Download Sample Statistics are used to estimate Population Parameters

Sample Statistics are used to estimate
Population Parameters
We will use two different statistics:
For categorical data: the sample
proportion, p
For numeric data: the sample mean
Sample Statistics

Have sampling distributions
• Shape: normal if you follow the ‘rules’
• Center: equal to the parameter we’re
•
estimating if we take a random sample
Spread: related to the population standard
deviations by a factor of 1/n
Sample Proportion, p
• Shape: Normal if n and n(1-) ≥10
• Center: (p) = 
• Spread:  (p) =(1-)/n
Example for sample proportions:

Toss a coin 30 times. The probability of
getting a head is 80%.
• This is a binomial trial because:
• Each toss is independent of all of the other tosses
• There is a fixed number of tosses, n = 30
• There is a fixed probability of success,  = 0.80
Example con’t

What is the distribution of our sample proportion?
•
•
•
Shape: Normal if n and n(1-) ≥10
• n = 30*0.80 = 24,
• n(1-) = 30*0.20 = 6,
• so we can’t say the shape is normal
Center: (p) = 
• (p) =  = 0.80
Spread: (p) =(1-)/n
• (p) =(1-)/n = (0.8*0.2/30) = 0.073
Sample Mean,
• Shape:
Normal if
• the original data is normal, X~N(x, x2), or
• n is large, at least 30
• Center: (
• Spread: (
) = x
) =/n
Example of a sample mean,

If X~N(15, 22), what is the distribution of X 36 ?
• Since X is normal X 36 is also normal
• The mean is the same, 15
• The standard deviation is 2/36 = 1/3
• So, X 36~ N( 15, (1/3)2)