Parameters and Statistics

1
Chapter 11

• Sampling Distributions

2
Parameters and Statistics

• Parameter a constant that describes a
  population or probability model, e.g., µ from a
  Normal distribution
• Statistic a random variable calculated from a
  sample e.g., x-bar
• These are related but are not the same!
• For example, the average age of the SJSU student
  population µ 23.5 (parameter), but the average
  age in any sample x-bar (statistic) is going to
  differ from µ

3
Example Does This Wine Smell Bad?

• Dimethyl sulfide (DMS) is present in wine causing
  off-odors
• Let X represent the threshold at which a person
  can smell DMS
• X varies according to a Normal distribution with
  µ 25 and s 7 (µg/L)

4
Law of Large Numbers
This figure shows results from an experiment
that demonstrates the law of large numbers (will
be discussed in class)
5
Sampling Distributions of Statistics

• The sampling distribution of a statistic predicts
  the behavior of the statistic in the long run
• The next slide show a simulated sampling
  distribution of mean from a population that has
  XN(25, 7). We take 1,000 samples, each of n 10,
  from population, calculate x-bar in each sample
  and plot.

6
Simulation of a Sampling Distribution of xbar
7
µ and s of x-bar
8
Sampling Distribution of Mean Wine tasting example
Population XN(25,7) Sample n 10 By sq. root
law, sxbar 7 / v10 2.21 By unbiased
property, center of distribution
µThusx-barN(25, 2.21)
9
Illustration of Sampling Distribution Does this
wine taste bad?

• What proportion of samples based on n 10 will
  have a mean less than 20?
• State Pr(x-bar 20) ?Recall x-barN(25,
  2.21) when n 10
• Standardize z (20 25) / 2.21 -2.26
• Sketch and shade
• Table A Pr(Z lt 2.26) .0119

10
Central Limit Theorem
No matter the shape of the population, the
distribution of x-bars tends toward Normality
11
Central Limit Theorem Time to Complete Activity
Example
Let X time to perform an activity. X has µ 1
s 1 but is NOT Normal
12
Central Limit Theorem Time to Complete Activity
Example

• These figures illustrate the sampling
  distributions of x-bars based on
• n 1
• n 10
• n 20
• n 70

13
Central Limit Theorem Time to Complete Activity
Example

• The variable X is NOT Normal, but the sampling
  distribution of x-bar based on n 70 is Normal
  with µx-bar 1 and sx-bar 1 / sqrt(70) 0.12,
  i.e., xbarN(1,0.12)
• What proportion of x-bars will be less than 0.83
  hours?
• (A) State Pr(xbar lt 0.83)
• (B) Standardize z (0.83 1) / 0.12 -1.42
• (C) Sketch right
• (D) Pr(Z lt -1.42) 0.0778