Lecture 20: Single Sample Hypothesis Tests: Population Mean and Proportion

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Lecture 20 Single Sample Hypothesis
TestsPopulation Mean and Proportion

• Devore, Ch. 8.2 - 8.3

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Topics

• Tests of Single Population Mean
• Normal Population w/ known s
• Large Sample Tests
• Normal Population w/ unknown s
• Tests of Single Population Proportion
• Large Sample Test
• Small Sample Test

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Recommended Steps in Hypothesis Testing

• Identify the parameter of interest and describe
  it in the context of the problem situation.
• Determine the null value and state the null
  hypothesis.
• State the alternative hypothesis.

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Hypothesis-Testing Steps, cond

• Give the formula for the computed value of the
  test statistic.
• State the rejection region for the selected
  significance level
• Compute any necessary sample quantities,
  substitute into the formula for the test
  statistic value, and compute that value.

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Hypothesis-Testing Steps-end
7. Decide whether H0 should be rejected and
state this conclusion in the problem context.
The formulation of hypotheses (steps 2 and 3)
should be done before examining the data.
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Single Population Mean Tests

• Identifying appropriate test statistic

Tests of Single Population Mean
Test
Case
USE
Formula
Statistic
known
, and normal
s
I
zo
population
s
large sample, unknown
knowing if normal not req'd (CLT)
II
zo
unknown
s
, but normal
III
to
population required
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Case I Mean - Normal, known s

• Null Hypothesis
• Test Statistic
• Alt Hypothesis Reject Region

Ho m mo
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Case II Large-Sample Tests
When the sample size is large, the z tests for
case I are modified to yield valid test
procedures without requiring either a normal
population distribution or a known
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Case II Mean - Large Sample

• Large Sample - rule of thumb n gt 40 -- for large
  samples, S will usually be close to s.
• Null Hypothesis
• Test Statistic
• Alt Hypothesis Reject Region

Ho m mo
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Case III Normal Population
If X1,,Xn is a random sample from a normal
distribution, the standardized variable
has a t distribution with n 1 degrees of
freedom.
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Case III Mean - Normal, unknown s

• Null Hypothesis
• Test Statistic
• Alt Hypothesis Reject Region

Ho m mo
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Examples Piston Rings

• A manufacturer of piston rings must produce rings
  with a target 80.995 mm. (Assume Normality)
• Suppose you take a sample of 15 rings and obtain
  a mean 80.996 and sample standard deviation of
  0.0019.
• Should you adjust your process to shift the mean
  closer to the target value? Assume a 0.05.
  (test if there is evidence to claim that the mean
  of the process is different than the target
  value)
• Should you adjust your process to shift the mean
  closer to the target value based on a historical
  (population) std dev of 0.0019? Assume a 0.05.

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b and sample size

• Hand calculations for Case I (normal, known s)
• For other cases, use Software (e.g., power and
  sample size feature in Minitab.)
• We will now examine some possible cases for Type
  II errors

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Type II errors for Case I Mean Test

• Type II error (conclude no difference, when a
  difference exists). Again, type II errors exist
  for any value in the alt hypothesis region
• P(Fail to Reject Ho when mm)
• There has been a mean shift (d) so that m m d
• b(m)

For Ha m gt mo
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Example Piston Rings

• What is the probability that you will fail to
  detect a shift in the mean from 80.995 to 80.997
  given a shift has occurred?
• assume a 0.05, n 15, s (known) 0.0019
• Assume Ha Reject if m gt mo
• Calculate b and power by hand and using Minitab

Note See Book for other b tests of other
alternative hypothesis
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Sample Size Calculation

• May want to know the sample size needed to detect
  a shift b(m) b for a level a test
• One-tail test
• Two-tail test
• For prior problem, what n is needed for a 0.05,
  b 0.1, diff 0.002 (80.995-80.887), s
  0.0019?
• Assume 1-side test

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A Population Proportion
Let p denote the proportion of individuals or
objects in a population who possess a specified
property.
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Large-Sample Tests
Large-sample tests concerning p are a special
case of the more general large-sample procedures
for a parameter
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Single Proportion Tests

• Typically, we only conduct proportion hypothesis
  test for large samples.
• For small samples, we may compute probabilities
  of Type I and Type II errors and compare with
  criteria (e.g., a0.05)

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Single Proportion - Large Sample

• Require np0 gt 10 and nqo gt 10 (Normal
  approximation)
• Null Hypothesis p po
• Test Statistic
• p-hat
• Alt Hypothesis Reject Region

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Example Proportion Test

• Suppose you produce injection molding parts.
• You claim that your process produces 99.9 defect
  free parts, or the proportion of defective parts
  is 0.001
• During part buyoff, you produce 500 parts of
  which 1 is defective.
• Note p-hat 1 / 500 0.002
• Compute Zo
• Za--gt (Zo gt Z0.01 2.33)
• Use a statistical test to demonstrate that your
  machine is not producing more defects than your
  advertised rates (assume a 0.01).

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Sample Size Determination

• Given a hypothesized defect rate of 0.001, how
  many samples would you need to detect that the
  defect rate increases to 0.002?
• Assume 1-side test with a 0.01 and b 0.01
• Solve using Minitab.

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Small-Sample Tests
Test procedures when the sample size n is small
are based directly on the binomial distribution
rather than the normal approximation.
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Small Sample Tests

• Examples
• Issues need to define a rejection region in
  terms of number of successes, c. then,
• Type I P(X gt c when XBin(n,po) )
• P(type I) 1 - B(c-1 n, po)
• Type II P(X lt c when X Bin(n,p) )
• P(type II when p p) B(c-1 n, p )