probability and the normal curve
TRANSCRIPT
Probability and the Normal Curve
• Common Probability Applications• Cards: chance of getting card or suit• Roulette, slot machine, PowerBall• Probability is known, we predict the sample
Population: All Possible Outcomes
What is the probability of throwinga “2” with a four-sided die?
“2” is ONE of FOUR possible
outcomes.
1 / 4 = 0.25 = 25%
What’s the probability of throwing a “4” with two four-sided dice?
1 2 3 4
1 2 3 4 5
2 3 4 5 6
3 4 5 6 7
4 5 6 7 8
“4” occurs in THREE out of
SIXTEEN possible outcomes.
3 / 16 = 0.1875 = 18.75%
When I know the frequency distribution, I can
compute a probability, even if it is not evenly
distributed as dice are. p(>4) = 2/10=0.20=20%
The exact shape of the normal distribution is specified by an equation relating each X value (score) with each Y value (frequency). The equation is shown above. ( and e are mathematical constants.)
The normal distribution is symmetrical with one mode. The frequency tapers off as you move farther from the middle in either direction.
22 2/)(
22
1
XeY
The frequency of scores in a particular range are
given by the areas of the Normal Curve.
No matter what the μ or σ, the proportions in the
areas are always the same.
If we want to know how many people scored 80
or more, when μ=68 and σ=6, we can compute
the z-score for 80, then find the area of the curve
above (further from the mean) than it.
2.28%
Practice Problems #1
Cassidy Jayne p(z > 1.25)
Heather Carlson p(z < -.50)
Hunter Bergerson p(z > 1.60)
Jesse Trutwin p(z < -1.30)
Kassidy Birdsall p(z > .75)
Shayna Schafter p(z < -1.65)
Tyler Bruggeman p(z > 1.96)
Practice Problems #2
Cassidy Jayne p(.25 < z < .60)
Heather Carlson p(-1.00 < z < 1.20)
Hunter Bergerson p(1.00 < z < 1.95)
Jesse Trutwin p(-1.00 < z < 1.25)
Kassidy Birdsall p(1.35 < z < 2.55)
Shayna Schafter p(-.70 < z < 2.05)
Tyler Bruggeman p(.35 < z < 2.10)
• Inferential Statistics Applications
– We know the characteristics of a sample
– We don’t know the parameters of the population
– We want to infer the population’s characteristics
– If we know the behavior of many samples, we
can use that to estimate the population
Sample
• Two dice are thrown where I cannot see them. If I am told they total “8” which is more likely to be the situation?
• The dice were 4-sided
• The dice were 6-sided
Sample• To answer, I need to ask two questions:
• What is the probability of “8” with 4-sided dice?
1 2 3 4
1 2 3 4 5
2 3 4 5 6
3 4 5 6 7
4 5 6 7 8
“8” occurs in ONE out of
SIXTEEN possible outcomes.
1 / 16 = 0.0625 = 6.25%
Sample• The other question I need to ask is
• What is the probability of “8” with 6-sided dice?
1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
“8” occurs in FIVE out
of THIRTY-SIX
possible outcomes.
5 / 36 = 0.1389 =
13.89%
Making an Inference
• If the dice were 4-sided, the chance of “8” is only about 6%
• If the dice were 6-sided, the chance of “8” ismore than double,almost 14%
• I think it is more likely (= I infer) that the dice were 6-sided dice. But I cannot be certain.
• Inferential statistics helps me make these decisions,and to know my chances of making an error.
• Week 7: Probability with a Normal Distribution
• Week 8: Distribution of Sample Means
• SCI 3777: The Logic of Testing a Hypothesis(inferring its accuracy)
