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STA 291 Summer 2010. Lecture 10 Dustin Lueker. z-Scores. The z-score for a value x of a random variable is the number of standard deviations that x is above μ If x is below μ, then the z-score is negative The z-score is used to compare values from different normal distributions

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sta 291 summer 2010

STA 291Summer 2010

Lecture 10

Dustin Lueker

z scores
z-Scores
  • The z-score for a value x of a random variable is the number of standard deviations that x is above μ
    • If x is below μ, then the z-score is negative
  • The z-score is used to compare values from different normal distributions
  • Calculating
    • Need to know
      • x
      • μ
      • σ

STA 291 Summer 2010 Lecture 10

z scores1
z-Scores
  • The z-score is used to compare values from different normal distributions
    • SAT
      • μ=500
      • σ=100
    • ACT
      • μ=18
      • σ=6
    • What is better, 650 on the SAT or 25 on the ACT?
      • Corresponding tail probabilities?
        • How many percent have worse SAT or ACT scores?

STA 291 Summer 2010 Lecture 10

sampling distribution
Sampling Distribution
  • Probability distribution that determines probabilities of the possible values of a sample statistic
    • Example
      • Sample Mean ( )
  • Repeatedly taking random samples and calculating the sample mean each time, the distribution of the sample mean follows a pattern
    • This pattern is the sampling distribution

STA 291 Summer 2010 Lecture 10

example
Example
  • If we randomly choose a student from a STA 291 class, then with about 0.5 probability, he/she is majoring in Arts & Sciences or Business & Economics
    • We can take a random sample and find the sample proportion of AS/BE students
    • Define a variable X where
      • X=1 if the student is in AS/BE
      • X=0 otherwise

STA 291 Summer 2010 Lecture 10

example1
Example
  • If we take a sample of size n=4, the following 16 samples are possible:

(1,1,1,1); (1,1,1,0); (1,1,0,1); (1,0,1,1);

(0,1,1,1); (1,1,0,0); (1,0,1,0); (1,0,0,1);

(0,1,1,0); (0,1,0,1); (0,0,1,1); (1,0,0,0);

(0,1,0,0); (0,0,1,0); (0,0,0,1); (0,0,0,0)

  • Each of these 16 samples is equally likely because the probability of being in AS/BE is 50% in this class

STA 291 Summer 2010 Lecture 10

example2
Example
  • We want to find the sampling distribution of the statistic “sample proportion of students in AS/BE”
    • “sample proportion” is a special case of the “sample mean”
  • The possible sample proportions are

0/4=0, 1/4=0.25, 2/4=0.5, 3/4 =0.75, 4/4=1

  • How likely are these different proportions?
    • This is the sampling distribution of the statistic “sample proportion”

STA 291 Summer 2010 Lecture 10

example3
Example

STA 291 Summer 2010 Lecture 10

example computer simulation
ExampleComputer Simulation

STA 291 Summer 2010 Lecture 10

sampling distribution example contd computer simulation
Sampling Distribution: Example (contd.), Computer Simulation

10 samples

of size n=4

STA 291 Summer 2010 Lecture 10

sampling distribution example contd computer simulation1
Sampling Distribution: Example (contd.), Computer Simulation

100 samples

of size n=4

STA 291 Summer 2010 Lecture 10

sampling distribution example contd computer simulation2
Sampling Distribution: Example (contd.), Computer Simulation

1000 samples

of size n=4

STA 291 Summer 2010 Lecture 10

computer simulations
Computer Simulations

10 samples

of size n=4

100 samples

of size n=4

1000 samples

of size n=4

Probability Distribution of the Sample Mean

STA 291 Summer 2010 Lecture 10

computer simulation
Computer Simulation
  • A new simulation would lead to different results
    • Reasoning why is same as to why different samples lead to different results
      • Simulation is merely more samples
  • However, the more samples we simulate, the closer the relative frequency distribution gets to the probability distribution (sampling distribution)
      • Note: In the different simulations so far, we have only taken more samples of the same sample size n=4, we have not (yet) changed n.

STA 291 Summer 2010 Lecture 10

reduce sampling variability
Reduce Sampling Variability
  • The larger the sample size, the smaller the sampling variability
  • Increasing the sample size to 25…

10 samples

of size n=25

100 samples

of size n=25

1000 samples

of size n=25

STA 291 Summer 2010 Lecture 10

interpretation
Interpretation
  • If you take samples of size n=4, it may happen that nobody in the sample is in AS/BE
  • If you take larger samples (n=25), it is highly unlikely that nobody in the sample is in AS/BE
  • The sampling distribution is more concentrated around its mean
  • The mean of the sampling distribution is the population mean
    • In this case, it is 0.5

STA 291 Summer 2010 Lecture 10

using the sampling distribution
Using the Sampling Distribution
  • In practice, you only take one sample
  • The knowledge about the sampling distribution helps to determine whether the result from the sample is reasonable given the model
  • For example, our model was

P(randomly selected student is in AS/BE)=0.5

    • If the sample mean is very unreasonable given the model, then the model is probably wrong

STA 291 Summer 2010 Lecture 10

effect of sample size
Effect of Sample Size
  • The larger the sample size n, the smaller the standard deviation of the sampling distribution for the sample mean
    • Larger sample size = better precision
  • As the sample size grows, the sampling distribution of the sample mean approaches a normal distribution
    • Usually, for about n=30, the sampling distribution is close to normal
    • This is called the “Central Limit Theorem”

STA 291 Summer 2010 Lecture 10