statistics lecture 7 (ch6)
DESCRIPTION
Sampling Distributions lectureTRANSCRIPT
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OBJECTIVES
• To understand concept of sampling distribution
• To understand concept of sampling error
• To determine the mean and std dev for the
sampling distribution of a sample mean
• To determine the mean and std dev for sampling
distribution of a sample proportion
• To calculate the probabilities related to the
sample mean and the sample proportion
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Sampling distributions
• Can be defined as the distribution of a sample
statistic.
• Scientific experiments are used to make
inferences concerning population parameters from
sample statistics.
• Need to know what is the relationship between the
sample statistic and its corresponding population
parameter.
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Sampling error
• Can be defined as the difference between the
calculated sample statistic and population
parameter.
• Sampling errors occur because only some of the
observations from the population are contained in
the sample.
• Sampling error:
sample statistic – population parameter
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Sampling error
• Size of the sampling error depends on the sample
selected.
• May be positive or negative.
• Should be kept as small as possible.
• For smaller samples the range of possible
sampling errors becomes larger.
• For larger samples the range of possible sampling
errors becomes smaller.
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CONCEPT QUESTIONS
• P201 QUESTIONS 1-4
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Sampling distribution of the mean
• Sample mean is often used to estimate the
population mean.
• Sampling distribution of the mean is the
distribution of sample means obtained if all
possible samples of the same size are selected
form the population.
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Sampling distribution of the mean
• If we calculate the average of all the sample
means, say we have m such samples, the result will
be the population mean:
• The standard deviation of all the sample means, will
be: referred to as the standard error of the mean
1
m
i
ix
x
m
xn
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Central Limit Theorem
• If the sample size becomes larger, regardless of the
distribution of the population from which the sample
was taken, the distribution of the sample mean is
approximately normally distributed:
– with
– and standard deviation
• The accuracy of this approximation increases as
the size of the sample increases.
• A sample of at least 30 is considered large enough
for the normal approximation to be applied.
x x
n
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Properties of the sampling distribution of
the sample mean
• For a random sample of size n from a population
with mean μ and standard deviation σ, the
sampling distribution of has:
– a mean
– and a standard deviation
x
xn
x
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Properties of the sampling distribution of
the sample mean
• If the population has a normal distribution, the
sampling distribution of will be normally
distributed, regardless the sample size.
• If the population distribution is not normal, the
sampling distribution of will be approximately
normally distributed, if the sample size ≥ 30.
•
x
x
2
;X Nn
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Example
• Marks for a semester test is normally distributed, with a mean of 60 and a standard deviation of 8. – X ~ N(60;82)
• A sample of 25 students is randomly selected:
– 2
2 2
;
8; 60;
25
x xX N
X N Nn
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Example • If we need to determine the probability that the
average mark for the 25 students will be between 58 and 63.
(58 63)
58 60 63 60
8 8
25 25
1,25 1,88
0,9699 0,8944 1 0,8643
P X
P Z
P Z
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INDIVIDUAL EXERCISE
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INDIVIDUAL EXERCISE
The past sales record for ice cream indicates the sales are right skewed, with the population mean of R13.50 per customer and a std dev of R6.50. A random sample of 100 sales records is selected. Find the probability of:-
1. Getting a mean of less than R13.25
2. Getting a mean of greater than R14.50
3. Getting a mean of between R13.80 and R15.20
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Solution
P205 - 207 of textbook
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WHICH EQUATION TO USE?
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Sampling distribution of
proportion • Categorical values such as number of
drivers that wear safety belts in Gauteng
or number of drivers who do not wear
safety belts
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Sampling distribution of the proportion
• Population proportion will be represented by p,
and the sample proportion by , where X is
the number of items with the characteristic and n
is the sample size.
• The standard error of the proportion is given as:
ˆ /p X n
(1 )p
p p
n
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Example
• Suppose that in a class of 100, 28 students fail a test.
• The population proportion of students who fail the test is:
28
ˆ 0,28100
Xp
n
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Example
• A sample of 50 students is randomly chosen
• What is the probability that more than 25% will fail the test?
ˆ 0,25
0,25 0,28
0,28(1 0,28)
50
0,47
1 0,6808 0,3192
P P
P Z
P Z
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Individual exercise/homework
• Read pages 195 – 211
• Self review test p 209
• Supplementary exercises p209
• Go to www.jillmitchell.net and view the following:-
• Video on sampling distributions
• Video on example of sampling distribution
• Video on central limit theorem
• Completely re-do the NUBE test using your textbook to
assist you.
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