Exam 2 - Probabilities and sampling distribution Flashcards Preview

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Flashcards in Exam 2 - Probabilities and sampling distribution Deck (53):
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Random phenomenon

A phenomenon where the outcome of one play is unpredictable, but the outcomes from many plays form a distribution

1

Probability of an outcome is the

Proportion of times that an outcome occurs in many, many repetitions(plays) of the random phenomenon.

2

In single random phenomenon the outcome is

Uncertain

Will the next flight to NY leave on time?

3

In many, many repetitions the proportion of specific outcomes is

Predictable

What proportion of flights to NY leave on time?

4

Randomness does NOT mean

Haphazard (disorganization)

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SRS imposes .... chance of selection for each individual in the population

Equal

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Sample space in probability is

The list of all possible outcomes of a random phenomen

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Event in probability is a

Single outcome or a subset of outcomes from the sample space

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Probability model is a

Mathematical description of a random phenomenon consisting of a sample space and a way of assigning probabilities to events.

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Probability explains only what happens in the .... run

Long

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If all probabilities are EQUALLY LIKELY, we need to count:
1.
2.
And that would be our probability

Count of outcomes in event of interest /
Over
Count of outcomes in sample space

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Probability rule 1

Probability must be a number

Between 0 and 1

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Probability rule 2:

The sum of probabilities from all

Possible outcomes must equal 1

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Probability rule 3

If two events cannot occur simultaneously, ...

The probability either one or the other occurs equals the sum of their probabilities

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Probability rule 4:

The probability that an event does not occur equals

1 minus the probability that the event does occur

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Disjoint Events

Two events that have no outcomes in common and, thus cannot both occur simultaneously.

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"Playing the game" or simulation means

Looking at the phenomena many many times

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Census

An examination of entire population

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Census is time consuming, very expensive and often impractical. What is the alternative?

1. Select SRS from population and compute x-bar(mean)

2. Make inference -

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Parameter

Values that represent whole population. In statistical practice, the value is not known because we cannot examine the entire population.
Mean (mu), sigma and Proportion P

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Parameter - mean

Mu - mean number of cigarets smoked per day by ALL teenagers

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Parameter of population P

Proportion of ALL teenagers who used tobacco in the last 30 days

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Statistic (think real world)

Values that come from a SAMPLE, statistics estimate parameters.

X-bar -sample mean
P-hat - sample proportion

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Sample mean

X-bar
Mean number of cigarettes smoked per day in a SAMPLE of teenagers

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Sample proportion

P- hat proportion of a SAMPLE of teenagers who used tobacco in the last 30 days

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In inference we use .... to estimate ..,

Statistics to estimate parameter

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Statistics
Mean -
Proportion -
Standard deviation -

X- bar
P- hat
S

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Parameter
Mean
Proportion
Standard deviation

Mu
P
Sigma

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What is statistical estimation?

Using sample statistics to estimate population parameter value

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Parameter is the result summarized from the

Entire population

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Statistic is any number result summarized from the

Sample

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If the response variable is quantitative we analyze ...

Mean x-bar

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If the response variable is categorical we analyze ...

Proportion p

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Law of Large Numbers

IF .....
Then ...

Draw observations at random from any population with finite mean mu. As the number of observations drawn increases, the mean x- bar of the observed values gets closer and closer to the mean mu of the population

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The larger the sample size, the .... the sample mean is to the population mean

Closer

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Sample statistic facts:
1. Value of statistic...
2. Value of statistic almost...
3. Statistic approaches...

1. Varies from sample to sample
2. Always differs from parameter values
3. Parameter value as sample size increases (the law of large numbers)

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How do we investigate the behavior of statistic?

By examining the sampling distribution of statistic

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Theoretical sampling distribution of x- bar is

The distribution of ALL x- bar values from ALL POSSIBLE SAMPLES of the same size from the same population

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Theoretical sampling distribution of x- bar
1. Take
2. Compute
3. Approximate

1. Take many, many SRSs
2. Compute x- bar for each
3. Approximate the theoretical sampling distribution of x- bar

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Approximate sampling distribution of x- bar is

The distribution of x- bar values obtained from repeatedly taking SRSs. Of the same size from the same population.

40

Approximate sampling distribution of x-bar can be modeled with .... curve

Normal

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How to determine how accurate is the sample mean as an estimator of mu?
1. Take...
2. Construct...
3. Note ...

1. Take many,many SRSs, compute x- bar for each sample
2. Construct histogram of x-bars to display the approximate sampling distribution of x- bar
3. Note center, spread and shape

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Mean of all sampling distributions of x- bar =

Mean of population

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As n increases, spread of sampling distribution of x- bar

decreases

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As n increases, shape of sampling distribution of x- bar becomes

More normal

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In sampling distributions
Center ... to population center regardless of sample size

Equal or X- bar=mu

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In sampling distributions as spread decreases n ...

Increases

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In sampling distributions the shape becomes ..... ....... as n increases

More normal

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How well does x-bar estimate mu?

Quite well for large SRSs

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Does x- bar vary about mu?

Yes

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Probability is measured on 0 to 1 scale , where 0 is .... And 1 is....

0 impossible , never occur
0.01 unlikely but occur once in a while in a long run
0.45 slightly less often than not
0.50 half of the time
0.55 slightly greater than one- half
0.99 greater than one half but less 1
1 - certain, will occurs every time

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Population distribution

The distribution of values of a variable among all individuals in the population

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Sampling distribution

The distribution of values taken by a statistic in all possible samples of the same size from the same population