Transcript PROBABILITY AND STATISTICS

PROBABILITY AND
STATISTICS
WEEK 9-10
Onur Doğan
The sampling distribution of the
sample statistics
Onur Doğan
The sampling distribution of the
sample statistics
Consider a population of N elements from which we can
obtain the following distinct data: {0, 2, 4, 6, 8}.
• Form samples of size 2 for this population.
• Define their means and figure the bar chart of the means.
• Define the sampling distribution of the sample ranges
and figure bar chart.
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The Central Limit Theorem
The mean is the most commonly used sample statistic and thus it is very
important. The central limit theorem is about the sampling distribution
of sample means of random samples of size n.
Let us establish what we are interested in when studying this
distribution:
1) Where is the center?
2) How wide is the dispersion?
3) What are the characteristics of the distribution?
The central limit theorem gives us an answer to all these questions.
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The Central Limit Theorem
Let µ be the mean and σ the standard deviation of a
population variable. If we consider all possible random
sample of size n taken from this population, the sampling
distribution of sample means will have the following
properties:
c) if the population is normally distributed the sampling distribution of the sample
means is normal; if the population is not normally distributed, the sampling distribution
of the sample means is approximately normal for samples of size 30 or more. The
approximation to the normal distribution improves with samples of larger size.
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The Central Limit Theorem
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The Central Limit Theorem
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The Central Limit Theorem
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Example
Consider a normal population with µ=100 and σ=25.
If we choose a random sample of size n = 36, what is
the probability that the mean value of this sample is
between 90 and 110?
In other words, what is P(90 < x < 110)?
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Example
The average male drinks 2L of water when
active outdoor s(with standard deviation of 0,7
L). You are planning a full day nature trip for 50
men and bring 110 L of water. What is the
probability that you will run out?
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Confidence Intervals
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Confidence Interval on the Mean of a
Normal Distribution, Variance Known
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Confidence Interval Formula
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Example
Suppose that the life length of a light bulb (X; unit: hour) follows
the normal distribution N(y, 402). A random sample of n = 30
bulbs is tested and the sample mean is found to be 780 hours.
•Construct a 95% two sided confidence interval on the mean life
length (µ) of a light bulb.
•Find a sample size n to construct a two-sided confidence interval
on µ with an error = 20 hours from the true mean life length.(use
α= 0.05)
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Example
• The diameter of a hole (X; unit: in.) for a cable
harness is normal with σ2= 0.012. A random
sample of n = 10 yields an average diameter of
1,5045 in.
• Construct a 99% upper-confidence bound on
the mean diameter (p) of the hole.
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Confidence Interval on the Mean of a Normal
Distribution,Variance Unknown (t distribution)
Small Sample CI?
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Example
You sample 36 apples from your farm’s harvest
of over 200.000 apples. The mean weight of the
sample is 112 grams (with a 40 gram sample
standart deviation). What is the probability that
the mean weight of all 200 000 apples is within
100 and 124 grams?
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A Large-Sample Confidence Interval for a
Population Proportion
Confidence Interval Formula:
Sample Size Selection
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Example
A sample of n = 40 bridges in a city is tested for
metal corrosion, and x = 28 bridges are found
corroded.
• Construct a 95% two-sided confidence interval on the
proportion of corroded bridges (p) in the county.
• Determine a sample size n to establish a 95%
confidence interval on p with an error = 0.05 from the
true proportion.
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Example
In a teaching district school management allows theacher
using omputer in their lessons. From the 6000 teachers in
district, 250 were randomly selected and asked if they fekt that
computers were an essential teaching tool for their calssrom.
Of those selected, 142 teachers felt that computers were an
essential teachin tool.
•Calculate a 99% confidence interal for the proportion of
teachers who felt that computers are an essential teaching tool
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Summary for Confidence Intervals
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Hypothesis Testing
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Test Regions
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Test Errors
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Hypothesis Testing Procedure
Tests on the Mean of a Normal Distribution,
Variance Known
• .
Example
For several years, a teacher has recorded his
students' grades, and the mean, µ for all these
students' grades is 72 and the standard deviation
is σ = 12. The current class of 36 students has an
average x = 75,2 (higher than µ = 72) and the
teacher claims that this class is superior to his
previous ones. Test the teacher’s claim for the
level of significance = α=0,05.
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Relationship Between
Hypothesis Test, CI and p-value
Tests on a Population Proportion
• .
Example
For the bridge example; a specialist claims that more than half of the
bridges gave been corroded in the city. Test the specialist’s claim with
%95 confidence.
Example
• Suppose that a factory is producing wheels for
airbuses. The manufacturer claims that they
produce wheels 3 meters diameter.
• The quality control department of the buyer
firm investigate a sample from the daily
product. They controlled 36 wheels and found
that average diamater is 2,92 and s.d. is 0,18.
• Test the claim at α=0.05 significance level
Example
• According to a recent poll 53% of Americans
would vote for the incumbent president. If a
random sample of 100 people results in 45%
who would vote for the incumbent, test the
claim that the actual percentage is 53%. Use a
0.10 significance level.
Example
• The national weather service says that the mean
daily high temperature for July in İzmir is 42°C.
A local weather service wants to test the claim
of 42°C because it believes it is different. A
sample of mean daily high temperatures for
October over the past 31 years yields =44°F and
s=3.8°C. Test the claim at α=0.01 significance
level.
Example
• In a clinical study of an allergy drug, 108 of
the 203 subjects reported experiencing
significant relief from their symptoms. At the
0.01 significance level, test the claim that more
than half of all those using the drug experience
relief.