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    S. Sampling Distribution

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    Brian Garcia
    The document discusses sampling distributions and their properties, including how the sampling distribution of the sample mean approaches a normal distribution as the sample size increases due to the central limit theorem. It provides examples of how to calculate the mean and standard deviation of sampling distributions and use them to find probabilities related to sample means. Different sampling techniques like simple random sampling are also covered.

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    S. Sampling Distribution

    Uploaded by

    Brian Garcia
    18 views28 pages
    The document discusses sampling distributions and their properties, including how the sampling distribution of the sample mean approaches a normal distribution as the sample size increases due to the central limit theorem. It provides examples of how to calculate the mean and standard deviation of sampling distributions and use them to find probabilities related to sample means. Different sampling techniques like simple random sampling are also covered.

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    S.-Sampling-Distribution

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    The document discusses sampling distributions and their properties, including how the sampling distribution of the sample mean approaches a normal distribution as the sample size increases due to the central limit theorem. It provides examples of how to calculate the mean and standard deviation of sampling distributions and use them to find probabilities related to sample means. Different sampling techniques like simple random sampling are also covered.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    18 views28 pages

    S. Sampling Distribution

    Uploaded by

    Brian Garcia
    The document discusses sampling distributions and their properties, including how the sampling distribution of the sample mean approaches a normal distribution as the sample size increases due to the central limit theorem. It provides examples of how to calculate the mean and standard deviation of sampling distributions and use them to find probabilities related to sample means. Different sampling techniques like simple random sampling are also covered.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
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    of 28

    SAMPLING

    DISTRIBUTIONS
    Sampling
    Distributions
    Sampling Sampling
    Distributions of the Distributions of the
    Sample Mean Sample Proportion

    Mean of the
    Sampling The Central Limit
    Theorem Mean of the
    Distribution of the Standard Error of the
    Sampling
    Sample Mean Sampling
    Distribution of the
    Sample Distribution of the
    Proportion Sample Proportion

    Standard Error of the


    Sampling Distribution
    of the Sample Mean
    BIASED SAMPLES – samples
    taken when the sampling
    process done doesn’t give
    every member of the
    population an equal chance of
    being selected.
    FOUR BASIC PROBABILITY
    SAMPLING TECHNIQUES
    1. Simple Random Sampling
    2. Systematic Sampling
    3. Stratified Sampling
    4. Cluster Sampling
    FOUR BASIC PROBABILITY SAMPLING TECHNIQUES
    1. SIMPLE RANDOM SAMPLING
    - You use a sample frame.
    - Sampling Frame is a list of individuals in
    the population that must first be defined.
    2. SYSTEMATIC SAMPLING
    - a sampling frame of the individuals in
    the population is also used. To obtain n
    members of the sample, the sampling
    frame will be divided into n groups and
    then one member from each group will be
    selected.
    FOUR BASIC PROBABILITY SAMPLING TECHNIQUES
    3. STRATIFIED SAMPLING
    - the population is group into strata so
    that each group will have a common
    characteristic, such as gender, age or
    grade level. Samples are then taken from
    each stratum using simple random
    sampling.
    4. CLUSTER SAMPLING
    - done by grouping the population into
    clusters that can be a pre-existing
    designation as to cities, towns or
    provinces.
    SAMPLING DISTRIBUTION
    OF SAMPLE MEANS

    Statistics – Measures computed using sample data


    Parameters - Measures computed using population
    data
    SAMPLING DISTRIBUTION
    - Probability distribution of Statistics
    SAMPLING WITH REPLACEMENT
    - sample selection process wherein a sample
    may be selected more than once.
    SAMPLING ERROR
    - difference between the sample mean and the
    population mean.
    SAMPLING DISTRIBUTION
    OF SAMPLE MEANS

    - frequency distribution
    using the sample means
    computed from all possible
    random samples of a specific
    size taken from a population.
    EXAMPLE 1.
    Consider a population consisting of
    observations 3, 6, 9, 12.
    Mean = 7.5 and Standard Deviation = 3.3541
    Suppose a random samples of size 2 with
    replacement are taken.
    SAMPLES SAMPLE MEAN

    * Construct a table and histogram for the


    sampling distribution.
    STANDARD ERROR
    OF THE MEAN
    - standard deviation of the
    sampling distribution of the sample
    mean or any statistic. The degree of
    accuracy of the sample mean as an
    estimation of the population mean. It
    measures the degree of accuracy of the
    sample mean as an estimate of the
    population.
    EXAMPLE 2. Samples of three cards are
    drawn at random from a population of
    eight cards numbered from 1-8.
    a. How many possible samples can
    be drawn?
    b. Construct the sampling
    distribution of sample means.
    c. Construct a histogram of the
    sampling distribution of the sample
    means.
    STEPS IN CONSTRUCTING THE SAMPLING
    DISTRIBUTION OF THE MEANS
    1. Determine the number of possible
    samples that can be drawn from the
    population using the formula.
    Where N = size of the population
    n = size of the sample mean
    2. List all the possible samples and compute
    the mean of each sample.
    3. Construct a frequency distribution of the
    sample means obtained in Step 2.
    ANSWER. Distribution of Sample Mean
    Statistics and Probability Book.
    Exercises. Page 107-108
    BOYS – #1 (a & b)
    #2, #4
    GIRLS – #1 (c & d)
    #3, #4
    FINDING THE MEAN AND VARIANCE
    OF THE SAMPLING DISTRIBUTION
    EXAMPLE 3. Consider a population consisting
    of 1, 2, 3, 4, and 5. Suppose samples of size 2
    are drawn from this population. Describe the
    sampling distribution of the sample means.
    Find:
    a. Mean and Variance of the Sampling
    Distribution of the Sample means
    b. Compare these values to the mean
    and variance of the population
    c. Draw the histogram of the sampling
    distribution of the population mean.
    PROPERTIES OF THE SAMPLING
    DISTRIBUTION OF SAMPLE MEAN
    If all possible samples of size n
    are drawn from a population size N
    with mean 𝝁 and variance, the
    properties are:
    1. The mean and the sampling
    distribution of the sample mean is
    equal to the population mean 𝝁.
    That is
    2. The variance of the sampling
    distribution of the sample means 𝝈 is given
    by:

    3. The standard deviation of the sampling


    distribution of the sample means is given by:
    DESCRIBING THE SAMPLING DISTRIBUTION
    OF THE SAMPLE MEANS FROM AN INFINITE
    POPULATION
    EXAMPLE 4. A population has a
    mean of 60 and a standard
    deviation of 5. A random sample of
    16 measurements is drawn from
    this population. Describe the
    sampling distribution of the
    samples means by computing its
    mean and standard deviation.
    DESCRIBING THE SAMPLING DISTRIBUTION OF
    THE SAMPLE MEANS FROM AN INFINITE
    POPULATION
    EXAMPLE 5. The heights of male
    college students are normally
    distributed with mean of 68 inches and
    standard deviation of 3 inches. If 80
    samples consisting of 25 students each
    are drawn from the population, what
    would be the expected mean and
    standard deviation of the resulting
    sampling distribution of the means?
    SOLVING PROBLEMS
    INVOLVING SAMPLING
    DISTRIBUTION OF THE
    SAMPLE MEANS
    THE CENTRAL LIMIT THEOREM
    If a random samples of size n
    are drawn from a population,
    then as n becomes larger, the
    sampling distribution of the
    mean approaches the normal
    distribution, regardless of the
    shape of the population
    distribution.
    EXAMPLE 6. The average lifetime of the
    batteries of cellular phone with a normal
    distribution is known to be 2.4 years with a
    standard deviation of 0.32 years. A random
    sample of size 16 will be taken.
    a. Determine the mean and the standard
    error of the sampling distribution of the
    means.
    b. What is the distribution of the
    sampling distribution of the sample mean?
    EXAMPLE 6. The average lifetime of the
    batteries of cellular phone with a normal
    distribution is known to be 2.4 years with a
    standard deviation of 0.32 years. A random
    sample of size 16 will be taken.
    c. What is the probability that a random
    sample of size 16 will have a sample mean
    greater than 2.5 years?
    d. What is the probability that a random
    sample of size 16 will have a sample mean
    less than 2.2 years?
    EXAMPLE 7. Students in a certain High School
    have an average height of 160cm with a
    standard deviation of 12 cm. Random
    samples of size 50 will be taken.
    a. Determine the mean and the standard
    error of the sampling distribution of the
    means.
    b. What is the distribution of the
    sampling distribution of the sample mean?
    EXAMPLE 7. Students in a certain High School
    have an average height of 160cm with a
    standard deviation of 12 cm. Random
    samples of size 50 will be taken.
    c. What percentage of the random
    samples will have a sample mean greater
    than 165 cm?
    d. What percentage of the random
    samples will have a sample mean within 2 cm
    of the population mean?
    ANSWER. Pages 132-133.
    BOYS – Odd numbers
    GIRLS – Even numbers

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