Exam Preparation Notes on Distribution of Sample Means

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Distribution of Sample Means (Chapter 7)

• Based on Chapter 7 in Gravetter & Wallnau.

Probability and Samples

• Initially, the discussion focuses on samples of size 1.
• Example: Given a normal distribution with $µ=68$ and $σ=6$, find the probability of selecting a person taller than 80 inches.
• Answer: Convert 80 inches to a z-score and find the proportion greater than that z-score in the unit normal table.
• P(X>80) = 0.0228 (for $z = 2.0$).
• Most research, however, involves samples with n>1.
• For $n = 1$, P(X>80) = 0.0228.
• For $n = 2$, P( (X1 + X2) / 2 > 80 ) = ???
• For $n = 100$, P( (X1 + X2 + ··· + X_{100}) / 100 > 80 ) = ???
• The probabilities are not equal; therefore, the single-score method must be modified for samples with n>1.
• Sampling error: Natural differences that exist by chance between a sample statistic and a population parameter.

Sampling Error: Review

• Population of UCSD students:
  • Population Parameters:
    • Average Age = 21.3 years
    • Average IQ = 112.5
    • 47% female, 48% male, 5% other
• Sample #1: Adam, Brad, Chelsea, Derrick, Elisa
  • Sample Statistics:
    • Average Age = 19.8
    • Average IQ = 104.6
    • 40% Female, 60% Male, 0% other
  • Sampling Error for #1:
    • 19.8 vs. 21.3 years
    • 104.6 vs. 112.5 IQ
    • 40 vs. 47% female
    • 60 vs. 48% male
    • 0 vs. 5% other
• Sample #2: Amy, Bryan, Chris, Deanna, Eric
  • Sample Statistics:
    • Average Age = 20.4
    • Average IQ = 114.2
    • 40% Female, 40% Male, 10% other
  • Sampling Error for #2:
    • 20.4 vs. 21.3 years
    • 114.2 vs. 112.5 IQ
    • 60 vs. 47% female
    • 40 vs. 48% male
    • 10 vs. 5% other

Probability and Samples

• Each independent sample from the population will exhibit some sampling error.
• Key questions:
  • How well does a sample (on average) represent the population from which it was drawn?
  • How likely is it that we draw a sample with particular characteristics?

Probability and Samples

• Detailed question: Given a population with a set $µ$ and $σ$, how likely is it to obtain a certain sample mean (M) when we take a sample of size n?
• Many possible samples can be obtained from a given population, each with different individuals, scores, and means.
• These possible samples form an orderly pattern: The Distribution of Sample Means (DSM).

Sample Means

• The distribution of sample means is the collection of the means of all the possible random samples of a particular size (n) that can be obtained from a population.
• This distribution differs from individual score distributions because it is composed of statistics (sample means), not individual scores.
• Referred to as a sampling distribution (or “Sampling distribution of M”).

Sample Means

• Example: A population of 4 scores: 2, 4, 6, 8. (X: 2, 4, 6, 8)

Sample Means

• Construct the distribution of sample means for samples of size $n = 2$.
• Population ($N = 4$): 2, 4, 6, 8
• Procedure:
  1. Write down all 16 possible samples and the sample mean (M) for each.
  2. Place all the obtained sample means in a frequency distribution and/or histogram.

Sample Values (n=2 from population 2, 4, 6, 8)

• Population: 2, 4, 6, 8
• Number of possible samples = $N^n = 4^2 = 16$
• Shows a table of all 16 samples, first score, second score, and sample mean.

Sample Means

• Things to note about the distribution:
  1. Mean of sample means = mean of population.
  2. Shape looks (sort of) normal.
  3. This distribution can be used to answer questions about probabilities of sample means.
• $µ = 5$

Sample Means

• We can use this distribution to answer questions about probabilities of sample means.
• If you take a sample of $n = 2$ scores from the original population, what is the probability of obtaining a sample mean greater than 6?
• In symbols: p(M > 6) = ?
• Probability = 3/16 = 0.1875 (3 of the 16 possible sample means are greater than 6)

Central Limit Theorem

• What about situations with larger populations and larger samples where calculating all possible sample means is unrealistic?
• Use the Central Limit Theorem:
• For any population with mean $µ$ and standard deviation $σ$, the distribution of sample means for sample size n will have a mean of $µ$, a standard deviation of $σ / </li> </ul> <p>√n$, and will approach a normal distribution as n approaches infinity.

  Central Limit Theorem

  • In table form:
    • Original Population (OP)
    • Distribution of Sample Means (DSM)
    • Sample Size (n)
    • | | OP | DSM |
    • | :---- | :----- | :----------- |
    • | Mean | $µ$ | $µ_M = µ$ |
    • | S.D. | $σ$ | $σ_M = σ / √n$ |
    • | Shape | any | normal if n >=30 or normal always normal |

  Central Limit Theorem: Mean of DSM

  • Mean of the distribution of sample means is $µ_M$ and always has a value equal to the mean of the population of scores, $µ$.
  • Mean of the distribution of sample means ($µ_M$) is called the expected value of M.
  • M is an unbiased statistic because $µ_M$, the expected value of M, is equal to the population mean, $µ$.

  Central Limit Theorem: S.D. of DSM

  • Variability of a distribution of scores is measured by the standard deviation ($σ$).
  • Variability of a distribution of sample means is measured by the standard deviation of the sample means, and is called the standard error of M and written as $σ_M$.
  • In journal articles or other textbooks, the standard error of M might be identified as “standard error,” “SE,” or “SEM”.

  Central Limit Theorem

  • Standard deviation: standard distance between a score X and the population mean $µ$.
  • Standard error: standard distance between a sample mean M and the mean of the distribution of sample means $µ_M$.

  Standard Error: $σ_M = σ / √n$

  • Magnitude determined by two factors.
    1. Size of sample
       • Law of large numbers: as the sample size increases, the error between the sample mean and the population mean should decrease.
    2. Population standard deviation:
       • Standard deviation is “starting point” for standard error.
       • $n=1: σ_M = σ$
       • n>1: σ_M < σ
       • The smaller the population variance (S.D.), the less error between M and $µ$.

  “Law of Large Numbers”

  • The larger a sample, the better its mean approximates the mean of the population (and thus the smaller $σ_M$ will be).
  • For a normal population with $µ = 10$, $σ = 5$, take 100 samples of size n, and plot the means of those 100 samples.

  Relationship between S.E. and n

  • Standard Error as a function of sample size.
  • The graph illustrates how standard error decreases as sample size (n) increases, given $σ = 10$.
  • Standard distance between a sample mean and the population mean.

  Population variance

  • The smaller the population variance, the better the sample mean (M) approximates the population mean ($µ$) (and thus the smaller $σ_M$ will be).

  Central Limit Theorem: Shape of DSM

  • The DSM is almost perfectly normal if either of the following two conditions are satisfied:
    • The population from which the samples are drawn is normal.
    • OR
    • The number scores in each sample (n) is 30 or more.

  Central Limit Theorem Proof: DSM Shape

  • Illustrates how the shape of the distribution of sample means (DSM) approaches a normal distribution as the sample size (n) increases.
  • Several graphs demonstrate the transition from non-normal to normal distributions as 'n' goes up.

  Central Limit Theorem Proof: DSM Shape

  • Illustrates how the shape of the distribution of sample means (DSM) approaches a normal distribution as the sample size (n) increases for different populations.
  • Several graphs demonstrate the transition from non-normal to normal distributions as 'n' goes up for different populations and their corresponding DSM.

  Central Limit Theorem Proof: DSM Shape

  • The mean of each sampling distribution is equal to 0.94
  • Population Parameters given by: $μ = 0.94$ and $σ = 0.05$
  • The standard deviation of each DSM gets smaller as n gets larger
    • When n = 2 : $μ<em>x = 0.94$ and $σ</em>x = 0.038$
    • When n = 4 : $μ<em>x = 0.94$ and $σ</em>x = 0.027$
    • When n = 16 : $μ<em>x = 0.94$ and $σ</em>x = 0.013$
    • When n = 64 : $μ<em>x = 0.94$ and $σ</em>x = 0.006$

  The Distribution Triad: Population, Single Sample, and DSM

  • (a) Original population of IQ scores.
    • $μ = 100$
    • $σ = 15$
  • (b) A sample of n = 25 IQ scores.
    • $M = 101.2$
    • $S=11.5$
  • (c) The distribution of sample means. Sample means for all the possible random samples of n = 25 IQ scores.

  Interactive learning!

  • https://shiny.rit.albany.edu/stat/sampdist/

  Learning Check

  • Question 1: A population of unknown shape has a mean of $μ = 60$ with $σ = 5$. The mean of the distribution of sample means for samples of size $n = 4$ selected from this population would have a value of .
    • A) 5
    • B) 15
    • C) 30
    • D) 60
  • Question 2: A population of unknown shape has a mean of $μ = 60$ with $σ = 5$. The distribution of sample means for samples of size $n = 4$ selected from this population would have a standard deviation of .
    • A) 1.25
    • B) 2.5
    • C) 5
    • D) 15
  • Question 3: A population of unknown shape has a mean of $μ = 60$ with $σ = 5$. The shape of the distribution of sample means for samples of size $n = 4$ selected from this population would be .
    • A) Normal
    • B) Positively Skewed
    • C) Negatively Skewed
    • D) Cannot determine from the information given
  • Use the password provided by Dr. Lowe to fill out the Canvas survey: Chapter 7 Question Set 1
  • Password: quentin

  Central Limit Theorem: Review

  • Original Population = OP
  • Distribution of Sample Means = DSM
  • Sample Size = n
  • | | OP | DSM |
  • | :---- | :----- | :----------- |
  • | Mean | $µ$ | $µ_M = µ$ |
  • | S.D. | $σ$ | $σ_M = σ / √n$ |
  • | Shape | any | normal if n >=30 or normal always normal |
  • Put this information on your cheat sheet!!!!!

  Probability and the Distribution of Sample Means

  • We can use the distribution of sample means to find out probabilities (= proportions!).
  • For example: Given a population, how likely is it to obtain a sample of size n with a certain M?

  Probability and the Sample Means

  • Single score vs Sample mean
  • $z = (X - µ) / σ$
    • X = score
    • µ = population mean
    • $σ$ = Standard Dev.
    • Interpretation: Given a population, how likely is it to obtain a score with a certain value X?
    • Or…proportion of individuals within our population with a score of X.
  • $z = (M - µ<em>M) / σ</em>M$
    • M = sample mean
    • $µ_M$ = mean of DSM ( = µ)
    • $σ_M$ = Standard Error
    • Interpretation: Given a population, how likely is it to obtain a sample of size n with a certain M?
    • Or…proportion of samples (with size n) with that mean out of all the total possible samples (with size n) from our population.

  Probability and the Sample Means

  • Example:
    • SAT-scores (normal, $μ=500, σ=100$).
    • Take a sample n=25.
    • What is p(M>540)?

  Example:

  • SAT scores are normally distributed and have $µ = 500, σ =100$
  • Probability of drawing one score > 540?
  • Probability of drawing one sample mean > 540? when n = 36
  • Probability of drawing one sample mean > 540? when n = 100

  Probability and the Sample Means

  • Another Example: SAT-scores (normal, $µ=500, σ=100$). Take sample $n=25$. What range of values for M can be expected 80% of the time (in other words, what are the boundaries of the middle 80%)?

  Review: Sampling Error

  • There will almost always be discrepancy between a sample mean and the true population mean
  • This discrepancy is called sampling error
  • The amount of sampling error varies across samples
  • The variability of sampling error is measured by the standard error of the mean

  Review: Standard Error & n

  • The standard error tells us how much error, on average, should exist between a sample mean and the population mean.
  • As the sample size n increases, the standard error decreases.

  Importance of large sample sizes

  • A meme is presented to demonstrate the importance of sample size.

  In the Literature

  • Journals vary in how they refer to the standard error but frequently use:
    • “SE”
    • “SEM”
  • Often reported in a table along with n and M for the different groups in the experiment
  • Example table:
    • | Group | n | Mean | SE |
    • | :---- | :- | :---- | :--- |
    • | A | 17 | 32.23 | 2.31 |
    • | B | 15 | 45.17 | 2.78 |

  In the Literature

  • Graphs often include error bars representing standard error.
  • Example: Mscore (±SE), M number of mistakes (±SE)

  Using the Standard Error

  • The standard error can help us decide which of the two alternatives is more likely.
  • Imagine an experiment:
    • Difference between sample and population:
      • due to treatment?
      • due to sampling error?

  Using the Standard Error

  • 95% of all the possible sample means (from the untreated population!) for n = 25 fall between 392.16 and 407.84.
  • Suppose we take a sample of n = 25 rats and obtain M = 404. Did the growth hormone work?
  • M = 404

  Using the Standard Error

  • 95% of all the possible sample means (from the untreated population!) for n = 25 fall between 392.16 and 407.84.
  • Suppose we take a sample of n = 25 rats and obtain M = 409. Did the growth hormone work?
  • M = 409

  Learning Check

  • Question 1: A population forms a normal distribution with $μ = 80$ and $σ = 20$. If a single score is selected from this population, how much distance, on average, would you expect between the score and the population mean?
    • A) 0
    • B) 5
    • C) 10
    • D) 20
    • E) 80
  • Question 2: A population forms a normal distribution with $μ = 80$ and $σ = 20$. If a sample of $n = 4$ is selected from this population, how much distance, on average, would you expect between the sample mean and the population mean?
    • A) 0
    • B) 5
    • C) 10
    • D) 20
    • E) 80
  • Use the password provided by Dr. Lowe to fill out the Canvas survey: Chapter 7 Question Set 2
  • Password: gone2

  More Examples…

  • For a population mean of $µ = 70$ and a standard deviation of $σ = 20$, how much error, on average, would you expect between the sample mean (M) and the population mean for each of the following sample sizes?
    • n = 4
    • n = 16
    • n = 25

  More Examples…

  • For a population with $σ = 12$, how large a sample is necessary to have a standard error that is:
    • less than 4 points?
    • less than 3 points?
    • less than 2 points?

  More Examples…

  • A normal distribution has a mean of $µ = 54$ and a standard deviation of $σ = 6$.
    • What is the probability of randomly selecting a score less than $X = 51$?
    • What is the probability of selecting a sample of $n = 4$ scores with a mean less than $M = 51$?
    • What is the probability of selecting a sample of $n = 36$ scores with a mean less than $M = 51$?