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Chapter 7.2 Binomial Distribution

- Binomial distribution special case of discrete probability distribution.

- The word binomial is derived from bi for “two” and nominal for “numbers”, suggesting a reference to trials (such

as coin fillips) that have only two possible values or numbers that could result.

- A “success” denotes the occurrence of an outcome in which we are interested, such as tossing a head, and a

“failure” denotes obtaining the opposite outcome.

- A binomial distribution can also be called a trial (sometimes called a Bernoulli trial), since a trail is a 2 possible

outcome random experiment of the sort described above.

- Each unique sequence of outcomes for all of the trials combined may define one possible outcome for the

overall experiment.

- Binomial experiment an experiment such as the two coin flips, that consists of a sequence of identical and

independent Bernoulli trials.

- Binomial random variable a variable that counts the number of successes (regardless of exact sequence)

among the trial results in the binomial experiment.

These conditions must be met:

1) The experiment consists of n identical trials.

2) There are only two, mutually exclusive, results possible for each trial.

3) All of the trials are independent.

4) The probabilities of success are constant for each trial.

Calculating the probabilities for a binomial random variable:

P(a) = The number of ways A can occur .

the number of different outcomes possible in the sample space.

N = fixed number of trials

x = specific number of success

p = probability of success in one trial

(1 – p) = probability of failure in one trial

P(x) = probability of getting exactly X successes among N trials

s

Example: Number of

Tails in 2 Tosses of Coin

n = 2; p = 0.5 X P(X)

0 1/4 = .25

1 2/4 = .50

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Binomial Probability Distribution Function

Minitab procedure:

1) Calc

2) Probability distributions

3) Binomial

4) Input the value for X into “Input Constant”

5) Click Probability

Finding Cumulative Binomial Probabilities

- Calculates the probabilities that X will fall below (or above, etc.) a certain value.

o Find P(X) for all X’s that meet the specified condition

o Add all the probabilities found in step 1.

In minitab:

1) Calc

2) Probability distribution

3) Binomial

4) Input the value for X into “Input constant”

5) Click Cumulative probability.

Result gives the probability of getting a count less than or equal to the X you type in.

Chapter 7 Materials

Chapter 8 Material

n

Miu (u)

P

Sigma (o)

X

X

Has gaps in between, underneath the curve

Has no gaps.

!1

!!

: probability of successes given and

: number of "successes" in sample 0,1, ,

: the probability of each "success"

: sample size

nX

X

n

P X p p

X n X

P X X n p

X X n

p

n

Example: Probability of 1 or more Tails

in 2 Tosses of Coin

n = 2; p = 0.5 X P(X)

0 1/4 = .25

1 2/4 = .50

2 1/4 = .25

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Chapter 8 Continuous Probability Distributions

8.1 The Normal Probability Distribution

- Normal can be applied to model the distributions of many real-world variables such as lengths, heights, weights

and other measured values of natural or manufactured objects.

- Most values are clustered near the mean, and the frequencies drop off after two or more standard deviations

from the mean.

- Each bar is drawn over a range of possible outcomes (a class) and a bar’s length represents the proportion of

outcomes that occur somewhere in that class.

- These percentages can be interpreted as probabilities and the total area under the curve (like the sum of

probabilities for a sample space) equals 1.

- Normal Curve/Bell Curve is such that for any class whose width is greater than zero, the area within the

curve lying above a class range for the variable does approximate, like a density scale histogram, the probability

of getting an outcome within that particular range of x-values.

- Probability density curve a probability distribution for a continuous variable, the normal curve is an example.

- Probability density function The ordinate value of the curve is determined for each x-value in the sample

space.

The normal Curve: The Mathematical Model

E = 2.71828

u = population mean

o = population standard deviation

X = value of random variable ( - infinite < x < infinite)

2

2

2

)(

2

1

)(

x

exf

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