Tests for proportions

1
Section 8.3

• Tests for proportions

2
Example 1, part a.

• A manufacturer of nickel hydrogen batteries
  randomly selects 100 nickel plates for test
  cells, cycles them a specified number of times,
  and determines that 14 of the plates have been
  blistered. Does this provide compelling evidence
  that more than 10 of all plates blister under
  such circumstances?
• What are the hypotheses being tested?

3
Test Statistic

• What is a statistic that can be used to test
  these hypotheses?
• What is the distribution of the test statistic?
• Give a standardized test statistic and the
  corresponding rejection region (use a 0.05).
• For our data draw a conclusion.

4
Example 1, part b.

• If it is really the case that 15 of all plates
  blister under these circumstances and a sample
  size of 100 is used, how likely is it that the
  null hypotheses will not be rejected by the 0.05
  test?

5
Example 1, part c.

• How many plates would have to be tested to have
  b(0.15)0.10 for the test of part a?

6
Summary of Example 1

• Example 1 is an example of a large sample test
  concerning a population proportion.
• The test statistic is
• The critical region for the upper-tailed test is
• Critical regions for lower-tailed tests and two
  sided tests will very in the usual way.
• Probability of type II error is calculated after
  correctly standardizing the test statistic.
• Sample size is determined by expressing type II
  error in terms of n and solving for n.

7
Example 2.

• Same as example 1, but only 10 plates are tested.
• A manufacturer of nickel hydrogen batteries
  randomly selects 10 nickel plates for test cells,
  cycles them a specified number of times, and
  determines that 2 of the plates have been
  blistered. Does this provide compelling evidence
  that more than 10 of all plates blister under
  such circumstances?

8
Test Statistic

• What is a statistic that can be used to test
  these hypotheses?
• What is the distribution of the test statistic?
• What is the rejection region (use a 0.05).
• For our data draw a conclusion.

9
Example 2, part b.

• If it is really the case that 15 of all plates
  blister under these circumstances and a sample
  size of 10 is used, how likely is it that the
  null hypotheses will not be rejected by the 0.05
  test?
• If this type II error is undesired, how would you
  solve for sample size?

10
Summary of Example 2

• Example 1 is an example of a small sample test
  concerning a population proportion.
• The test statistic is the number of successes
  in the sample, and the rejection region is
  determined directly from the binomial
  distribution.
• The critical region for the upper-tailed test has
  the form
• Critical regions for lower-tailed tests and two
  sided tests have the usual form as well.
• Probability of type II error by a binomial
  calculation.
• Sample size calculations usually lead to large
  sample tests and so can be done as in example 1.
  If you determine that a small sample is
  sufficient double check the error probabilities
  using binomial calculations.

11
Comparison of two tests.

• A company that sells these plates advertises to
  their customers that only 10 will blister when
  cycled this number of times. The customer is
  beginning to believe that the true proportion is
  higher and is interested in proving this if the
  true proportion is at least 15?
• Which type of error is the company concerned
  about?
• Which type of error is the customer concerned
  about?
• Who is benefited by the larger sample size?
• Who should most concerned if the decision is only
  based on a sample of size 1?