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Midterm

Statistics Midterm 2 Notes

20 Pages
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School
UOIT
Department
Business
Course Code
BUSI 1450U
Professor
William Goodman

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Statistics Midterm 2 Information
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
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
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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Description
Statistics Midterm 2 InformationChapter 72 Binomial DistributionBinomial distributionspecial case of discrete probability distributionThe 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 resultA success denotes the occurrence of an outcome in which we are interested such as tossing a head and a failure denotes obtaining the opposite outcomeA 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 aboveEach unique sequence of outcomes for all of the trials combined may define one possible outcome for the overall experimentBinomial experimentan experiment such as the two coin flips that consists of a sequence of identical and independent Bernoulli trialsBinomial random variablea 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 Pa The number of ways A can occur the number of different outcomes possible in the sample space Nfixed number of trials xspecific number of success pprobability of success in one trial 1pprobability of failure in one trial Pxprobability of getting exactly X successes among N trialsExample Number of Tails in 2 Tosses of Coinn2 p05 X PX 01425s 12450 Binomial Probability Distribution FunctionnnXX1PXpp XnX Minitab procedureprobability ofsuccesses givenand PXXnp1 Calc 2 Probability distributionsnumber of successes in sample 01XXn3 Binomial 4 Input the value for X into Input Constant 5 Click Probabilitythe probability of each successpsample sizenFinding Cumulative Binomial ProbabilitiesCalculates the probabilities that X will fall below or above etc a certain value o Find PX for all Xs that meet the specified condition o Add all the probabilities found in step 1 Example Probability of 1 or more Tails in 2 Tosses of Coinn2 p05 X PX 01425 12450 21425In minitab 1 Calc 2 Probability distribution 3 Binomial 4 Input the value for X into Input constant 5 Click Cumulative probabilityResult gives the probability of getting a count less than or equal to the X you type in Chapter 7 MaterialsChapter 8 Material n Miu u P Sigma o X X Has gaps in between underneath the curve Has no gaps
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