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Stats 11 (Fall 2004) Lecture Note Instructor: Hongquan XuIntroduction to Statistical Methods for Business and EconomicsChapter 10 Data on a Continuous VariableWe will cover only Sections 10.1.1–10.1.3.Section 10.1 One-Sample IssuesIn Chapters 8 and 9, we learned CIs and significance tests for a population mean µ. Both require that wehave a from a population.If the sample size is large, the assumption of a normal distribution is not so crucial. (Why?)However, if we do not have much data, this assumption will be important to check. So we have the followingquestion.How to judge if data follow approximately a normal distribution?1. First look at a histogram or stem-leaf plot - check for non-normal features such as gaps, outliers,and strong skewness.If roughly symmetric, unimodal, bell-shaped - then we can turn to a tool that is more s ensitive forassessing normality.2. Normal Quantile Plot (aka Q-Q Plot or Normal Probability Plot)Big Idea: Plot percentiles of a standard normal distribution against the corresponding percentiles of thedata.If the observations follow a normal distribution, the resulting plot should beDeviations from this would indicate possible departures from a normal distribution:• Outliers appears as points that are far away from the overall pattern of the plot.• In a positively skewed distribution, the largest observations fall distinctly above a line drawnthrough the main body of p oints.• In a negatively skewed distribution, the observations fall distinctlythe line.NOTE: Real data almost always show some departure from the “theoretical” normal distribution — don’toverreact to minor wiggles in the plot (see Fig. 10.1.3 on page 413).Examples of Normal Quantile Plots:1(a) (b)(c)The histograms for (a) and (b) are2Note:• If the as sumptions are false, the results of the analysis may be meaningless.• The t- tests and CIs are to presence of outliers.• The two-sided t-tests and CIs are against the departure from the normality as-sumption if there are no apparent outliers.• One-sided tes ts are more s ensitive to skewness.• Do not use t-proce dures for small to moderate samples (n ≤ 40) if there are outliers.• Al ways plot your data before using formal tools of analysis (tests and confidence inte rvals).• Nonparametric methods (to be introduced later) do not assume any particular distribution and areless sensitive to outliers.Section 10.1.2 Paired ComparisonsRecap: We have discussed the one-sample procedures for estimating a population mean and testinghypotheses about the population mean. One sample designs have a major drawback – no comparisongroup – and thus are subject to various biases (placebo effect, confounding, etc.)Matched pairs designs are one way to introduce comparison into a study. One matched pairs designhas all subjects receive just one treatment, but responses are recorded be fore and after the treatment. Thedifferences in responses are examined for learning ab out the eff ec t (if any) of the treatme nt.In these paired designs, it is the differences that we are interested in analyzing. By focusing on thedifferences we again have just one sample of observations (the differences) and are able to use the ”one-sample” inference methods to form a confidence interval of the mean difference or to test hypotheses aboutthe mean difference.For paired data, analyze the differences.Examples: Do piano lessons improve spatial-temporal reasoning of preschool children?Data: The changes (= after piano lessons - before piano lessons) in reasoning scores for n=34 preschoolchildren:2 5 7 -2 2 7 4 1 0 7 3 4 3 4 9 4 5 2 9 6 0 3 6 -1 3 4 6 7 -2 7 -3 3 4 4The larger values, the better reasoning.Here is Stata T-Test OutputOne-sample t test------------------------------------------------------------------------------Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]---------+--------------------------------------------------------------------score | 34 3.617647 .5239618 3.055196 2.551639 4.683655------------------------------------------------------------------------------Degrees of freedom: 333Ho: mean(score) = 0Ha: mean < 0 Ha: mean ~= 0 Ha: mean > 0t = 6.9044 t = 6.9044 t = 6.9044P < t = 1.0000 P > |t| = 0.0000 P > t = 0.0000(a) Give a 95% CI for the mean improvement in reasoning scores.(b) State the hyp otheses :(c) What is the observed test statistic?(d) What are the degrees of freedom?(e) What is the p-value?(f) What is the decision?(g) What is the conclusion?Section 10.1.3 A Nonparametric TestWe will learn one Nonparametric Test, called sign test. It does not assume the data are from normalpopulation, but still assume our data are a random sample.For nonparametric tests, the hypotheses are about the population median ˜µ. The null hypothesis would beH0: ˜µ = 0. The alternative can be one-sided or two-sided.Examples: Do piano lessons improve spatial-temporal reasoning of preschool children?H0: ˜µ = 0.H1:4If H0were true, we would expect that the changes are roughly equally likely to be negative and positive. Ifthere are much more positive numbers than negative numbers, we would reject the null. We could computethe P-value indeed.Counting positive and negative numbers can be done simply by looking at the sign of the changes. So thename ofThe Sign TestFor each numbers, use + and − for p ositive and negative numbers, respectively. For our example, we have+ + + - + + + + 0 + + + + + + + + + + + 0 + + - + + + + - + - + + +Let X =# of ”+”. What is the distribution of X if H0were true?For our data, there are 34 observations and two ties (change=0). We can ignore the ties. So X hasdistribution if H0were true.How many “+” are there?How to find the P-value? Recall thatP-value = the probability of gett ing a test statistic as extreme or more extreme than theactual observed test statistic, assuming H0is true. (The direction of extreme matches H1.)How likely will we have as many as or more “+” if H0were true? OrHow likely will we have as many as or more heads if a fair coin is tossed 34 times?The P-value=Decision:Conclusion:Comments on signed tests and nonparametric methods• les s s ensitive to outliers• do not assume any particular distribution for the original observations• do ass ume random samples from the populations of interest• me asure of ce nter is the median rather than the mean• tend to be som ewhat les s effe ctive at detecting departures from a null hypothesis• tend to give wider confidence


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