Statistics 200 Lecture #14 Thursday, October 6, 2016 - PowerPoint Presentation

Slide1

Statistics 200

Lecture #14 Thursday, October 6, 2016Textbook: Sections 8.4, 8.5, 8.6

• Recognize the four conditions for a binomial random variable• Calculate the mean and standard deviation for a binomial random variable• Use probability notation for continuous random variables and relate this notation to area under a density function.• Standardize any normal distribution and then use tables or a computer to find probabilities

Objectives:

Slide2

pick a card

from shuffled deck

Look for an

ace

repeat three

more times

Put

card back in the

deck;

reshuffle the deck

Example:

What do you notice?

X:

number of

aces

in 4 tries

Slide3

A binomial random variable is:

X = number of ___________ in n ________________ trials of a random circumstance in which p = probability of success is __________.Your job: Recognize when an experiment satisfies the 4 conditions for a binomial experiment. These 4 conditions are…

successes independent constantIn other words, X counts the successes in a __________________. binomial experiment

Slide4

Conditions for a binomial experiment

1

234There are n “trials”, where n is fixed and known in advanceWe can define two possible outcomes for each trial: “Success” (S) and “Failure” (F)

The outcomes are independent; no single outcome influences any other outcome

The probability of “Success” is the same for each trial. We use “p” to write P(Success).

Slide5

If X= # of aces in 4 tries, is X a binomial random variable?

Must confirm that our set-up satisfies all four conditions.

1Fixed # trials: __________234S / F outcomes: ____________Independent trials: _____P(success) constant: _____Conclusion:

Yes: n=4

Yes: S = ace

Yes

Yes Yes! n = 4, p = 4/52

Slide6

Slight Change: Is this still a Binomial?

pick a card

from

shuffled deck

Look for an

ace

repeat three

more times

Put card aside;

reshuffle the deck

X:

number of

aces

in 4 tries

Slide7

If X= # of aces in 4 tries, is X a binomial random variable?

Must confirm that our set-up satisfies all four conditions.1

Fixed # trials: __________234S / F outcomes: ____________Independent trials: _____P(success) constant: _____Conclusion:

Yes: n=4

Yes: S = ace

No

No NOT a binomial

Slide8

Review of confidence intervals

Suppose we take a sample of 1009 adults and ask them a yes or no question. Of them, 646 answer yes, and the rest answer no. What is the value of p-hat?

646 – 363646 / 363

1009 – 646

646 / 1009

1/

sqrt

(1009)

Slide9

Review of confidence intervals 2

Suppose we take a sample of 1009 adults and ask them a yes or no question. Of them, 646 answer yes, and the rest answer no. What is the value of the margin of error for a 95% confidence interval?

646 – 363646 / 363

1009 – 646

646 / 1009

1/

sqrt

(1009)

Slide10

• Initial

Survey Question:

How many cups of coffee, if any, do you drink on an average day?

Example:

Americans' Coffee Consumption Is Steady, Few Want to Cut Back

Coffee shops are reportedly the fastest-growing segment of the restaurant industry, yet the percentage of Americans who regularly drink coffee hasn't budged.

Sixty-four percent

of U.S. adults report drinking at least one cup of coffee on an average day, unchanged from 2012.

Results for this Gallup poll are based on telephone interviews conducted July 8-12, 2015, with a random sample of

1,009

adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia.

Slide11

11

Is the Gallup

polling process a binomial experiment? 1. n = 1009 trials 2. Success = “drink at least one cup of coffee a day” Failure = “don’t drink any coffee” 3. Independent trials: 4

.

p remains constant:

Conclusion

:

(is a binomial) (not a binomial

)Thus, if X = number who drink at least one cup of coffe a day in a sample of 1009 U.S. national adults, X is a binomial random variable. check!

check!close enough! (We don’t sample with replacement but the population is huge)

Slide12

Which binomial condition is not met?

• An

airplane flight is considered on time if it arrives within 15 minutes of its scheduled arrival time. At O’Hara Airport, in Chicago, 300 flights are scheduled to arrive on one day in January. X = number of the 300 flights that arrive on time. n = ________ success = ______ independent trials: _____ same probability for each trial: _____________ 300 on time arrival no!

probably not

Slide13

Which binomial condition is not met?

• A football team plays 12 games in its regular season where i

t is determined whether or not the team wins each game. X = number of games the team wins during the regular season of 12 games n = ________ success = ______ independent trials: _____ same probability for each trial: _____________ 12 winning game doubtful no!

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Which binomial condition is not met?

•

A woman buys a lottery ticket every week. She continues to buy tickets until she wins. Let X = number of tickets that she buys until she finally wins. n = ________ success = ______________ independent trials: _____ same probability for each trial: _____________ ??? winning lottery yes

yes

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Mean and standard deviation for binomial random variables

Mean:

Standard deviation:

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Final exampleConsider our 3-coin example with X = # heads. This is a binomial random variable

with n = ____, p = ______Thus the mean number of heads is _______ = ________The standard deviation is _______________ = _______________

3 0.5 3 × 0.5 0.866 1.5

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Continuous Random Variable

Assumes

a range of values covering an interval. _____________.

M

ay

be limited by instrument’s

accuracy / decimal points,

but still

continuous.

• We can’t find probabilities for exact outcomes.• For example: P(X = 2) = 0.

• Instead we can find probabilities for a range of values.

Find probabilities using a probability density function, which is a curve. Calculate probabilities by finding the area under the curve.is this area

Slide18

Recall – we calculate probabilities by finding area under the curve.

Probability density function

This is the density for a chi-square random variable.The density is larger for smaller values of X. To calculate a probability, we must find an __________. area

Slide19

Recall – we calculate probabilities by finding area under the curve.

Probability density function

Red area here is P(X>1.5)The shaded part has area _________The area under the entire curve is __________1.00.2207

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=

=

==

0.68

0.15

0.79

0.05

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An important probability

P(X=5) = ________A ______ has ___ area

Rule: P( X=a ) = ___ for any value a0lineno0

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Return to the normal distribution

Used Empirical Rule to make histogram

We’ve already seen the normal distribution MeanStandard deviationEmpirical rule

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Goal: Standardization

One Standard Normal Distribution

Limitless

number

of Normal Distributions

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Normal Distributions: Bell-Shape

(General) Normal:

General normal distribution:Standard normal distribution:

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How

to find Normal Probabilities

Use calculus – integrateRead normal probability tablesUse a probability calculatorMinitabinternethttp://davidmlane.com/normal.html

Total area under the curve = _____

1

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http://davidmlane.com/normal.html

Default Screen

Forward

Backwards

Probability Density Function

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What is

the probability that Z is:

greater than 1? at least 1? exactly 1?

Slide28

28

P(Z > 1) =

Forward

0.16

Slide29

Minitab: Graph> Probability Distribution Plots

Slide30

Minitab: (Density) Probability Distribution Plot

P(Z > 1) = 0.1587

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What is

the probability that Z is:

greater than 1? at least 1? exactly 1?

0.16

0.16

0

Slide32

Standard normal example:

32

Z is a standard normal

random variable.

Which is the entirely correct

picture for: P(Z > 1.72)?

Slide33

Clicker Question of understanding:

Z is a standard normal random variable.

Consider the probability statement P(-1.5 < Z < 2.0). Which is the only possible probability for this statement?0.6800 B. 0.9104 C. 0.9500 D. 0.9654 E 0.9970

Slide34

How to relate all this to Z-scores

We can standardize values from any normal distribution to relation them to the standard normal distribution.

Slide35

Z-score 1:

Z-score 2:

Thus,P(7<X<10) = P( __ < Z < ___ )-10

Slide36

If you understand today’s lecture…8.11, 8.41, 8.43, 8.47, 8.49, 8.63, 8.65, 8.67, 8.71, 8.81a, 8.83, 8.85 (for the normal distribution problems, sketch a picture!)

Objectives:

• Recognize the four conditions for a binomial random variable• Calculate the mean and standard deviation for a binomial random variable• Use probability notation for continuous random variables and relate this notation to area under a density function.• Standardize any normal distribution and then use tables or a computer to find probabilities