CVEXAMPLES X = blood serum cholesterol level C) Run The Experiment And Collect The DataTopic (9) – Concepts of Inferential Statistics 9 - 1 Topic (9) – CONCEPTS OF INFERENTIAL STATISTICS Recall that we have only incomplete information about the population about which we wish to make a statement. Hence, the sample or experiment results are needed to make inferences about that population. For example: Statistics Parameters Mean x μ Standard Deviation s σ Coefficient of Variation CV CV Median m M Correlation Coefficient r ρ Proportion of Successes p π There are two types of inferences: estimation and hypothesis testing Estimation After we have collected the data, there are several ways we could report the results: We could report a Point Estimate, which is a single number from the sample. The statistics listed above are point estimators.Topic (9) – Concepts of Inferential Statistics 9 - 2 The problem is that they have little context or meaning unless additional information is provided. For example, when reporting a sample mean, for the reader to evaluate it, requires knowledge of the mean’s sampling variability, such as including the sample size n and the sample standard deviation s. So, other possibilities for reporting information about a mean include: 1. Report x and s BUT not useful since s is not the standard deviation of x 2. Report x and nσσx= BUT usually we can’t because we don’t know the value of σ 3. Report x and nsSEM =. • is the Standard Error Of The Mean SEM• is an unbiased estimate of SEMxσ (under random sampling) • one common report is to list SEMx± but it’s not the best choice as we’ll see later We would like to report a range or interval of plausible values for the population parameter we’re estimating.Topic (9) – Concepts of Inferential Statistics 9 - 3 Such a report is called an Interval Estimate of the population parameter. When done right, the interval estimator includes our point estimate, an estimate of the precision of the point estimator, and some measure of our “comfort or confidence” with the estimate we are providing. These are called Confidence Interval Estimates. Example: surveys that report an observed proportion and a margin of error for that observed proportion such as an AP poll that showed that “67% of registered Democrats prefer Clinton over Obama. The survey had a margin of error of 4% with 95% confidence.” 67% = point estimate 4% = measure involving the precision of the point Estimate and our confidence level 95% = level of confidence Interval estimators can be calculated for several parameters related to population characteristics, including the mean (as we just saw), the variance 2σ, the median, percentiles, the population proportion of a category, etc.Topic (9) – Concepts of Inferential Statistics 9 - 4 An alternative to estimation is hypothesis testing, where it is of interest to make a decision or reach a conclusion about some characteristic of a population. Hypothesis Testing Recall the definition of Scientific Method: 1. knowledge is obtained in a systematic and objective manner in order to extend our understanding. 2. Based on this knowledge we form a HYPOTHESIS – a tentative or postulated explanation of the phenomenon. Hence it is a statement about a population characteristic (eg, a mean μ or proportion π or the difference between population means 21μμ−) 3. To evaluate the hypothesis we DESIGN and execute an objectively planned experiment or observational study. 4. The resulting data are TESTED to determine if they support or do not support the hypothesis. A) Constructing Hypotheses Almost all statistical testing procedures are based on testing two competing claims: the null and alternative hypotheses.Topic (9) – Concepts of Inferential Statistics 9 - 5 Defn: Ho is the NULL HYPOTHESIS. This is the status quo, i.e. it is the truth until disproven by testing. HA is the ALTERNATIVE HYPOTHESIS. This is the competing claim made by the researcher*. *There are exceptions, usually for testing equivalency or goodness of fit to a probability distribution. The testing procedure results in either 1) rejecting the null hypothesis in favor of the alternative hypothesis because the data support the alternative, or 2) failing to reject the null hypothesis because there is insufficient evidence to show it is wrong Note the similarity to jury trials. We will discuss this further latter on when we talk about how to make a decision as to which is true. But first, we need to set up the hypotheses that we want to test: Step 1) State Your Claim In WordsTopic (9) – Concepts of Inferential Statistics 9 - 6 EXAMPLES: A. In a study of the effect of the new drug for reducing serum cholesterol levels in men at risk of heart disease, the company’s claim is that the drug reduces serum cholesterol levels in the target population. So we might write: Ho: the drug does not reduce serum cholesterol levels HA: the drug does reduce serum cholesterol levels B. An entomologist believes that there is sexual dimorphism in body size of the periodical cicada. So her hypothesis might be stated: Ho: there is no sexual dimorphism in the periodical cicada HA: there is sexual dimorphism in the periodical cicada Note that these are informal statements that need to be clarified and made more rigorous Step 2) State The Hypotheses In Terms Of The Relevant Population Characteristics (parameters) EXAMPLESTopic (9) – Concepts of Inferential Statistics 9 - 7 A. In the study of the effect of the new drug for reducing serum cholesterol levels we had Ho: the drug does not reduce serum cholesterol levels HA: the drug reduces serum cholesterol levels What variable(s) are being measured? X = blood serum cholesterol level What population characteristic is being modified by the drug? If the drug leads to a reduction in blood levels, the population mean should go down. (They believe that the variability doesn’t change) What is the value of the characteristic without the drug? Without the drug, the target population has a mean blood serum level of 250. So we can
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