CHAPTER 5_Part A
Sampling Distributions For
Counts And Proportions
• COUNT STATISTIC AND THE BINOMIAL
PROBABILITY MODEL
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QUESTION
• WHAT DOES THE PREFIX BI MEAN?
• THE PREFIX BI MEANS “TWO.”
• WE WILL BE CONSIDERING RANDOM
EXPERIMENTS IN WHICH THERE ARE
ONLY TWO OUTCOMES.
• THE TWO OUTCOMES WILL BE
CALLED SUCCESS OR FAILURE
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BINOMIAL PROBABILITY MODELS
BERNOULLI TRIAL
A RANDOM EXPERIMENT WITH TWO
COMPLEMENTARY OUTCOMES, ONE CALLED
SUCCESS (S), AND THE OTHER CALLED
FAILURE (F), IS CALLED A BERNOULLI TRIAL.
P(SUCCESS) = p
P(FAILURE) = q = 1 - p
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EXAMPLES
• TOSSING A FAIR COIN 20 TIMES
SUCCESS = HEADS WITH p = 0.5 AND
FAILURE = TAILS WITH q = 1 – p = 0.5
NOTE
• EACH TOSS OF THE COIN IS A TRIAL
• LET THE RANDOM VARIABLE X BE THE
NUMBER OF SUCCESSES IN THE 20
TOSSES. IS X BERNOULLI?
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• PRODUCTS COMING OUT OF A PRODUCTION LINE
SUCCESS = DEFECTIVE ITEMS
FAILURE = NON-DEFECTIVE ITEMS
DEFINE Y THE NUMBER OF DEFECTIVE ITEMS
COMING OUT OF THE PRODUCTION LINE. IS Y
BERNOULLI?
• TAKING A MULTIPLE CHOICE EXAM UNPREPARED
SUCCESS = CORRECT ANSWER
FAILLURE = WRONG ANSWER
DEFINE Z THE NUMBER OF CORRECT ANSWERS
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CONDITIONS FOR A BINOMILA PROBABILITY
EXPERIMENT
(1) THE TRIALS MUST BE BERNOULLI, THAT IS, THE
RANDOM EXPERIMENT MUST HAVE TWO
COMPLEMENTARY OUTCOMES – SUCCESS OR
FAILURE;
(2) THE TRIALS MUST BE INDEPENDENT. THIS
MEANSTHE OUTCOME OF ONE TRIAL WILL NOT
AFFECT THE OUTCOME OF THE OTHER TRIAL.
(3) THE PROBABILITY OF SUCCESS IS THE SAME
FOR EACH TRIAL.
(4) THE NUMBER OF TRIALS IS FIXED OR THE
EXPERIMENT IF CONDUCTED A FIXED NUMBER OF
TIMES.
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REMARK ON INDEPENDENCE OF BERNOULLI
TRIALS
• THE 10% CONDITION
• BERNOULLI TRIALS MUST BE
INDEPENDENT. IF THAT ASSUMPTION IS
VIOLATED, IT IS STILL OKAY TO PROCEED
AS LONG AS THE SAMPLE IS SMALLER
THAN 10% OF THE POPULATION.
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EXAMPLES: DETERMINING WHETHER A RANDOM
VARIABLE IS BINOMIAL
• Determine which of the following are binomial
random variables. For those that are binomial, state
the two possible outcomes and specify which is a
success. Also state the values of n and p.
1. A fair coin is tossed ten times. Let X be the number
of times the coin lands heads
2. Five basketball players each attempt a free throw.
Let Y be the number of free throws made.
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ARE YOU INTERESTED IN THE NUMBER OF
SUCCESSES IN BERNOULLI TRIALS?
• QUESTION: WHAT IS THE NUMBER OF
SUCCESSES IN A SPECIFIED NUMBER OF
TRIALS?
• THE BINOMIAL PROBABILITY MODEL
ANSWERS THIS QUESTION, THAT IS, THE
PROBABILITY OF EXACTLY k SUCCESSES IN
n TRIALS.
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BINOMIAL PROBABILITY MODEL
• LET n = NUMBER OF TRIALS
p = PROBABILITY OF SUCCESS
q = PROBABILITY OF FAILURE
X = NUMBER OF SUCCESSESS IN n TRIALS
 n  k nk
P ( X  k )    p q
k 
n
n!
  
 k  k!( n  k )!
where,
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TI83/84 PLUS For Computing Binomial
Expressions
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Example 1
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Example 2
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Example 3
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Mean And Standard Deviation Of A Binomial
Variable X
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EXAMPLE 4
• AN OLYMPIC ARCHER IS ABLE TO HIT THE BULL’SEYE 80% OF THE TIME. ASSUME EACH SHOT IS
INDEPENDENT OF THE OTHERS. IF SHE SHOOTS 6
ARROWS, WHAT’S THE PROBABILITY THAT
• (1) SHE GETS EXACTLY 4 BULL’S-EYES? 0.246
• (2) SHE GETS AT LEAST 4 BULL’S-EYES? 0.901
• (3) SHE GETS AT MOST 4 BULL’S-EYES? 0.345
• (4) SHE MISSES THE BULL’S-EYE AT LEAST ONCE?
•
0.738
• (5) HOW MANY BULL’S-EYES DO YOU EXPECT HER
TO GET?
4.8 BULL’SEYES
• (6) WITH WHAT STANDARD DEVIATION? 0.98
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EXAMPLE 5 – Extra Credit Problem
• ASSUME THAT 13% OF PEOPLE ARE LEFT-HANDED.
IF WE SELECT 5 PEOPLE AT RANDOM, FIND THE
PROBABILITY OF EACH OUTCOME BELOW.
• (1) THERE ARE EXACTLY 3 LEFTIES IN THE GROUP.
• 0.0166
• (2) THERE ARE AT LEAST 3 LEFTIES IN THE GROUP.
• 0.0179
• (3) THERE ARE NO MORE THAN 3 LEFTIES IN THE
GROUP. 0.9987
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Normal Approximation of Binomial
Distribution
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Example 6
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Example 7 – (Self-read)
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How Large is “Large Enough”
•The Success/Failure Condition
– A Binomial is approximately Normal if we
expect at least 10 successes and 10
failures.
np  10, nq  10
– This comes from the binomial being
skewed for a small number of successes
or failures expected.
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How Large Is “Large Enough”
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How Large Is “Large Enough”
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