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Ch8 Sampling, Sampling Methods, and the Central Limit Theorem - Coggle…

- - - - The cost of studying all the items in a population may be prohibitive. (P251/P234, P279/P262)
      - The physical impossibility of checking all items in the population. (P251/P234, P279/P262)
      - To contact the whole population would be time consuming. (P251/P234, P279/P262)
      - The destructive nature of some tests. Often testing destroys the sampled item and it cannot be returned to the population.(P251/P234, P279/P262)
  - - - Simple Random Sampling （簡單隨機抽樣）(P252/P235)
        
        A sample selected so that each item or person in the population has the same probability or chance of being included. (P252/P235, P279/262)
        
        The selection process (P252/P235)
        
        Drawing lots. (P252/P235)
        
        Using a table of random numbers such as the one in Appendix B.4. (P252/P235)
        
        Choose a starting point in the table(P252/P235)
        
        To close your eyes and point at a number in the table.
        
        To randomly pick a column and row.
        
        Statistical software often includes random number generators to compute a list of random numbers. (P252/P236)
        
        The most widely used sampling method.(P252/P235) Be used in the selection of the days, the times, and the starting point. (P254/P239)
      - Systematic Random Sampling (系統隨機抽樣) (P254/P238)
        
        The selection process.
        
        S1: Create a Sampling Frame
        
        S2: Determine the Sample Size and Interval (k) k= Sample Size/Population Size
        
        S3 Randomly Select a Starting Point
        
        S4 Select Samples at Fixed Intervals
        
        A Sampling Frame is a list or data set that includes all members of a population from which a sample can be drawn.
        
        It serves as a fundamental tool in the sampling process to ensure that every member has a chance of being selected.
        
        Limitations: Should carefully observe the physical order of the population. When the physical order is related to the population characteristic, then systematic random sampling should not be used because the sample could be biased. (P254/P239)
        
        A random starting point is selected, and then every kth member of the population is selected.(P254/P239, P279/262)
        
        Be used to select the actual customer.(P254/P239)
      - Stratified Random Sampling (分層隨機抽樣) (P255/P239)
        
        When a population can be clearly divided into groups based on some characteristic. (P255/P239)
        
        The selection process.(P255/P239)
        
        S1:Usually the strata are formed based on members’ shared attributes or characteristics
        
        S2: A random sample from each stratum is taken in a number proportional to the stratum’s size when compared to the population.
        
        S3: Once the strata are defined, we apply simple random sampling within each group or stratum to collect the sample.
        
        A population is divided into subgroups(群/組), called strata(層), and a sample is randomly selected from each stratum. (P255/P239, P279/P262)
        
        It guarantees each group is represented in the sample. (P255/P239)
        
        The advantage of more accurately reflecting the characteristics of the population than does simple random or systematic random sampling.(P255/P239)
      - Cluster Sampling (叢式抽樣)(P256/P240)
        
        The selection process.
        
        S1: Define the Population and Divide it into Clusters
        
        S2: Randomly Select Clusters.
        
        S3: Survey All Members Within the Selected Clusters
        
        Often employed to reduce the cost of sampling a population scattered over a large geographic area. (P256/P240)
        
        Useful when the population is geographically dispersed or when a complete sampling frame is difficult to establish.
        
        A population is divided into clusters(叢, also called primary units(基本單位) or intermediate unit(中間單位).) using naturally occurring geographic or other boundaries. Then, clusters are randomly selected and a sample is collected by randomly selecting from each cluster. (P256/P240, P279/262)
        
        Limitations: If members within clusters are too similar, it may reduce the representativeness of the sample.
    - - Convenience sampling （便利抽樣）
        
        The sample may not represent the entire population, the results are prone to bias.
        
        This method is often used for preliminary research or when resources are limited.
        
        A non-probability sampling method where researchers select samples based on ease of access or availability.
- - - - If the population distribution is symmetrical (but not normal), you will see the normal shape of the distribution of the sample mean emerge with samples as small as 10.
      - if you start with a distribution that is skewed or has thick tails, it may require samples of 30 or more to observe the normality feature
      - If the population follows a normal probability distribution, then for any sample size the sampling distribution of the sample mean will also be normal.
    - - To create confidence intervals for the population mean (described in Chapter 9)
      - To perform tests of hypothesis (described in Chapter 10)
      - The central limit theorem states that, for large random samples, the shape of the sampling distribution of the sample mean is close to the normal probability distribution. The central limit theorem says that if sample size increases, the sampling distribution of sample means continues to approximate the normal distribution. (Chart 8-3, P265/P250)
    - - The central limit theorem is true for all population distributions. (P265/P249)
  - - - Proportions are used when measuring a random variable with the nominal scale of measurement and the outcomes are limited to two values. Observations are classified into one of two mutually exclusive groups. (P275/P259)
      - Proportion（比例）
        
        The fraction, ratio, or percent indicating the part of the sample or the population having a particular trait of interest. (P276/P260)
        
        The sampling distribution of the sample proportion is a probability distribution of all possible sample proportions of the same sample size.(P279/P263)
      - Sample Proportion (樣本比例)
        
        The sample proportion is the number of “successful” outcomes relative to the total number of outcomes. (P279/P263)
        
        (8-3)
        
        p is the sample proportion; x is the count or number of “successful” outcomes; n is the sample size or the total count or number of outcomes.
        
        As a statistic, the sample proportion, p, estimates the true population proportion, π. (P276/P260)
        
        The sample proportion, p, is a random variable.(P276/P260)
        
        The two requirements that determine if the sample size is large enough:(P276/P260)
        
        P is described with a probability distribution.
      - Mean(比例樣本平均數)(P276/P260)
        
        For this normal distribution, the mean is equal to the population proportion, π.
        
        For a given sample size, the mean of all possible sample proportions selected from a population is equal to the population proportion, π. (P279/P263)
      - Standard Error of the Proportion (比例的標準誤)(P276/P260)
        
        The standard error of the proportion measures the variation in the sampling distribution of the sample proportion.(P279/P263)
        
        (8-4) (P276/P260)
      - The probability Distribution: Z-Value(P276/P260)
        
        If the sample size is large enough, we use the standard normal distribution to determine the probability that a sample proportion falls in a particular range. (P279/P263)
        
        (8-5) (P276/P260)
    - - The distribution of all possible sample means, calculated from all possible random samples, is normally distributed. This distribution is called the sampling distribution of the sample mean.(P261/P245)
        
        A probability distribution of all possible sample means of a given sample size..(P261/P245)
      - Using sampling and knowing about the sampling distribution of the sample mean ensures that large sampling errors are unlikely.(P261/P245)
      - The important relationships between the population distribution and the sampling distribution of the sample mean: (P262/P247)
        
        2 The dispersion of the sampling distribution of the sample mean is narrower than the population distribution.
        
        There is less variation in the distribution of the sample mean than in the population distribution. (P279/P259)
        
        3 The sampling distribution of the sample mean tends to become bell shaped and to approximate the normal probability distribution.
        
        1 The mean of the sample means is exactly equal to the population mean.
        
        For a given sample size, the mean of all possible sample means selected from a population is equal to the population mean. (P279/P259)
      - Standard Error of The Mean, also called "The standard deviation of the sampling distribution of the sample mean" (樣本平均數的標準誤) (P271/P255)
        
        The standard error of the mean measures the variation in the sampling distribution of the sample mean. (P279/P262)
        
        (8-1)
        
        Note the relationship between the standard error and sample size, n. As the sample size increases, the dispersion of the sampling distribution of the mean decreases and the shape of the distribution narrows around the population mean. (P271/P255)
      - The Sample Mean (樣本平均數) (P271/P255)
        
        The mean of the sampling distribution is exactly equal to the population mean. (P271/P255)
      - To determine the probability that a sample mean falls in a particular region:(P273/P256, P279/P263)
        
        Review:The sampling distribution will be normally distributed under two conditions:(P273/P257)
        
        When the samples are taken from populations known to follow the normal distribution, the size of the sample is not a factor.
        
        When the shape of the population distribution is not known, sample size is important. In general, the sampling distribution will be normally distributed as the sample size approaches infinity. In practice, a sampling distribution will be close to a normal distribution with samples of at least 30 observations.
        
        (8-2) (P273/P257)
  - - - The bar over the x is a sample mean and the subscript 1 indicates it is the mean of the first sample. (P260/P244)
  - - - As an estimate of the population mean, the sample mean will be different from the population mean. This is the concept of sampling error. (P260/P245)
        
        The sampling error is the difference between a population parameter and a sample statistic. (P279/P262)