Statistik bagi sains gunaan

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Statistik bagi sains gunaan

• MTH3003 PJJ
• SEM I 2015/2016

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Assessment

• ASSIGNMENT 25
• Assignment 1 (10)
• Assignment 2 (15)
• Mid exam 30
• Part A (Objective)
• Part B (Subjective)
• Final Exam 40
• Part A (Objective)
• Part B (Subjective - Short)
• Part C (Subjective Long)

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Chapter 1Describing Data with Graphs

• Definition
• Graphing

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Chapter 2Describing Data with Numerical Measures

• MEASURES OF CENTER- Arithmetic Mean or
  Average- Median- ModeGroup and ungrouped
  data

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Measures of Variability

• Range
• Interquartile Range
• Variance
• Standard Deviation
• Group an ungrouped data

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Box plot

• interpret
• Calculate
• Q1, Q2 and Q3, IQR, Upper fence, lower fence,
  outlier

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Interquartiles Range (IQR Q3 Q1)

• The lower and upper quartiles (Q1 and Q3), can be
  calculated as follows
• The position of Q1 is
  

• The position of Q3 is

once the measurements have been ordered. If the
positions are not integers, find the quartiles by
interpolation.
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Example

• The prices () of 18 brands of walking shoes
• 60 65 65 65 68 68 70 70
• 70 70 70 70 74 75 75 90 95

Position of Q1 0.25(18 1) 4.75 Position of
Q3 0.75(18 1) 14.25
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Chapter 3Probability and Probability
Distributions

• Basic concept
• The probability of an event - how to find prob
• Counting rules
• Calculate probabilities

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Calculate probabilities

• Event Relations Union, Intersection, Complement
• Calculating Probabilities for
• Unions
• The Additive Rule for Unions
• A Special Case Mutually Exclusive
• Complements
• Intersections
• Independent and Dependent Events
• Conditional Probabilities
• The Multiplicative Rule for Intersections

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Chapter 4RANDOM VARIABLES

• Probability Distributions forDiscrete Random
  Variables
• Properties for Discrete Random Variables
• Expected Value and Variance

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Properties for Discrete Random Variables

• The properties for a discrete probability
  function (PMF) are
• Cumulative Distribution Function (CDF)

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Example

• Toss a fair coin three times and define X
  number of heads.

x 3 2 2 2 1 1 1 0
X p(x)
0 1/8
1 3/8
2 3/8
3 1/8
P(X 0) 1/8 P(X 1) 3/8 P(X 2)
3/8 P(X 3) 1/8
1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8
HHH
HHT
HTH
THH
HTT
THT
TTH
TTT
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Chapter 5Several Useful Discrete Distributions

• Discrete distributions
• The binomial distribution
• The Poisson distribution
• The hypergeometric distribution
• To find probabilities
• formula
• cumulative table

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Key Concepts

• I. The Binomial Random Variable
• 1. Five characteristics n identical independent
  trials, each resulting in either success S or
  failure F probability of success is p and
  remains constant from trial to trial and x is
  the number of successes in n trials.
• 2. Calculating binomial probabilities
• a. Formula
• b. Cumulative binomial tables
• 3. Mean of the binomial random variable m np
• 4. Variance and standard deviation s 2 npq
  and

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Example
A marksman hits a target 80 of the time. He
fires five shots at the target. What is the
probability that exactly 3 shots hit the target?
P(x 3) P(x ? 3) P(x ? 2) .263 - .058
.205
Check from formula P(x 3) .205
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Key Concepts

• II. The Poisson Random Variable
• 1. The number of events that occur in a period
  of time or space, during which an average of m
  such events are expected to occur. Examples
• The number of calls received by a switchboard
  during a given period of time.
• The number of machine breakdowns in a day
• 2. Calculating Poisson probabilities
• a. Formula
• b. Cumulative Poisson tables
• 3. Mean of the Poisson random variable E(x) m
• 4. Variance and standard deviation s 2 m and

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Example
 
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Key Concepts

• III. The Hypergeometric Random Variable
• 1. The number of successes in a sample of size n
  from a finite population containing M
  successes and N - M failures
• 2. Formula for the probability of k successes in
  n trials
• 3. Mean of the hypergeometric random variable
• 4. Variance and standard deviation

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Example
A package of 8 AA batteries contains 2 batteries
that are defective. A student randomly selects
four batteries and replaces the batteries in his
calculator. What is the probability that all four
batteries work?
Success working battery N 8 M 6 n 4
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Chapter 6The normal probability distribution

• The Standard Normal Distribution
• 1. The normal random variable z has mean 0 and
  standard deviation 1.
• 2. Any normal random variable x can be
  transformed to a standard normal random
  variable using
• 3. Convert necessary values of x to z.
• 4. Use Normal Table to compute standard normal
  probabilities.

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Example
The weights of packages of ground beef are
normally distributed with mean 1 pound and
standard deviation 0.1. What is the probability
that a randomly selected package weighs between
0.80 and 0.85 pounds?
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The Normal Approximation to the Binomial

• We can calculate binomial probabilities using
• The binomial formula
• The cumulative binomial tables
• When n is large, and p is not too close to zero
  or one, areas under the normal curve with mean
  np and variance npq can be used to approximate
  binomial probabilities.

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Approximating the Binomial

• Make sure to include the entire rectangle for the
  values of x in the interval of interest. That is,
  correct the value of x by This is called
  the continuity correction.
• Standardize the values of x using

• Make sure that np and nq are both greater than 5
  to avoid inaccurate approximations!

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Example
Suppose x is a binomial random variable with n
30 and p .4. Using the normal approximation to
find P(x ? 10).
n 30 p .4 q .6 np 12 nq 18
The normal approximation is ok!
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Example
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Chapter 7Sampling Distributions

• Sampling Distributions
• Sampling distribution of the sample mean
• Sampling distribution of a sample proportion
• Finding Probabilities for the
• Sample Mean
• Sample Proportion

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The Sampling Distribution of the Sample Mean

• A random sample of size n is selected from a
  population with mean m and standard deviation s.
• The sampling distribution of the sample mean
  will have mean m and standard deviation
  .
• If the original population is normal, the
  sampling distribution will be normal for any
  sample size.
• If the original population is non normal, the
  sampling distribution will be normal when n is
  large.

The standard deviation of x-bar is sometimes
called the STANDARD ERROR (SE).
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Finding Probabilities for the Sample Mean

• If the sampling distribution of is normal
  or approximately normal, standardize or rescale
  the interval of interest in terms of
• Find the appropriate area using Z Table.

Example A random sample of size n 16 from a
normal distribution with m 10 and s 8.
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The Sampling Distribution of the Sample Proportion
The standard deviation of p-hat is sometimes
called the STANDARD ERROR (SE) of p-hat.
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Finding Probabilities for the Sample Proportion

• If the sampling distribution of is normal
  or approximately normal, standardize or rescale
  the interval of interest in terms of
• Find the appropriate area using Z Table.

If both np gt 5 and np(1-p) gt 5
Example A random sample of size n 100 from a
binomial population with p 0.4.