Chapter 3: Sampling Distribution

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Chapter 3 Sampling Distribution

• 3.1 Population and Sampling Distribution
• 3.2 Sampling Distribution of
• 3.3 Sampling Distribution of
• 3.4 Sampling Distribution of s2

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3.1 Population and Sampling Distribution

• 3.1.1 Motivation and Definitions
• 3.1.2 Population and Sampling Distribution

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Motivation and Definitions

• Statistical Inference
• Arriving at conclusions concerning a population
  parameter.
• Impossible or impractical to observe the entire
  set of observation that makes up the population,
  therefore we must depend on a subset of
  observations from population (or called the
  sample) that help us make inferences concerning
  that same population.

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Population Sampling Distribution

• Population is a collection or set, of all
  individuals, objects, or items of interest.
• Population Parameters
• Mean ?,
• Variance ?2 ,
• Proportion, p
• Population Distribution
• The probability distribution of the population
  data.
• A sample is a portion, or part, of the population
  of interest.
• A statistic is any quantity whose value can be
  calculated from a sample data. Sample statistics
  includes
• Sample mean,
• Sample variance, s2
• Sample proportion,

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Population Sampling Distribution

• Sampling Distribution
• The sampling distribution of a sample statistic
  is its probability distribution.
• E.g. The sampling distribution of the sample
  mean, , is the probability distribution of
  .
• Standard Error
• The standard error of a statistic is the standard
  deviation of its sampling distribution.
• Sampling Error
• The difference between the value of a sample
  statistics and the value of the corresponding
  population parameter.
• Nonsampling Error
• The errors that occur in the collection,
  recording and tabulation of data.

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3.2 Sampling Distribution of

• 3.2.1 Shape of the Sampling Distribution of
• 3.2.2 Central Limit Theorem
• 3.2.3 Sampling Distribution of the Sample Mean
• 3.2.4 Sampling Distribution of the Difference
  between Two Mean
• 3.2.5 Sampling Distribution of Small Sample Mean

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• The probability distribution of the mean of all
  the possible random sample of size n that could
  be selected from a given population.
• It lists the various values that can assume
  and the probability of each value of .
• The mean and standard deviation of the sampling
  distribution of are called the mean and
  standard deviation of and are denoted by
  and respectively.
• The mean of the sampling distribution of is
  always equal to the mean of the population.
  (Unbiased estimator)
• The standard deviation of the sampling
  distribution of is
• where is the standard deviation of the
  population and n is the sample size. This
  formula is used when n/N 0.05, where N is the
  population size.

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Steps in tabulating the sampling distribution of

• Draw many samples of a specified size.
• Sample size n 2
• Number of possible samples 10
• Calculate the value of a statistic for each
  sample.
• Sample statistic
• Tabulate the frequency distribution of
• Tabulate the probability distribution of
• (probability relative frequency)

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Example

• Tabulate the sampling distribution of
• Population 2 4 6 8 10
• Population size N 5
• Population mean
• Population standard deviation
• Steps
• Sample size , n 3
• No of possible sample 10
• Value of a statistic for each sample

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• Tabulate the frequency distribution and
  probability distribution of
• (probability relative frequency)

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• Mean of the sampling distribution of
• The standard deviation of the sampling
  distribution of
• Because n/N is greater than 0.05, we have to use
  the correction
• factor and get
• Normally in most practical applications the
  sample size would be smaller than the population
  size, thus we can use the formula below to
  calculate the standard deviation
• Note
• As the sample size (n) increases, the value of
  the standard deviation of ( ) decreases.
  (Consistent estimator)
• Standard deviation

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Exercise

• The mean wage per hour for all 5000 employees who
  work at a large company is 13.50 and the
  standard deviation is 2.90. Let be the mean
  wager per hour for a random sample of certain
  employees selected from this company. Find the
  mean and standard deviation of for a sample
  size of
• 30
• 75
• 200

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• a) Mean
• Standard deviation
• b) Mean
• Standard Deviation
• c) Mean
• Standard Deviation

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Shape of the Sampling Distribution of

• The shape of the sampling distribution of
  relates to the two following cases
• The population from which the sample are drawn
  has a normal distribution
• The population from which the sample are drawn
  does not have a normal distribution
• If the population from which the samples are
  drawn is normally distributed ,
  then the sampling distribution of will also
  be normally distributed with mean, , and
  standard deviation,
• irrespective of the sample size.
• Note must be 0.05 for to
  be true.

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Example

• The speeds of all cars traveling on a stretch of
  Interstate Highway are normally distributed with
  a mean of 68 miles per hour and a standard
  deviation of 3 miles per hour. Let be the
  mean speed of a random sample of 20 cars
  traveling on this highway.
• Calculate the mean and standard deviation of
  and describe the shape of its sampling
  distribution.
• Answer
• miles per hour
• Normal distribution

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Central Limit Theorem

• Most of the time the population from which the
  samples are selected is not normally distributed.
  
• In such case, the shape of the sampling
  distribution of is inferred from a very
  important theorem called the central limit
  theorem.
• If a random sample of n observations is drawn
  from a population with unknown distribution,
  either finite or infinite, then when n is
  sufficiently large (n 30) , the sampling
  distribution of the sample mean can be
  approximated by a normal density function.
  Irrespective of the shape of the population
  distribution.

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Central Limit Theorem
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Example

• The GPAs of all 5540 students enrolled at a
  university have an approximate normal
  distribution with a mean of 3.02 and a standard
  deviation of 0.29. Let be the mean GPA of a
  random sample of 48 student selected from this
  university.
• Find the mean and standard deviation of and
  comment on the shape of its sampling
  distribution.
• Answer
• Approximately normal distribution

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Sampling Distribution of the Sample Mean

• If we take all possible samples of the same
  (large) size from a population parameter and
  calculate the mean for each of these samples,
  then about
• 68.26 of the sample means will be within one
  standard deviation of the population mean.
• 95.44 of the sample means will be within two
  standard deviation of the population mean.
• 99.74 of the sample means will be within three
  standard deviation of the population mean.

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• Sampling distribution of is its probability
  distribution. The area under the distribution
  curve represent this probability.
• In order to find the probability, you have to
  compute the z-value for the respective value.
• Then, find the value of z under the standard
  normal curve.

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Example

• The insider diameter of a randomly selected
  piston ring is a random variable with mean value
  12 cm and standard deviation 0.04 cm. Supposed
  the distribution of diameter is normal.
• If is the sample mean diameter for a random
  sample of n 16 rings, find the mean and the
  standard deviation of the
• distribution?
• Answer the question posed in part (a) for a
  sample size of
• n 64 rings.
• Calculate when n
  16.
• How likely is that the sample mean diameter
  exceeds 12.01 when n 25?

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Example

• Sam and Janet Evening wants to estimate the
  average dollar amount of the orders filled by
  their South Pacific Catering Company. They obtain
  their estimate by selecting a simple random
  sample of 49 orders. Their order are normally
  distributed with and .
  
• Whats the value of the standard error?
• Whats the chance that the sample mean will fall
  within one standard deviation of the population
  mean?
• Whats the probability the sample mean will lie
  between 116.50 and 124.00?
• Within what range of values does the have a
  94.5 chance of falling?

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• Standard error standard deviation
• Probability falls within one standard deviation
  of the
• 120 3, or between 117.00 and 123.00.
• Find the corresponding z-value
• for 117.00,
• for 123.00,
• P (117 123) P (z 1) 1 - P( z
  1 )
• 0.8413 1 0.8413
• 0.6826

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• Probability the sample mean will lie between
  116.50 and 124.00
• for 116.50,
• for 124.00,
• P ( lt 116.5) P (z -1.17 )
• 0.1210
• P ( lt 124) P (z 1.33)
• 0.9082
• P (116.5 lt lt 124) P (-1.17 z 1.33)
• 0.9082 0.1210
  0.7872

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• Range of values does the have a 94.5 chance
  of falling
• 94.5 of the falls within two standard
  deviation of the population mean
• Thus, the range of values must be within two
  standard deviation of the mean.
• 120 2(3) 120 2(3)
• 114 126

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Sampling Distribution of the Difference between
Two Means
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Example

• The television picture tubes of manufacturer A is
  normally distributed with a mean lifetime of 6.5
  years and a standard deviation of 0.9 year, while
  those of manufacturer B is normally distributed
  with a mean lifetime of 6.0 years and a standard
  deviation of 0.8 year. What is the probability
  that a random sample of 36 tubes from
  manufacturer A will have a mean lifetime that is
  at least 1 year more than the mean lifetime of a
  sample of 49 tubes from manufacturer B?

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• Solution
• Given
• ?A 6.5 ?A0.9 nA 36,
• ?B 6.0 ?B 0.8 nB 49
• In order to determine
  , write
• Thus

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Example

• The elasticity of a polymer is affected by the
  concentration of a reactant. When low
  concentration is used, the true mean elasticity
  is 55, and when high concentration is used the
  mean elasticity is 60. The standard deviation of
  elasticity is 4, regardless of concentration. If
  two random samples of size 16 are taken, find the
  probability that mean elasticity of high
  concentration is at least 2 more than the mean
  elasticity of low concentration.

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• Solution
• Given
• ?high 60, ?high 4 nhigh 16 and
• ?low 55, ?low 4 nlow 16
• In order to determine
  , write
• Thus

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Sampling Distribution of Small Sample Mean

• Students t-distribution is used in the inference
  of population mean or comparative samples when
  the population standard deviation ? is unknown.
• In practice, when is the mean, s2 is
  variance of a random sample of size n obtained
  from population satisfying a normal distribution
  with mean ? , then the sample statistic
• follows the t-distribution with (n ?1) degree of
  freedom.

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• If the sample size is large (n ? 30),
  distribution of T does not differ considerably
  from the standard normal.
• When the sample size is small (n lt 30), the
  values of s2 fluctuate considerably from sample
  to sample and the distribution of T deviates
  appreciably from that of standard normal
  distribution (see figure below).
• The probability distribution of T is described by
  the Student's t-distribution.
• Let random variables Z N(0,1) and
  , and if Z and V are independent, then the
  distribution of
• is referred to as students t-distribution with
  (? n ? 1) degree of freedom (df).

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• A typical way to refer to the t-distribution
  table is by the Critical Values, ? for
  t-distribution,
• P(T gt t?,? ) ?

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Example

• For a random sample of size 24 selected from a
  normal distribution,
• Find k such that .
• Find k such that .

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Example

• A random sample of size 9 is drawn from a normal
  population with unknown variance and a mean of
  20. Find the probability that the sample mean is
  greater than 24 if the sample standard deviation
  is 4.1436.

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• Solution

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Example

• Supposing you know that average Malaysian man
  exercises for 60 minutes a week and a random
  sample of 25 means you have drawn from the
  population has a mean of 65 minutes per week with
  a standard deviation of 10 minutes.
• What is the probability of getting a random
  sample of this size with a mean less than or
  equals 65 if the actual population mean is 60?

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• Solution

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3.3 Sampling Distribution of

• 3.3.1 Central Limit Theorem
• 3.3.2 Sampling Distribution of the Sample
  Proportion
• 3.3.3 Sampling Distribution of the Difference
  between Two Proportions

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• The concept of proportion is the same as the
  concept of relative frequency.
• The population proportion is obtained by taking
  the ratio of the number of elements in a
  population with a specific characteristic to the
  total number of elements in the population.
• The sample proportion gives a similar ratio for a
  sample.

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Example

• Suppose a total of 789,654 families live in a
  city and 563,282 of them own homes. Then
• Take a sample of 240 families from this city and
  158 of them are homeowners. Then

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• Similar to sample mean, the sample proportion is
  a random variable. Hence, it possesses a
  probability distribution, which is called its
  sampling distribution.
• This distribution gives the various values that
  can assume and their probability.
• The mean of the sample proportion is denoted by
  and is equal to the population proportion, p.
  Thus,
• The standard deviation of the sample proportion
  is denoted by and is given by the
  formula
• p population proportion
• q 1 p
• This formula is used when n/N 0.05.

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Example

• BCC Consultant has five employees. The table
  below list their names and information regarding
  their knowledge of statistics.
• Population proportion,

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• Steps
• Sample size , n 3
• No of possible sample 10
• Value of a statistic for each sample

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• Frequency, relative frequency and sampling
  distribution
• Calculate the mean and standard deviation

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Central Limit Theorem

• The shape of the sampling distribution of is
  inferred from the central limit theorem.
• According to the central limit theorem, the
  sampling distribution of is approximately
  normal for a sufficiently large sample size.
• In this case, the sample size is considered to be
  sufficiently large if np and nq are both greater
  than 5, that is if
• np gt 5 and nq gt 5

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Example

• Suppose that in a local school, 20 of all
  students in grades 7 through 12 are afraid of
  being attacked in their school. Let be the
  proportion of students in a random sample of 50
  seventh-through-twelfth-grade students who fear
  such attacks.
• Calculate the mean and standard deviation of
  and comment on the shape of its sampling
  distribution.
• Answer
• Approximately normal distribution

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Sampling Distribution of the Sample Proportion

• We use the concept of mean, standard deviation,
  shape of the sampling distribution of to
  determine the probability that the value of
  computed from one sample falls within a given
  interval.
• The z-value for a value of
• with limiting distribution Z N(0,1) as n??.

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Example

• A college had a business entering freshman class
  of 528 students last year. Of these, 211 have
  brought their own personal computers to campus. A
  random sample of 120 entering freshman was taken.
• What is the standard deviation of the sample
  proportion bringing own personal computer to
  campus?
• What is the probability that the sample
  proportion is less than 0.33?
• What is the probability that the sample
  proportion is between 0.4 and 0.5?

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• Solution

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Example

• Maureen Webster, who is contesting for the
  mayors position in a large city, claim that she
  is favored by 53 of all eligible voters of that
  city. Assume that this claim is true.
• What is the probability that in random sample of
  400 registered voters taken from this city, less
  than 49 will favor Maureen Webster?
• Answer
• Calculate p and q
• p 0.53, q 1 p 1.0 0.53 0.47

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• Calculate the mean and standard deviation
• Calculate the z-value
• Find the required probability
• P( lt 0.49) P(z lt -1.60)
• 0.0548

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Sampling Distribution of the Difference between
Two Proportions
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Example

• Election returns showed that a certain candidate
  received 65 of the votes. Find the probability
  that two random samples, sample A consists of 200
  voters and sample B consists of 150 voters,
  indicated more than a 10 difference in the
  proportions who voted for the candidate.

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• Solution

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Example

• For the freshman class in Example 3.8, 20 of
  female students and 23 of male students have
  brought their own personal computer to campus. A
  random sample of 50 female and 55 male students
  been taken. Find the probability that the
  difference for proportion of female and male
  students who brought their own personal computer
  to campus is less than 0.1

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• Solution

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3.3 Sampling Distribution of s2
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• Sample variance s2 and the chi-square
  distribution is often used to determine the
  interval estimate of the actual (population)
  variance sits.
• Typically one need to find probability such as
  P(X2 gt ?2?) and one can use the table of upper
  critical values of chi-square distribution as
  shown below.

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Example

• Find the value of for the following
• (Refer to the table of Critical Area Chi-Square
  Distribution)
• a) 0.99 when v 4
• b) 0.025 when v 19
• c) 0.045 when v 25

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Example

• The claim that the variance of a normal
  population is is to be rejected
  if the variance of a random sample of size 16
  exceeds 54.668 or is less than 12.102. What is
  the probability that this claim will be rejected
  even though ?
• Solution

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Example

• A random sample of 10 observations is taken from
  a normal population having the variance 42.5.
  find the approximate probability of obtaining
  sample variance between 9.8596 and 79.9236
• Solution
• n 10, s2 42.5, thus v 10 1 9

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Sampling Distribution of the Ratio of Two
Variances

• F-Distribution used to compare two or more sample
  variances.
• The statistics F is defined to be the ratio of
  two independent Chi-squared random variables,
  each divided by its number of degrees of freedom.
  
• If and are independent,
  then the random variable
• defines the F-distribution with ?1 and ?2
  degrees of freedom.
• Writing fa (v1, v2) for fa with v1 and v2 degrees
  of freedom, we obtain

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Example

• For a F-distribution find
• a) f0.01 with v1 24 and v2 20
• b) f0.01 with v1 20 and v2 24
• c) f0.95 with v1 15 and v2 20
• c) f0.99 with v1 30 and v2 12

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Example

• ,
• ,
• The heat producing capacity of two coal mines,
  Mine A and Mine B, are normally distributed with
  the variances sA2 10920 and sB2 15750. If
  sA2 and sB2 are the variances of independent
  random samples of size nA 5 and nB 6 taken
  from these two mines, find the probability that
  the ratio of the two variances are more than 3.6.