Runs Test for Randomness

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Lesson 15 - 2

• Runs Test for Randomness

2
Objectives

• Perform a runs test for randomness
• Runs tests are used to test whether it is
  reasonable to conclude data occur randomly, not
  whether the data are collected randomly.

3
Vocabulary

• Runs test for randomness used to test claims
  that data have been obtained or occur randomly
• Run sequence of similar events, items, or
  symbols that is followed by an event, item, or
  symbol that is mutually exclusive from the first
  event, item, or symbol
• Length number of events, items, or symbols in a
  run

4
Runs Test for Randomness

• Assume that we have a set of data, and we wish to
  know whether it is random, or not
• Some examples
• A researcher takes a systematic sample, choosing
  every 10th person who passes by and wants to
  check whether the gender of these people is
  random
• A professor makes up a true/false test and
  wants to check that the sequence of answers is
  random

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Runs, Hits and Errors

• A run is a sequence of similar events
• In flipping coins, the number of heads in a row
• In a series of patients, the number of female
  patients in a row
• In a series of experiments, the number of
  measured value was more than 17.1 in a row
• It is unlikely that the number of runs is too
  small or too large
• This forms the basis of the runs test

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Runs Test (small case)

• Small-Sample Case If n1 20 and n2 20, the
  test statistic in the runs test for randomness is
  r, the number of runs.
• Critical Values for a Runs Test for Randomness
  Use Table IX (critical value at a 0.05)

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Runs Test (large case)
Large-Sample Case If n1 gt 20 or n2 gt 20 the test
statistic in the runs test for randomness is
2n1n2 µr -------- 1
n
r - µr z ------- sr
where
Let n represent the sample size of which there
are two mutually exclusive types. Let n1
represent the number of observations of the first
type. Let n2 represent the number of observations
of the second type. Let r represent the number of
runs.
Critical Values for a Runs Test for
Randomness Use Table IV, the standard normal
table.
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Hypothesis Tests for Randomness Use Runs Test
Step 0 Requirements 1) sample is a sequence of
observations recorded in order of their
occurrence 2) observations have two mutually
exclusive categories. Step 1 Hypotheses
H0 The sequence of data is random. H1 The
sequence of data is not random. Step 2 Level of
Significance (level of significance determines
the critical value) Large-sample case
Determine a level of significance, based on the
seriousness of making a Type I error.
Small-sample case we must use the level of
significance, a 0.05. Step 3 Compute Test
Statistic Step 4 Critical Value Comparison
Reject H0 if Step 5 Conclusion Reject or
Fail to Reject
r - µr z0 ------- sr
Small-Sample r Large Sample
Small-Sample Case r outside Critical
interval Large-Sample Case z0 lt -za/2 or z0 gt
za/2
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Example 1
The following sequence was observed when flipping
a coin H, T, T, H, H, T, H, H, H, T, H, T, T,
T, H, H The coin was flipped 16 times with 9
heads and 7 tails. There were 9 runs observed.
Values n 16 n1 9 n2 7 r 9
Critical values from table IX (9,7) 4, 14
Since 4 lt r 9 lt 14, then we Fail to reject and
conclude that we dont have enough evidence to
say that it is not random.
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Example 2
The following sequence was observed when flipping
a coin H, T, T, H, H, T, H, H, H, T, H, T, T,
T, H, H, H, TT, T, T, H, H, T, T, T, H, T, T, H,
H, T, T, H, T, T, T, T The coin was flipped 38
times with 16 heads and 22 tails. There were 18
runs observed.
Values n 38 n1 16 n2 22 r 18
2n1n2 µr -------- 1
19.5263 n
r - µr z --------- ?r
z -0.515
Since z (-0.515) gt -Za/2 (-2.32) we fail to
reject and conclude that we dont have enough
evidence to say its not random.
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Example 3, Using Confidence Intervals

• Trey flipped a coin 100 times and got 54 heads
  and 46 tails, so
• n 100
• n1 54
• n2 46

r - µr z --------- ?r
We transform this into a confidence interval,
PE /- MOE.
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Using Confidence Intervals

• The z-value for a 0.05 level of significance is
  1.96
• LB 50.68 1.96 4.94 41.0
• to
• UB 50.68 1.96 4.94 60.4
• We reject the null hypothesis if there are 41 or
  fewer runs, or if there are 61 or more
• We do not reject the null hypothesis if there are
  42 to 60 runs

13
Summary and Homework

• Summary
• The runs test is a nonparametric test for the
  independence of a sequence of observations
• The runs test counts the number of runs of
  consecutive similar observations
• The critical values for small samples are given
  in tables
• The critical values for large samples can be
  approximated by a calculation with the normal
  distribution
• Homework
• problems 1, 2, 5, 6, 7, 8, 15 from the CD