Transcript lesson18-sample n population distribution

Aim: How do we use sampling
distributions for proportions?
HW5: complete last slide
Recap Definitions
1. Variable: a characteristic or attribute that can assume
different values
2. Random Variable: a variable whose values are
determined by chance
3. Discrete Variable: have a finite number of possible
values or an infinite number of values tat can be
counted
4. Continuous Variable: variables that can assume all
values in the interval between any two given values
- they are obtained from data that can be measured
rather than counted
Statistical Inference
• Statistical inference draws conclusions about a
population or process on the basis of data
• Data is summarized by statistics such as means,
proportions, and slopes of regression lines
• When data are produced by random sampling, a
statistic is a random variable that obeys the laws
of probability theory
• Sampling distributions of statistics provides the
link between probability and data
Sample Distribution
• Shows how a statistics would vary in repeated
data production.
• It is a probability distribution that answers the
question “What would happen if we did this
many times?”
• A statistic from a random sample or
randomized experiment is a random variable.
The probability of the statistics is its sample
distribution.
Population Distribution
• The population distribution of a variable is
the distribution of its values for all members
of the population. The population distribution
is also the probability distribution of the
variable when we choose one individual at
random from the population.
Discrete Probability Distribution
• Discrete Probability Distribution: consists of
the values a random variable can assume and
the corresponding probabilities of the values.
The probabilities are determined theoretically
or by observation.
Example
• A sample survey asks 2000 college students
whether they think that parents put too much
pressure on their children. We would like to
view the responses of these students as
representative of a larger population of
students who hold similar beliefs. That is, we
will view the responses of the sampled
students as an SRS from a population.
SRS
• SRS = Stratified Random Sample
– First divide the population into groups of similar
individuals called strata
– Then choose a separate SRS in each stratum and
combine these SRSs to form the full sample
How do we read the example?
• When there are only two possible outcomes for a
random variable, we can summarize the results
by giving the count for one of the possible
outcomes.
• We let n represent the sample size and we use X
to represent the random variable that gives the
count for the outcome of interest.
– In our sample survey of college students, n = 2000 and
X is the number of students who think that parents
put too much pressure on their children.
• Suppose X = 840. The random variable of interest is X and its
value is 840.
Choosing the meaning of
the random value X
• In our example, we chose the random variable
X to be the number of students who think that
parents put too much pressure on their
children.
• We could have chosen X to be the number of
students who do not think that parents put
too much pressure on their children.
• The choice is yours!
Interpreting Random Value X
• To interpret the meaning of the random
variable X in this setting, we need to know the
sample size n.
• The conclusion we would draw about student
opinions in our survey would be quite
different if we had observed X = 840 from a
sample of size n = 1000.
Sample Proportion
• When a random variable has two possible
outcomes, we can use the sample
proportion,
as a summary.
X
p
n
• The sample proportion of students surveyed
who think that parents put too much pressure
on their children is
Example
• In a random sample of 150 seniors, 45 report
taking transportation to school.
• Give n, X and p for this setting
– Solution: n = 150
X = transportation 45
45
p
 .30
150
OR
n = 150
X = no transportation 105
105
p
 .70
150
Class Work #2
• Worksheet
Homework
1.
Explain what is wrong in each of the following scenarios.
A.
B.
C.
2.
If you toss a fair coin three times and a head appears each time, then the
next toss is more likely to be a tail than a head.
If you toss a fair coin three times and a head appears each time, then the
next toss is more likely to be a head than a tail.
Sample proportion is one of the parameters for a binomial distribution.
A poll of 1500 college students asked whether or not they had used the
Internet to find a place to live sometime within the past year. There were
525 students who answered “Yes”; the other 975 answered “No.”
A.
B.
C.
D.
What is n?
Choose one of the two possible outcomes to define the random variable, X.
Give a reason for your choice.
What is the value of X?
Find the sample proportion.