Download Chapter 9 Day 2

Chapter 9
Day 2
Warm-up
 If
students picked numbers completely at
random from the numbers 1 to 20, the
proportion of times that the number 7
would be picked is 0.05. When 15 students
picked a number “at random” from 1 to
20, 3 of them picked the number 7.
Identify the parameter and
accompanying statistic in this situation.
Homework Solutions
1.
parameter: μ = 2.5003; statistic: 𝑥 = 2.5009
2.
Statistic: 𝑝 = 7.2%
3.
Parameter: ρ = 52%; statistic: 𝑝 = 48%
4.
Statistic for both: 𝑥 = 335 and 𝑥 = 289
5.
We will finish this together!
Example



Let’s simulate drawing simple random
samples from the population of all U.S.
residents. Suppose that in fact 60% of the
population find clothes shopping timeconsuming and frustrating. Then the true
value of the parameter we want to estimate
is ρ = .6
We can simulate the population by using our
calculators letting 0 to 5 stand for people who
find shopping frustrating and 6 to 9 stand for
those who do not.
Use your calculator to find a simple random
sample of 50 people and determine the
statistic 𝑝. Repeat 10 times.

Make a histogram of the 10 values of 𝑝 you
found.

Describe the overall shape of the distribution.

Now, let’s use the class’ data to find 100 SRSs.

We will make the histogram together.

Describe the overall shape of the distribution.
How does this compare to your earlier
histogram?
The Bias of a Statistic
 The
statistic used to estimate a parameter
is unbiased if the mean of its sampling
distribution is equal to the true value of
the parameter being estimated.
 Find
the mean of the 100 observations of
𝑝. Mark the mean on the histogram to
show its center. Does the statistic 𝑝
appear to have a large or small bias as
an estimate of the population proportion?
The variability of the statistic
 The
variability of a statistic is described by
the spread of its sampling distribution. This
spread is determined by the sampling
design and size of the sample. Larger
samples give smaller spread.
The variability of the statistic
 As
long as the population is much larger
than the sample (say, at least 10 times as
large), the spread of the sampling
distribution is approximately the same for
any population size.
 So in other words... A scoop of 50 M&Ms
would have the same distribution
regardless of if it was a sample for a
truckload of M&Ms or a bucket of M&Ms
Bias vs. Variability
RESULTS - Summary
 If
n is “large” enough (n is approaching N,
n is getting bigger)
 Then three things occur…



1. The shape of the distribution become
approximately normal (bell shaped)
2. The mean of the sampling distribution
equals the mean of the population.
3. The standard deviation of the sampling
distribution will become smaller
(determined by your choice of n)