Download Large Sample Tests – Non

Large Sample Tests – Non-Normal population
• Suppose we have a large sample from a non-normal population and
we are interested in conducting a hypotheses test for a single mean.
• First, we need to assume that all the observations are independent
and identically distributed with finite mean and variance.
• Then we can apply the CLT to the sample mean.
• The test is conducted using the standard normal (Z) distribution.
• Note, in this case we do not require σ to be known, since a large
sample implies that the sample standard deviation s will be close to
σ for most variables.
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Example
• Do middle-aged male executives have different average blood
pressure than the general population? The national center for Health
Statistics reports that the mean systolic blood pressure for males 35
to 44 years of age is 128. The medical director of a company looks
at the medical records of 72 company executives in this age group
and finds that the mean systolic blood pressure in this sample is x  126.07
and the standard deviation is 15. Is there evidence that the executives
blood pressure differ from the national average?
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Small Sample Tests for a Single Mean
• Suppose we have a small sample and we are interested in conducting
a hypotheses test for a single mean.
• First, we need to assume that all the observations are independent
and identically normally distributed with unknown finite mean and
variance.
• The CLT does not apply to the sample mean.
• The test is conducted using the t distribution with n-1 degrees of
freedom.
• Note, to be confident in our test results we need to check the normality
assumption.
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Example
• In a metropolitan area, the concentration of cadmium (Cd) in leaf
lettuce was measured in 6 representative gardens where sewage sludge
was used as fertilizer. The following measurements (in mg/kg of dry
weight) were obtained.
Cd 21 38 12 15 14
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Is there strong evidence that the mean concentration of Cd is higher
than 12.
Descriptive Statistics
Variable
Cd
N
6
Mean
18.00
Median
14.50
TrMean
18.00
StDev
10.68
SE Mean
4.36
• The hypothesis to be tested are:
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• The test statistics is:
The degrees of freedom are:
Conclusions: (Using RR and P-value)
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Large Sample Tests–Normal population
• Suppose we have a large sample from a normal population and we
are interested in conducting a hypotheses test for a single mean.
• First, we need to assume that all the observations are independent
and identically normally distributed with unknown finite mean and
variance.
• The CLT is not necessary.
• The test is conducted using the t distribution with n-1 degrees of
freedom.
• Note, if n is large the t distribution with n-1 degrees of freedom
converges to the N(0,1) distribution.
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Example
• The GE Light Bulb Company claims that the life of its 2 watt bulbs
normally distributed with a mean of 1300 hours. Suspecting that the
claim is too high, Nalph Rader gathered a random sample of 161
bulbs and tested each. He found the average life to be 1295 hours
and the standard deviation 20. Test the company's claim using  = 0.01.
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Large Sample Tests for a Binomial Proportion
• Suppose we have a large sample from a Bernoulli(θ) distribution.
• That is, we assume that all the observations are independent and
identically Bernoulli trails.
• The sample proportion is in fact the sample mean.
• The CLT applies to the sample proportions.
• The test is conducted using the standard normal (Z) distribution.
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Example
• Statistics Canada records indicate that of all the vehicles undergoing
emission testing during the previous year, 70% passed on the first
try. A random sample of 200 cars tested in a particular county
during the current year yields 124 that passed on the initial test.
Does this suggest that the true proportion for this county during the
current year differs from the previous nationwide proportion? Test
the relevant hypothesis using α = 0.05.
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