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PARAMETRIC STATISTICAL INFERENCE. PARAMETER ESTIMATION Confidence intervals Estimate a population parameter Mean HYPOTHESIS TESTING To assess evidence provided by data about some claim concerning a population SIGNIFICANCE TESTING
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PARAMETRIC STATISTICAL INFERENCE PARAMETER ESTIMATION • Confidence intervals • Estimate a population parameter • Mean HYPOTHESIS TESTING • To assess evidence provided by data about some claim concerning a population • SIGNIFICANCE TESTING • 2 ways of hypothesis testing – testing for statistical significance: • Classical hypothesis tests • P-value method REASONING OF TESTS OF SIGNIFICANCE • An outcome that would rarely happen if a claim were true is good evidence that the claim is not true
PARAMETRIC STATISTICAL INFERENCE:HYPOTHESIS TESTING • Example 1: Suppose we wish to know whether the mean number of weekly shopping trips made by households in a particular neighborhood of an urban area differs from the mean value for the urban area as a whole • Example 2: We measure carbon monoxide concentration from different residential neighborhoods in some city and we wish to compare the levels (are levels in one area significantly larger or smaller than levels observed in the other areas) Logic behind hypothesis testing: • Examine the likelihood (probability) that a alternate claim (or outcome) would occur if an existing claim were true • If the probability is high, then we say that there is not enough evidence to accept the alternate claim and we retain the original claim • If the probability is low, then we say that there is enough evidence against the original claim in support of the alternate claim • An outcome that would rarely happen if a claim were true is good evidence that the claim is not true
PARAMETRIC STATISTICAL INFERENCE:HYPOTHESIS TESTING Example:The mean household size in a certain city is 3.2 persons with a standard deviation of =1.6. A firm interested in estimating weekly household expenditures on food takes a random sample of n=100 households. To check whether the sample is truly representative, the firm calculates the mean household size of the sample to be 3.6 persons. What is the probability the firm’s sample is not representative of the city with respect to household size, such that the sample mean is actually greater than the mean for the entire city.
VOCABULARY OF HYPOTHESIS TESTS 1. Statement of hypotheses • Claims concern a population, so we express them in terms of population parameters • What is the population parameter being tested? • NULL HYPOTHESIS (Ho) – original claim - the statement being tested in a statistical test • Test is designed to assess the strength of evidence against the null hypothesis • Statement of “no effect” or “no difference” • ALTERNATIVE HYPOTHESIS (Ha) – alternate claim – the statement to the contrary of Ho • 3 possible combinations of Ho and Ha (a) Ho: =o; Ha: o (b) Ho: =o; Ha: >o (c) Ho: =o; Ha: <o
PARAMETRIC STATISTICAL INFERENCE More detail on hypothesis statements: One-sided versus two-sided alternative ·One-sided: we are interested only in deviations from the null hypothesis in one direction ·Two-sided: no direction is specified • Example: State the null and alternate hypotheses for each of the following problems: 1. The diameter of a spindle in a small motor is supposed to be 5 mm. If the spindle is either too small or too large, the motor will not work properly. The manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has moved away from the target. 2.Census Bureau data show that the mean household income in the area served by a shopping mall is $52,000 per year. A market research firm questions shoppers at the mall. The researchers suspect the mean household income of mall shoppers to be higher than that of the general population.
