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QNT 531 Advanced Problems in Statistics and Research Methods

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WORKSHOP 1

By

Dr. Serhat Eren

University OF PHOENIX

- 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.

- The formula for a 95% confidence interval can thus be expressed as:
Sample proportion margin of error

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.

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.

Notation for Population and Sample Proportions

- Sample size = n
- Population proportion = p
- Sample 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.

Formula for a (1-) 100% Confidence Interval for a

Proportion

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

- 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).
- What is the point estimate of the proportion of all adults who planned to spend less money on gifts during the 1998 holiday season?

- Using 95% confidence, what is the margin of error associated with this estimate and what is the confidence interval?

- What should be the sample size if the desired marginal error is 0.03?

- What should be the sample size if the desired marginal error is 0.02?

- 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.

- 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.

- 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 “a” 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.

- 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.

Two-Tail Test of the Mean: Small Sample

- Let's first look at two-tail tests of the mean when is unknown.

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 ttest instead of a Z test statistic, and the rejection region cutoff values must be found from the t-table rather than the Z-table.

- 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 thettable 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.

- 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 Hoif the calculated t statistic is too small. Thus, we have the rejection region shaded in Figure 12.1.

- For an upper-tail test we want to reject Hoif the calculated tstatistic is too large.
- The rejection region for this type of test is shaded inFigure 12.2.