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Unformatted text preview: INTRODUCTION TO HYPOTHESIS TESTING Apr 21’10 INTERVAL ESTIMATION TECHNIQUES vs HYPOTHESIS TESTING Let us consider in comparison two statistical inference procedures: interval estimation techniques and hypothesis testing. Both procedures imply that we are given a population with unknown mean µ (or maybe, unknown proportion) and with known (or unknown) standard error σ . We are given also a single random sample of size n drawn from that population for which we can compute a sample mean x * = x mean * and a corresponding standard deviation (designated as s (or σ s )) (or compute the sample proportion * and the standard deviation σ ). We are suggested to reach a probabilistic conclusion about a certain statistical parameter of a population (for example, a mean or proportion) on the basis of the information contained in a single random sample drawn from that population. Both inferential procedures are based on the known properties of the sampling distribution (see handout #9 ) and they use the same initial steps . Let us speak for certainty about the sampling distribution of the sample means. We already studied the statistical inference technique known as an estimation of a confidence interval for a population mean (see handout #10 ). We are given (or may compute directly) some statistical parameters of a single ra ndom sample drawn from a parent population: a sample size n , a sample mean, x * = x mean * , a sample variance, s 2 (sometimes a population variance, σ 2 , also may be known) and we should find a symmetric confidence interval centered at the point x mean * on the x mean- axis such that the probability to find a population mean µ in this interval is (1 – α) where α is a small number given to us. The Central Limit Theorem states (see handout #9 ) that a mean µ x_mean for such a sampling distribution is equal to the population mean, µ , and a standard deviation ( standard error of the sample mean) σ x_mean is equal to σ/n 1/2 , where σ is the population standard deviation. A population of the sample means is normally distributed when the parent population is normally distributed or it is approximately normally distributed for the samples of size n > 30 when the parent population is not normally distributed. (The approximation to the normal distribution improves with samples of larger size.) Thus, we may introduce for our sampling distribution of the sample means the normal curve (see (8.2); the graph is given in handouts #8 and #10 ) _ __ _ f(x) = [ 1/(σ x_mean √2π )] * exp[ –(x – µ x_mean ) 2 /(2σ x_mean 2 )] (*) _ where x means a continuous random variable representing a possible value of the sample mean for an arbitrary random sample drawn from the parent population. Then we may introduce the statistic _ _ _ z = (x – µ x_mean )/σ x_mean = (x – µ x_mean )/(σ/√n) (#) and work with the standard normal curve __ f(z) = ( 1/√2π ) * exp( –z 2 /2 ) (8.2b) 1 (The curve (8.2b) has a maximum at z = 0...
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- Spring '05
- Normal Distribution, Null hypothesis, Hypothesis testing