1 QNT 531 Advanced Problems in Statistics and Research Methods WORKSHOP 1 By Dr. Serhat Eren University OF PHOENIX
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QNT 531Advanced Problems in Statistics and
Research Methods
QNT 531Advanced Problems in Statistics and
Research Methods
WORKSHOP 1
By Dr. Serhat Eren
University OF PHOENIX
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SECTION 1
ESTIMATING PROPORTIONS WITH CONFIDENCE SECTION 1
ESTIMATING PROPORTIONS WITH CONFIDENCE
The most commonly reported information that can be used to construct a confidence interval is the margin of error. To use that information, you need to know this fact.
To construct a 95% confidence interval for a population proportion, simply add and subtract the margin of error to the sample proportion.
The margin of error is often reported using the symbol " " which is read plus or minus.
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SECTION 1
ESTIMATING PROPORTIONS WITH CONFIDENCESECTION 1
ESTIMATING PROPORTIONS WITH CONFIDENCE
The formula for a 95% confidence interval can thus be expressed as:
Sample proportion margin of error
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SECTION 1
CONSTRUCTING A CONFIDENCE INTERVAL FOR A PROPORTION
SECTION 1
CONSTRUCTING A CONFIDENCE INTERVAL FOR A PROPORTION
Developing the Formula for 95% Confidence Interval The formula for 95% confidence interval for a
population proportion:
Sample proportion 2(S.D.)
To be exact, we would actually add and subtract 1.96(S.D.) instead of 2(S.D.) because 95% of the values for a bell-shaped curve fall within 1.96 standard deviations of the mean.
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Other Levels of Confidence Intervals Although 95% confidence intervals are by far the
most common, you will sometimes see 90% or 99% intervals as well.
To construct those, you simply replace the value 2 in the formula with 1.645 for a 90% confidence interval or with the value 2.576 for a 99% confidence interval.
SECTION 1
CONSTRUCTING A CONFIDENCE INTERVAL FOR A PROPORTION
SECTION 1
CONSTRUCTING A CONFIDENCE INTERVAL FOR A PROPORTION
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SECTION 1
FOR THOSE WHO LIKE FORMULASSECTION 1
FOR THOSE WHO LIKE FORMULAS
Notation for Population and Sample ProportionsSample size = nPopulation proportion = pSample proportion =
Notation for the Multiplier for a Confidence Interval We specify the level of confidence for a confidence
interval as (1-)100%. For example, for a 95% confidence interval, = 0.05.
P̂
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Formula for a (1-) 100% Confidence Interval for aProportion
Common Values of 1.0 for a 68% confidence interval 1.96 or 2.0 for a 95% confidence interval 1.645 for a 90% confidence interval 2.576 for a 99% confidence interval 3.0 for a 99.7% confidence interval
n
ppZP
)ˆ1(ˆˆ2/
2/Z
SECTION 1
FOR THOSE WHO LIKE FORMULASSECTION 1
FOR THOSE WHO LIKE FORMULAS
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Problem 40 (Page 38 or 75) Problem 40 (Page 38 or 75) An Associated Press poll of 1018 adults found 255
adults planned to spend less money on gifts during the 1998 holiday season compared to the previous year (ICR Media survey, November 13–17, 1998).
a) What is the point estimate of the proportion of all adults who planned to spend less money on gifts during the 1998 holiday season?
25.01018
255ˆ n
pP
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Problem 40 (Page 38 or 75) Problem 40 (Page 38 or 75)
b) Using 95% confidence, what is the margin of error associated with this estimate and what is the confidence interval?
223.0,277.0027.025.0)ˆ1(ˆˆ
:
027.0)0136.0(96.11018
75.025.096.1
)ˆ1(ˆ
2/
2/
n
ppZP
isthenIntervalConfidence
E
n
ppZE
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Problem 40 (Page 38 or 75) Problem 40 (Page 38 or 75)
c) What should be the sample size if the desired marginal error is 0.03?
800
)03.0(
)75.0(25.0)96.1()ˆ1(ˆ
)ˆ1(ˆ
2
2
2
22/
2/
n
E
ppZn
n
ppZE
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Problem 40 (Page 38 or 75) Problem 40 (Page 38 or 75)
d) What should be the sample size if the desired marginal error is 0.02?
1801
)02.0(
)75.0(25.0)96.1()ˆ1(ˆ
)ˆ1(ˆ
2
2
2
22/
2/
n
E
ppZn
n
ppZE
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Problem 40 (Page 38 or 75) Problem 40 (Page 38 or 75)
d) What happens to the sample size as the desired marginal error changes?
The sample size increases when the desired marginal error decreases or sample size decreases when the desired marginal error increases.
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SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
More often the population standard deviation is not known. In this case the sample standard deviation, s, must be used to calculate an estimate of the unknown population standard deviation,.
If the sample is sufficiently large, n >30, then you can use the Z test statistic to do a hypothesis test on the mean. However, very often you have a small sample.
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Now let us think about the implications of having a small sample on our hypothesis test. The steps of any hypothesis test are listed below:Step 1: Set up the null and alternative hypotheses.Step 2: Pick the value of “” and find the rejection
region.Step 3: Calculate the test statistic.Step 4: Decide whether or not to reject the null
hypothesis.Step 5: Interpret the statistical decision in terms of the
stated problem.
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
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The t-test statistic is calculated as follows:
Notice that the calculation for the t statistic is just the same as Z with replaced with s.
ns
Xt
/
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
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Two-Tail Test of the Mean: Small SampleLet's first look at two-tail tests of the mean
when is unknown.
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
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One-Tail Test of the Mean: Small Sample In the previous section we learned that the two-tail
hypothesis testing procedure for , is affected in two major ways by the lack of knowledge about .
The test statistic becomes a t test instead of a Z test statistic, and the rejection region cutoff values must be found from the t-table rather than the Z-table.
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
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The same can be said about one-tail tests of the mean when is unknown.
The only step in the procedure that we need to update is finding the rejection region using the t table for one-tail tests.
The form of the rejection region is the same as when is known. The only difference is in finding the cutoff values.
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
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Remember that we constructed the rejection regions by following a series of logical arguments as to what values of X-bar would lead us to reject the null hypothesis. These arguments still apply.
For a one-tail test we want to reject Ho if the calculated t statistic is too small. Thus, we have the rejection region shaded in Figure 12.1.
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
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For an upper-tail test we want to reject Ho if the calculated t statistic is too large.
The rejection region for this type of test is shaded in Figure 12.2.
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
SECTION 1
HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE
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