Transcript Confidence Intervals

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GrowingKnowing.com © 2011
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Binomial probabilities
 Your choice is between success and failure
 You toss a coin and want it to come up tails
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Tails is success, heads is failure
 Although you have only 2 conditions: success or failure, it
does not mean you are restricted to 2 events
 Example: Success is more than a million dollars before I’m 30
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Clearly there are many amounts of money over 1 million that would
qualify as success
 Success could be a negative event if that is what you want

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Success for a student is find an error in the professor’s calculations
If I am looking for errors, then I defined “success” as any event in
which I find an error.
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Conditions for binomials
 The outcome must be success or failure
 The probability of the event must be the same in every trial
 The outcome of one trial does not affect another trial.
 In other words, trials are independent
 If we take a coin toss, and you want tails for success.
 Success is tails, failure is heads
 Probability on every coin toss is 50% chance of tails
 It does not matter if a previous coin toss was heads or tails,
chance of tails is still 50% for the next toss. Independent.
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Don’t forget zero
 Would you like to clean my car or clean my shoes?
 Don’t forget zero as an option
 There are 3 possible outcomes: clean car, shoes, or nothing.
 If I toss a coin 3 times, what is the sample space?
 A sample space lists all the possible outcomes
 You could get tails on every toss of 3 (TTT).
 You could get tails twice and heads once (TTH)
 You could get tails once, and heads twice (THH)
 Do not forget you may get tails zero in 3 tries. (HHH)
 So the sample space is 3T 2T, 1T, and always include 0T
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Formula
P(x) = nCxpx(1-p)(n-x)
 n is the number of trials.
 How often you tried to find a success event
 x is the number of successes you want
 p is the probability of success in each trial
 Remember that nCx is the combination formula
 many calculators give nCx with the push of a button
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How to calculate
 Let’s use an example to demonstrate.
 You are taking a multiple choice quiz with 4 questions.
You want to know if you guess every question, what’s
probability you guess 3 questions correctly. There are
4 choices for each question of which one is correct.
 Probability (p) to guess a question correctly is ¼ = .25
 n is 4 because we have 4 trials. (questions on the quiz)
 x is 3, you are asked the probability of guessing 3
successfully.
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 Formulas is nCxpx(1-p)(n-x)
 From the last example: n=4, p=.25, x = 3
 x=0 4C0p0(1-p)(4-0) = 1 (.250 (1-.25)4 = 1(.75) 4
 x=1
1(1-p)(4-1) = 4(.251 (1-.25)4-1 = 1(.75) 3
C
p
4 1
= .31640625
= .421875
 x=2 4C2p2(1-p)(4-2) = 6(.252 (.75)4-2 =6(.0625(.5625) = .2109375
 x=3 4C3p3(1-p)(4-3) = 4(.253 (.75)1 = .0625(.75) 3 = .046875
 x=4 4C4p4(1-p)(4-4) = 1(.254 (.75)0 = .003906(1) = .003906
 Probability of guessing 3 successfully (x=3) is .046875
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Sample questions
 Let’s use the findings from the example to examine Calculations from
popular binomial questions.
 Exact number of successes
 What’s probability of guessing 3 questions
correctly?

x=3, p = .047
 What’s probability of guessing 2 questions
the example :
x=0, p = .316
x=1, p = .422
x=2, p = .211
x=3, p = .047
x=4, p = .004
correctly?
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X=2, p = .211
 What’s probability of guessing 0 questions
correctly?
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X=0, p = .316
So we have 32% chance we’d guess no questions right
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 Less
 What’s probability of guessing less than 2
questions correctly?
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Add x=0 + x=1 so (.316 + .422) = .738
 What’s probability of guessing 2 or less
questions correctly?
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x=0 + x=1 + x=2 (.316 + .422 + .211) = .949
 What’s probability of guessing less than 1
Calculation from
the example :
x=0, p = .316
x=1, p = .422
x=2, p = .211
x=3, p = .047
x=4, p = .004
question correctly?

X=0, p = .316
 Notice what is included and what is excluded.
 Guessing “2 or less” we include x = 2.
 Guessing “less than 2” we exclude x = 2.
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 More
Calculation from
 What’s probability of guessing more than 2 the example :
x=0, p = .316
questions correctly?
x=1, p = .422
 Add x=3+ x=4 so (.047 + .004) = .051
x=2, p = .211
 What’s probability of guessing 2 or more
x=3, p = .047
questions correctly?
x=4, p = .004
 x=2 + x=3 + x=4 (.211 + .047 + .004) = .262
 Notice what is included and what is
excluded.
 Guessing “2 or more” we include x = 2.
 Guessing “ more than 2” we exclude x = 2.
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 More
 What’s probability of guessing at least 1
question correctly?
Calculation from
the example :
x=0, p = .316
x=1, p = .422
 Add x=1 + x=2 + x=3 + x=4
x=2, p = .211
 Note: ‘at least’ is a more-than question
x=3, p = .047
 some students confuse ‘at least’ with ‘less-than’
x=4, p = .004
 You can always save time with the
complement rule
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Calculate x for the small group, and if you
subtract that probability by 1, you will get the
rest of the grouping for x.
1-.316 =.684 is probability x=1 to 4
Double-check .422+.211+.047+.004 = .684
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 Between
 What’s probability of guessing between 2
and 4 (inclusive) questions correctly?
Calculation from
the example :
x=0, p = .316
x=1, p = .422
 We are told to include x=4
x=2, p = .211
 W want x=2, 3 and 4 so .211+ +.047 + .004 = .262
x=3, p = .047
 What’s probability of guessing between 1
x=4, p = .004
and 4 question correctly?
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Assuming x=4 is inclusive, we want x=1+2+3+4
x between 1 and 4 = .422+.211+.047+.004 = .684
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 You need to practice as many of the ways of asking for
binomials can be confusing until you’ve done a few
 Examples
 At least 4,
 Not less than 4
 Greater than 4
 None
 No more than 2
 Binomials can take a long time with high numbers of trials,
so use the complement rule to avoid excessive work.
 If trials are 30, probability is .5 per trial, what is the
probability of less than 29 successes?
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Calculate x=30 and 29, add them, then subtract 1 to get x=0 to 28
Or, in 3 hours calculate each value from x=0 to x=28 and add them.
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