Chapter 9.1 Sampling Distributions
Learning Objectives
Know the difference between a statistic and a parameter
Understand that the value of a statistic varies between samples
Be able to describe the shape, center and spread of a given sampling distribution
Understand how bias and variability of a statistic affects the sampling distribution
Parameter -
Statistic Examples
Is the boldfaced number a parameter or a statistic?
1. 60,000 members of the labor force were interviewed of whom 7.2% were unemployed
2. A lot of ball bearings has a mean diameter of 2.5003 cm. A 100 bearings are selected from
the
lot and have a mean diameter of 2.5009 cm.
3. A telemarketing firm in Los Angeles randomly dials telephone numbers. Of the first 100
numbers dialed 48% are unlisted. This is not surprising because 52% of all Los Angeles
residential phones are unlisted.
Sampling Variability
Sample proportion: p̂
Example
A poll found that 1650 out of 2500 randomly selected adults agreed with the statement that
shopping is frustrating. What is the proportion of the sample who agreed?
Sampling variability -
Applet: http://www.rossmanchance.com/applets/Reeses/ReesesPieces.html
With a sample size of 25, I repeatedly
Generated samples.
What is the mean of the distributuion?
What is the spread of the
entire distribution?
With a sample size of 100, I repeatedly
Generated samples.
What is the mean of the distributuion?
What is the spread of the
entire distribution?
So what do we notice:
1-
2-
Sampling Distribution
Describing Sampling Distributions
1.
2.
3.
4.
Unbiased Statistic - a statistic is unbiased if the mean of the sampling distribution is _________
to the true value of the parameter being estimated
Variability of a statistic
1. Is described by the spread of its sampling distribution.
2. This spread is determined by the sampling design and the size of the sample.
3. Larger samples give smaller spread.
High bias; low variability
Low bias; high variability
High bias; high variability
Low bias; low variability
Formulas we need: To calculate the standard deviation for proportion
𝑝(1 − 𝑝)
𝜎=√
𝑛
Example
60% of people find clothes shopping frustrating.
Find the proportion of people that fall within 2 standard deviations of the mean for samples of
size
n =100
Step 1:
Step 2:
n = 2500
Step 1:
Step 2:
Why does the size of the population have little influence on how statistics from a random
sample behave?
Ch 9.2 Sample Proportions
Learning Objectives
Know the characteristics of the sampling distribution of p̂
Know when to use the normal approximation for p̂
Be able to solve problems using the normal approximation for p̂
Sampling distribution of p̂
Choose an SRS of size n from a large population, then:
1.
2.
3.
In order to analyze data ,we have to make some assumptions before we begin our
calculations!
Assumption 1
-The data was taken from a random sample
Assumption 2
-The standard deviation _________ for ______ can only be used when the population is at
least 10 times as large as the sample
Assumption 3
-We can say that the sampling distribution of p̂ is approximately normal when ___________
and ____________.
Example
There are 1.7 million first-year college students of those, 1500 first-year college students are
asked whether they applied for admission to any other college. In fact 35% of all first-year
students applied to a college other than the one they are attending. What is the probability that
your sample will give a result within 2 percentage points of this true value?
Step 1: Check your assumptions
Step 2: Calculate σ ,write in terms of the problem, convert to z-score, draw curve…..
Complete 9.15 pg. 477