Chapter 8: Sampling Distributions

Section 8.1: Distribution of the Sample Mean

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. Section 1.1:
      • quantitative data
    2. Section 3.1:
      • mean
      • median
    3. Section 3.2:
      • standard deviation
      • variance
      • Empirical Rule (a.k.a., 68-95-99.7 Rule)
    4. Section 3.4:
      • z-score
      • percentiles
    5. Section 5.1:
      • probability
      • outcome
      • random event
      • probability rules
      • unusual event
    6. Section 5.2:
      • addition rule for probability
      • complement of an event, Ec
      • complement rule for probabilities
      • keywords for probability:
        1. or
        2. and [sometimes you will need to identify and even though word is not used]
        3. not
    7. Chapter 7:
      • all
  2. procedural knowledge
    1. Section 5.1:
      • verify probability models
    2. Section 5.2:
      • how to calculate the probability of an event using the addition rule
    3. Chapter 7:
      • all
  3. conditional knowledge
    1. Section 1.1:
      • how to identify the difference between qualitative data and quantitative data
      • how to identify the difference between discrete quantitative data and continuous quantitative data
    2. Section 5.1:
      • know when it is appropriate to use the concept of probability
      • know the importance of the concept of randomness or chance in probability
      • interpret value of probability
      • identify when an event is considered unusual
    3. Section 5.2:
      • how to determine which probability rule to use for a given problem
      • how to determine if two events are disjoint
      • how to identify the complement of an event
      • know that probability, proportion, and percentage are equivalent
    4. Chapter 7:
      • all
  4. important notes
    1. you are required to use the correct symbols for mean of sampling distribution of sample mean, μx-bar, in all answers to questions
    2. you are required to use the correct symbols for standard deviation of sampling distribution of sample mean, σx-bar, in all answers to questions

Learning Goals

  1. declarative knowledge (definitions)
    1. sampling distribution
    2. sampling distribution of the sample mean
    3. law of large numbers
    4. central limit theorem (CLT) for mean: Suppose a random variable X has any distribution with mean μ and standard deviation s. Then, as the sample size n is increased, the sampling distribution of the sample mean will become approximately normal with mean μ and standard deviation, σ/√n.
  2. procedural knowledge
    1. identify the sampling distribution of x bar, i.e., shape, center, and spread
    2. calculate probabilities based on normal curve using the empirical rule
    3. calculate mean, μx-bar, and standard deviation (a.k.a., standard error of the mean), σx-bar for a given sample size, n
    4. calculate probabilities based on sample means and the sampling distribution using the TI83/84
  3. conditional knowledge
    1. identify the effect sample size, n, has on the shape of the sampling distribution
    2. interpret the central limit theorem
    3. identify how to use normalcdf appropriately for each type of question: normal distributions, and sampling distributions
    4. identify when to use normalcdf for sampling distributions: random sample with n ≥ 30 or population is normally distributed
    5. understand that sample statistics vary from sample to sample and confidence intervals are based on this sampling distribution
    6. explain difference between standard deviation and standard error of the mean
    7. understand that confidence intervals are random quantities, varying from sample to sample. Sometimes the confidence interval includes the true population parameter and sometimes they do not. The probability the confidence interval includes the parameter is called the confidence level
    8. given σx-bar and n, find σ
    9. determine the effect sample size,n, has on the probability when using a sampling distribution [be careful]
    10. explain the trade-off between confidence level and reliability of the confidence interval

Section 8.2: Distribution of the Sample Proportion

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. Section 1.1:
      • qualitative data
    2. Section 3.1:
      • mean
      • median
    3. Section 3.2:
      • standard deviation
      • variance
      • Empirical Rule (a.k.a., 68-95-99.7 Rule)
    4. Section 3.4:
      • z-score
      • percentiles
    5. Section 5.1:
      • probability
      • outcome
      • random event
      • probability rules
      • unusual event
    6. Section 5.2:
      • addition rule for probability
      • complement of an event, Ec
      • complement rule for probabilities
      • keywords for probability:
        1. or
        2. and [sometimes you will need to identify and even though word is not used]
        3. not
    7. Chapter 7:
      • all
    8. Section 8.1:
      • sampling distribution
      • law of large numbers
      • central limit theorem (CLT)
  2. procedural knowledge
    1. Section 5.1:
      • verify probability models
    2. Section 5.2:
      • how to calculate the probability of an event using the addition rule
    3. Chapter 7:
      • all
    4. Section 8.1:
      • calculate probabilities based on normal curve using the empirical rule
  3. conditional knowledge
    1. Section 1.1:
      • how to identify the difference between qualitative data and quantitative data
      • how to identify the difference between discrete quantitative data and continuous quantitative data
    2. Section 5.1:
      • know when it is appropriate to use the concept of probability
      • know the importance of the concept of randomness or chance in probability
      • interpret value of probability
      • identify when an event is considered unusual
    3. Section 5.2:
      • how to determine which probability rule to use for a given problem
      • how to determine if two events are disjoint
      • how to identify the complement of an event
      • know that probability, proportion, and percentage are equivalent
    4. Chapter 7:
      • all
    5. Section 8.1:
      • identify the effect sample size, n, has on the shape of the sampling distribution
      • interpret the central limit theorem

Learning Goals

  1. declarative knowledge (definitions)
    1. sample proportion, p-hat
  2. procedural knowledge
    1. identify the sampling distribution of x bar, i.e., shape, center, and spread
    2. convert the number of successes, x, into proportion, x bar
    3. calculate mean, μp-hat, and standard deviation, σp-hat for a given sample size, n
    4. calculate probabilities based on sample proportions and the sampling distribution using the TI83/84 [make sure to convert the number of successes, x, into proportion, x bar]
  3. conditional knowledge
    1. identify how to use normalcdf appropriately for each type of probability question: normal distributions, sampling distribution of means, and sampling distributions of proportions
    2. identify when to use normalcdf to calculate a probability using the sampling distribution of proportions [random sample; and two conditions given below]
  4. important notes
    1. you are required to use the correct symbols for mean of sampling distribution of sample proportion, μp-hat, in all answers to questions
    2. you are required to use the correct symbols for standard deviation of sampling distribution of sample proportion, σp-hat, in all answers to questions

Chapter 8: Required Formulas – Need to Know for Tests

  1. Mean of the Sampling Distribution of x bar: μx bar = μ
  2. Standard Deviation of the Sampling Distribution of x bar [a.k.a., Standard Error of the Mean]: Standard Error of the Mean
  3. Standardizing a Normal Random Variable [a.k.a., z-Score for x bar]: Standard Error of the Mean
  4. Sample Proportion: Sample Proportion
  5. Mean of the Sampling Distribution of x bar: μx bar = p

Chapter 8: Required Formulas – Will be Given on Tests

  1. Standard Deviation of the Sampling Distribution of x bar: Standard Error of the Proportion
  2. One Condition Required for Sampling Distribution of x bar to be Approximately Normally-Distributed: p-hat conditions
  3. Another Condition Required to Construct a CI for p: n ≤ 0.05N OR 20n ≤ N