1 QNT 531 Advanced Problems in Statistics and Research Methods WORKSHOP 1 By Dr. Serhat Eren University OF PHOENIX

2 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.

3 SECTION 1 ESTIMATING PROPORTIONS WITH CONFIDENCE  The formula for a 95% confidence interval can thus be expressed as: Sample proportion  margin of error

4 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.

5 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 for a 90% confidence interval or with the value for a 99% confidence interval. SECTION 1 CONSTRUCTING A CONFIDENCE INTERVAL FOR A PROPORTION

6 SECTION 1 FOR THOSE WHO LIKE FORMULAS 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.

7 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  for a 90% confidence interval  for a 99% confidence interval  3.0 for a 99.7% confidence interval SECTION 1 FOR THOSE WHO LIKE FORMULAS

8 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?

9 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?

10 Problem 40 (Page 38 or 75) c)What should be the sample size if the desired marginal error is 0.03?

11 Problem 40 (Page 38 or 75) d)What should be the sample size if the desired marginal error is 0.02?

12 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.

13 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.

14  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

15  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. SECTION 1 HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE

16 Two-Tail Test of the Mean: Small Sample  Let's first look at two-tail tests of the mean when  is unknown. SECTION 1 HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE

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22 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

23  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

24  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 SECTION 1 HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE

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26  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 SECTION 1 HYPOTHESIS TEST OF THE MEAN: SMALL SAMPLE

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