Transcript 4.6 -4.8 PowerPoint

Sta220 - Statistics
Mr. Smith
Room 310
Class #14
Section 4.6 and 4.8
Section 4.6
We learn how to make inferences about the
population on the basis of information
contained in the sample. Several of these
techniques are based on the assumption that
the population is approximately normally
distributed. It will be important to determine
whether the sample of data come from a normal
population before we can apply these
techniques properly.
Procedure
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Definition
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Example 4.24
The EPA mileage ratings on 100 cars are
reproduced in the following table. Numerical
and graphical descriptive measures for the data
are shown on the StatCrunch and SPSS printouts
presented in Figure 4.26. Determine whether
the EPA mileage ratings are from an
approximate normal distribution.
Table 4.6
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Summary statistics:
Column
MPG
n
Mean
100
Variance Std. dev. Std. err. Median
36.994 5.84622 2.41789 0.24178
63
71
971
37
Range
14.9
Min
30
Max
44.9
Q1
Q3
35.65
38.35
#1: Histogram or Stem-and-leaf Display
• Clearly, the mileages fall
into an approximately
mound shaped,
symmetric distribution
centered around the
mean of about 37 mpg.
• Therefore, check #1 in
the box indicates that
the data are
approximately normal.
#2: Compute the Intervals
• We obtain 𝑥= 37 and s =
2.4 from the summary
from StatCrunch. The
intervals are shown in
Table 5.3 as is the
percentage of mileage
ratings that fall into
each interval.
• These percentages
agree almost exactly
with those from a
normal distribution.
#3: Ratio IQR/s
• From the Summary
Statistics, the 25th
percentile (labeled Q1)
is 𝑄𝐿 = 35.65 and 75th
percentile (labeled
Q3)is 𝑄𝑈 = 38.35.
• Then IQR = 2.7 and the
ratio is
𝐼𝑄𝑅
2.7
=
= 1.13
𝑠
2.4
• Since the value is
approximately equal to
1.3, we have further
confirmation that the
data are approximately
normal.
SPSS normal probability plot
for gas mileage data
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• An SPSS normal probability
plot of the mileage data is
shown in Figure 4.26.
• Notice that the ordered
mileage values fall
reasonably close to a
straight line when plotted
against the expected values
from a normal distribution.
• These suggest that EPA
mileage data are
approximately normally
distributed.
Conclusion
The checks for normality are simple, yet
powerful, techniques to apply, but they are only
descriptive in nature. Thus, we should be
careful not to claim that the 100 EPA mileage
ratings are, in fact, normally distributed. We
can only stat that it is reasonable to believe that
the data are from a normal distribution.
Section 4.8
In previous sections, we assumed that we knew
the probability distribution of a random
variable, and using this knowledge, we were
able to compute the mean, variance, and
probabilities associated with the random
variable. However, in most practical
applications, the true mean and standard
deviation are unknown quantities that have to
be estimated.
Definition
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We will often use the information contained in
these sample statistics to make inferences
about the parameters of a population.
Table 4.8
• Note that the term statistic refers to sample
quantity and the term parameter refers to a
population quantity.
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Before being able to use the sample statistics to
make inferences about population parameters, we
need to be able to evaluate their properties.
Does one sample statistic contain more information
than another about a population parameter?
On what basis should we choose the ‘best’ statistic
for making inferences about a parameter?
For example, if we wanted to estimate a
parameter of a population– say, the population
mean 𝜇 – we can use a number of sample
statistics for our estimate. Two possibilities are
the sample mean 𝑥 and the sample Median M.
Which of these do you think will provide a
better estimate of 𝜇?
Lets consider the following example:
Toss a fair die and let x equal the number of dots
showing on the up face.
Suppose the die is tossed three times, producing
the sample measurements 2, 2, 6. The sample
mean is 𝑥 = 3.33, and the sample median is M =
2.
Since the population mean is 𝜇 = 3.5, you can see
that, for this sample of three measurements, the
sample mean 𝑥 provides an estimate that falls
closer to 𝜇 than does the sample median M.
Now suppose we toss the die three more times
and obtain the sample measurements 3, 4, 6.
Then the mean and median of this ample are 𝑥 =
4.33 and M = 4, respectively. This time M is
closer to 𝜇.
This illustrates an important point: Neither the
sample mean or the sample median will always fall
closer to the population mean. We cannot compare
these two sample statistics or , in general, any two
sample statistics on the basis of their performance
with a single sample.
We recognize that sample statistics are themselves
random variables, because different samples can
lead to different values for a sample statistics.
Last, as random variables, sample statistics must
be judged and compared on the basis of their
probability distribution.
This means the collection of values and
associated probabilities of each statistics that
would be obtained if the sampling experiment
were repeated a VERY LARGE NUMBER OF TIME.
Definition
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In actual practice, the sampling distribution of
statistic is obtained mathematically or (at least
approximately) by simulating the sample on a
computer, using a procedure similar to that just
described.
Say that you have two statistics, A and B, for
estimating the same parameter and the following
graph below represents their sampling distribution.
Which would you prefer and why?
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Remember that, in practice, we will not know
the numerical value of the unknown parameter
𝜎 2 , so we will not know whether statistic A or
statistic B is closer to 𝜎 2 for a particular sample.
Example 4.26
Consider the popular casino game of craps, in which a
player throws two dice and bets on the outcome (the sum
total of the dots showing on the upper faces of the two
dice). Let’s say that if the sum total of the die is 7 or 11, the
roller wins $5; if the total is 2, 3, or 12, the roller loses $5;
and for any other total (4, 5, 6, 8, 8, 9, or 10) no money is
lost or won on the roll. Let x represent the result of the
come-out roll wager (-$5, $0, or +$5). The following table is
the actual probability distribution of x is:
Outcome of
Wager, x
-5
0
5
p(x)
1/9
6/9
2/9
Now, consider a random sample of n = 3 comeout rolls.
a. Find the sampling distribution of the same
mean, 𝑥
b. Find the sampling distribution of the same
median, M.
c. Then use the sampling distribution for 𝑥 to
find the expected value of 𝑥.
Table 6.2
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a. From the table, you can see that 𝑥 can
assume the values -5, -3.33, -1.67, 0, 1.67,
3.33 and 5. Because 𝑥 = -5 occurs in one
sample, P(𝑥 = -5) = 1/729 ≈ .0014.
Calculating the probabilities of the remaining
values of 𝑥 and arranging them in a table, we
obtain the following probability distribution.
𝑥
p(𝑥)
-5
-3.33
1/729 ≈ 18/729
.0014
≈.0247
-1.67
0
1.67
3.33
114/729 288/729 228/729 72/729
≈.1564 ≈.3951 ≈.3127 ≈.0988
5
8/729
≈.0110
This is the sampling distribution for 𝑥 because it
specifies the probability associated with each
possible value of 𝑥. You can see that the mostly
likely mean outcome after 3 randomly s3lected
come-out rolls is 𝑥 = $0; this result occurs with
probability .3951
b. From the table, you can see that 𝑀 can
assume the values -5, -0, and 5. Because 𝑀 = -5
occurs in seven samples, P(M= -5) = 25/729 ≈
.0343.
Calculating the probabilities of the remaining
values of 𝑀 and arranging them in a table, we
obtain the following probability distribution.
𝑥
p(𝑥)
-5
0
5
25/729 ≈
.0343
612/729≈ 92/729
.8395
≈.1262
Once again, the most likely median outcome
after 3 randomly selected come-out rolls M = $0,
a result that occurs with probability .8395.
c. The expected value E(𝑥) =
. 5558
Though the following example demonstrates the
procedure for finding the exact sampling distribution of a
statistic when the number of different samples that could
be selected from the population is relative small. In the
real world, populations often consist of large number of
different values, making samples difficult to count.
When this occurs, we choose to obtain the approximate
sampling distribution for a statistic by simulating the
sampling over and over again and recording the
proportion of times different values of the statistic occur.
4.8 Homework due Wednesday
4.9 Notes on Monday
Chapter 4 Test Next Thursday