Chapter 7 The Normal Probability Distribution

1
Chapter 7The Normal Probability Distribution

• 7.5
• Sampling Distributions
• The Central Limit Theorem

2
(No Transcript)
3
(No Transcript)
4
Illustrating Sampling Distributions Step 1
Obtain a simple random sample of size n.
5
Illustrating Sampling Distributions Step 1
Obtain a simple random sample of size n. Step 2
Compute the sample mean.
6
Illustrating Sampling Distributions Step 1
Obtain a simple random sample of size n. Step 2
Compute the sample mean. Step 3 Assuming we are
sampling from a finite population, repeat Steps 1
and 2 until all simple random samples of size n
have been obtained.
7
EXAMPLE Illustrating a Sampling
Distribution The following data represents the
ages of 6 individuals that are members of a
weekend golf club. 23, 25, 49, 32, 38, 43 Treat
these 7 individuals as a population. (a) Compute
the population mean. (b) List all possible
samples of size n 2 and determine the sample
mean age. (c) Construct a relative frequency
distribution of the sample means. This
distribution represents the sampling distribution
of the sample mean.
8
EXAMPLE Illustrating a Sampling
Distribution The weights of pennies minted after
1982 are approximately normally distributed with
mean 2.46 grams and standard deviation 0.02
grams. Approximate the sampling distribution of
the sample mean by obtaining 200 simple random
samples of size n 5 from this population.
9
The data on the following slide represent the
sample means for the 200 simple random samples of
size n 5. For example, the first sample of n
5 had the following data 2.493 2.466
2.473 2.492 2.471
10
(No Transcript)
11
The mean of the sample means is 2.46, the same as
the mean of the population. The standard
deviation of the sample means is 0.0086, which is
smaller than the standard deviation of the
population. The next slide shows the histogram of
the sample means.
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
EXAMPLE Finding the Area Under a Normal
Curve The weights of pennies minted after 1982
are approximately normally distributed with mean
2.46 grams and standard deviation 0.02 grams
Approximate the sampling distribution of the
sample mean by obtaining 200 simple random
samples of size n 15 from this population of
pennies minted after 1982.
17
(No Transcript)
18
The mean of the 200 sample means is 2.46 (the
mean of the population). The standard deviation
of the 200 sample means is 0.0049. Notice that
the standard deviation of the sample means is
smaller for the larger sample size.
19
(No Transcript)
20
(No Transcript)
21
(No Transcript)
22
(No Transcript)
23
EXAMPLE Describing the Distribution of the
Sample Mean The weights of pennies minted after
1982 are approximately normally distributed with
mean 2.46 grams and standard deviation 0.02
grams. What is the probability that in a simple
random sample of 10 pennies minted after 1982, we
obtain a sample mean of at least 2.465 grams?
24
EXAMPLE Sampling from a Non-normal
Population The following distribution represents
the number of people living in a household for
all homes in the United States in 2000. Obtain
200 simple random samples of size n 4 n 10
and n 30. Draw the histogram of the sampling
distribution of the sample mean.
25
(No Transcript)
26
(No Transcript)
27
(No Transcript)
28
(No Transcript)
29
(No Transcript)
30
(No Transcript)
31
EXAMPLE Using the Central Limit Theorem Suppose
that the mean time for an oil change at a
10-minute oil change joint is 11.4 minutes with
a standard deviation of 3.2 minutes. (a) If a
random sample of n 35 oil changes is selected,
describe the sampling distribution of the sample
mean. (b) If a random sample of n 35 oil
changes is selected, what is the probability the
mean oil change time is less than 11 minutes?
32
EXAMPLE Using the Central Limit Theorem (c )
If a random sample of n 50 oil changes is
selected, what is the probability the mean oil
change time is less than 11 minutes? (d) What
effect did increasing the sample size have on the
probability?
33
Chapter 7The Normal Probability Distribution

• 7.6
• The Normal Approximation to the Binomial
  Probability

34
Criteria for a Binomial Probability Experiment An
experiment is said to be a binomial experiment
provided 1. The experiment is performed a fixed
number of times. Each repetition of the
experiment is called a trial. 2. The trials are
independent. This means the outcome of one trial
will not affect the outcome of the other
trials. 3. For each trial, there are two mutually
exclusive outcomes, success or failure. 4. The
probability of success is fixed for each trial of
the experiment.
35

• Notation Used in the Binomial Probability
  Distribution
• There are n independent trials of the experiment
• Let p denote the probability of success so that
  1 p is the probability of failure.
• Let x denote the number of successes in n
  independent trials of the experiment. So, 0 lt x
  lt n.

36
As the number of trials n in a binomial
experiment increase, the probability distribution
of the random variable X becomes symmetric and
bell-shaped. As a general rule of thumb, if np(1
p) gt 10, then the probability distribution will
be approximately symmetric and bell-shaped.
37
(No Transcript)
38
P(X 18) P(17.5 lt X lt 18.5)
39
P(X lt 18) P(X lt 18.5)
40
Summary EXACT APPROXIMATE BINOMIAL NORMALP(X
a) P(a - 0.5 lt X lt a 0.5) P(X lt
a) P(X lt a 0.5) P(X gt a) P(X gt a -
0.5) P(a lt X lt b) P(a - 0.5 lt X lt b 0.5)
NOTE For P(X lt a) P(X lt a - 1), so rewrite P(X
lt 5) as P(X lt 4).
41
EXAMPLE Using the Binomial Probability
Distribution Function According to the United
States Census Bureau, 18.3 of all households
have 3 or more cars. (a) In a random sample of
200 households, what is the probability that at
least 30 5 have 3 or more cars? (b) In a random
sample of 200 households, what is the probability
that less than 15 have 3 or more cars? (c)
Suppose in a random sample of 500 households, it
is determined that 110 have 3 or more cars. Is
this result unusual? What might you conclude?