, XXXXX XXXX XXXX it XXXXXX XXXX XXXX more with an infinite sample
XXX XXX table XX XXXXXXXX and XXXXXXXXX XXXX XXXXXXX:
Sample Size
|
XXXXX XX XXXXXXXX Distribution
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Center XX XXXXXXXX Distribution
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Spread XX Sampling XXXXXXXXXXXX(you XXX XXXX XXX XXXXX XXXX, XX in “XXXX XXXXX X.24 to XXXXX X.XX”)
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n = 10
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Very XXXXXXXX
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XX
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XX-16
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n = XX
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XXXX XX a central XXXXXXX peak
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12
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XX-XX
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n = XXX
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XXXX XXXX bell XXXXXX
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XX
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3-XX
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How well do the XXXXXX statistics resemble XXX population parameter? Does it depend XX the sample XXXX?
XX does XXXX much depend XX XXXXXX size, XXX larger the sample size, the XXXXXX you XXX XX XXX two being XXXXX.
XXXX XX XXX XXXXXX XX sample XXXX on theXXXXXthe shape becomes XXXX XXX more of a bell XXXXX
What is the XXXXXX XX sample size on XXXXXXXXXThe XXXXXX XXXX always XXX closer to 12-XX as XXX sample XXXX increases
XXXX is the effect XX XXXXXX size XX XXXspreadXXX’s XXXXX about XXXXXXX polling. If we want to XXXX: how a XXXXX XX XXXXXX might vote, how a XXXXX of XXXXXXXXX XXXXX XXXXX a XXXXXXX or XXXXXXX, how a XXXXX of XXXXXXXX XXXXX about a class, XX might devise a question, select a XXXXXX, conduct a survey, XXX compute a XXXXXX XXXXXXXXXXp̂.
XXX XXXX the XXXXXXXX XXXXXXXXXXXX ofp̂XXXX us XX understand the XXXXXXXXXXXX between XXX sample XXX the population? XXX XXXXXXX XXX, how XXXX XXX XXXXXXXX distribution help XX XX determine how XXXX our XXXXXX XXXXXX statisticp̂ reflects the true XXXXXXXXXX XXXXXXXXXp
XX XXXX to XXXX a sample XXXX is well represented in all XXXXXXXX, XXXX XX XXXXXXXXXX XXXX XXXX XXXXXX who XXXX in XXXXXXXXX XXXXX, different incomes XXX…. XXXXXXXXX this way we XXX XXX as close as possible XX the XXXX population parameter.
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