Statistical Inference: Other One-Sample Test Statistics

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Chapter 12 Statistical Inference Other
One-Sample Test Statistics I One-Sample z Test
for a Population Proportion,
p A. Introduction to z Test for a Population
Proportion
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1. The binomial function rule can be
used to determine the probability of r
successes in n independent trials.
2. When n is large, the normal distribution
can be used to approximate the probability
of r or more successes. The approximation
is excellent if (a) the population is at
least 10 times larger than the sample and
(b) np0 gt 15 and n(1 p0) gt 15, where p0 is
the hypothesized proportion.
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B. z Test Statistic for a Proportion
1.
sample estimator of the population
proportion
p0 hypothesized population
proportion n size of the sample used
to compute
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population standard error of a proportion,

where p denotes the population proportion.
C. Statistical Hypotheses for a Proportion
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D. Computational Example 1. Student
Congress believes that the proportion of
parking tickets issued by the campus police
this year is greater than last year. Last
year the proportion was p0 .21. 2. To
test the hypotheses they obtained a random
sample of n 200 students and found that
the proportion who received tickets this
year was
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z.05 1.645 3. The null hypothesis can
be rejected the campus police are issuing
more tickets this year.
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E. Assumptions of the z Test for a Population
Proportion 1. Random sampling from the
population 2. Binomial population 3. np0
gt 15 and n(1 p0) gt 15 4. The population is
at least 10 times larger than the sample
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II One-Sample Confidence Interval for a
Population Proportion, p A. Two-Sided
Confidence Interval
2. is an
estimator of the
population standard error of a proportion.
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B. One-Sided Confidence Interval 1. Lower
confidence interval
2. Upper confidence interval
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C. Computational Example Using the Parking
Ticket Data 1. Two-sided 100(1 .05)
95 confidence interval
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2. One-sided 100(1 .05) 95 confidence
interval
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3. Comparison of the one- and two-sided
confidence intervals Two-sided
interval
One-sided interval
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• D. Assumptions of the Confidence Interval for a
• Population Proportion
• 1. Random sampling from the population
• 2. Binomial population
• 3. np0 gt 15 and n(1 p0) gt 15
• 4. The population is at least 10 times larger
  than the
• sample

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III Selecting a Sample Size, n A. Information
needed to specify n 1. Acceptable margin of
error, m, in estimating p. m is usually
between .02 and .04. 2. Acceptable
confidence level usually .95 for z.05 or
z.05/2 3. Educated guess, denoted by p,
of the likely value of p
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B. Computational Example for the Traffic
Ticket Data 1. One-sided confidence
interval, let m .04, z.05 1.645, and
p .27
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C. Conservative Estimate of the Required Sample
Size 1. If a researcher is unable to
provide an educated guess for m, a
conservative estimate of n is obtained by
letting p .50.
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IV One-Sample t Test for Pearsons Population
Correlation A. t Test for ?0 0
(Population Correlation Is Equal to
Zero) 1. Values of r that lead to
rejecting one of the following null
hypotheses are obtained from Appendix Table
D.6.
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Appendix Table D.6. Critical Values of the
Pearson r
8 0.549 0.632 0.716 0.765 10 0.497 0.576 0.658
0.708 20 0.360 0.423 0.492 0.537 30 0.296 0.34
9 0.409 0.449 60 0.211 0.250 0.274 0.325 100 0
.164 0.195 0.230 0.254
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1. Table D.6 is based on the t distribution
and t statistic
B. Computational Example Using the Girls
Basketball Team Data (Chapter 5) 1.
r .84, n 10, and r.05, 8
.549 2. r.05, 8 .549 is the one-tailed
critical value from Appendix Table D.6.
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1. Because r .84 gt r.05, 8 .549, reject
the null hypothesis and conclude that
players height and weight are positively
correlated.
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C. Assumptions of the t Test for Pearsons
Population Correlation Coefficient 1. Rando
m sampling 2. Population distributions of X
and Y are approximately normal. 3. The
relationship between X and Y is linear.
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4. The distribution of Y for any value of X is
normal with variance that does not depend
on the X value selected and vice versa.
V One-Sample Confidence Interval for
Pearsons Population Correlation A. Fishers
r to Z? Transformation 1. r is bounded by 1
and 1 Fishers Z? can exceed 1 and 1.

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Appendix Table D.7 Transformation of r to Z?
0.200 0.203 0.400 0.424 0.600 0.693 0.800 1.099
0.225 0.229 0.425 0.454 0.625 0.733 0.825 1.172
0.250 0.255 0.450 0.485 0.650 0.775 0.850 1.256
0.275 0.282 0.475 0.517 0.675 0.820 0.875 1.354
0.300 0.310 0.500 0.549 0.700 0.867 0.900 1.472
0.325 0.337 0.525 0.583 0.725 0.918 0.925 1.623
0.350 0.365 0.550 0.618 0.750 0.973 0.950 1.832
0.375 0.394 0.575 0.655 0.775 1.033 0.975 2.185
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• B. Two Sided Confidence Interval for ? Using
• Fishers Z? Transformation
• 1. Begin by transforming r to Z?. Then obtain
• a confidence interval for Z?Pop

2. A confidence interval for r is obtained by
transforming the lower and upper confidence
limits for Z?Pop into r using Appendix
Table D.6 .
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C. One-Sided Confidence Interval for
? 1. Lower confidence limit
2. Upper confidence limit
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D. Computational Example Using the Girls
Basketball Team Data (Chapter 5) 1. r .84,
n 10, and Z? 1.221
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2. Graph of the confidence interval for ?
3. A confidence interval can be used to test
hypotheses for any hypothesized value of
?0. For example, any hypothesis for which ?0
.54 could be rejected.
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E. Assumptions of the Confidence Interval for
Pearsons Correlation Coefficient
1. Random sampling 2. ? is not too close
to 1 or 1 3. Population distributions of X
and Y are approximately normal 4. The
relationship between X and Y is linear
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5. The distribution of Y for any value of X is
normal with variance that does not depend
on the X value selected and vice
versa. VI Practical Significance of Pearsons
Correlation A. Cohens Guidelines for
Effect Size ? r .10 is a small strength of
association ? r .30 is a medium strength
of association ? r .50 is a large strength
of association