Statistical Inference: Sampling Distributions and Hypothesis Testing - Prof. Dongmei An, Study notes of Statistics

Download Statistical Inference: Sampling Distributions and Hypothesis Testing - Prof. Dongmei An and more Study notes Statistics in PDF only on Docsity! 1 ST3123 Introduction of Statistics II Review A: Basic Concepts in Inferential Statisticsc • Population and Sample(simple random sample) Population: _____________________________ Sample: ________________________________ • Simple random sampling • Parameter (____________) and statistic(______________) • Statistics(______________statistics and _______________statistics) • Inferential statistics (___________________and____________________) • Central Limit Theorem and sampling distribution of sample mean 1. (CLT) When sample size n is sufficiently large (________), the sampling distribution of sample mean x will be approximately a _________ distribution with mean _______________ and standard deviation______________, regardless the population distribution. The larger the sample size, the better will be the normal approximation to the sampling distribution of x . 2. When sample size n is small ( 30n < ), and the sample drawn from an approximately normally distributed population with meanμ and population standard deviation is known , 3. When sample size n is small ( 30n < ), and the sample drawn from an approximately normally distributed population with meanμ and population standard deviation is unknown, 2 B: two tables • Standard Normal table (z-table, Table IV, p894) 1. the total area under the z-curve is____; 2. symmetric and bell shaped; 3. the ________z falls in a specific range ( 1 2,z z ) equals the associated ____under the z-curve. • Critical Values of t- table (Table VI, p896) 1. the total area under the t-curve is_____; 2. symmetric and bell shaped (with heavy tails compared to z-curve); 3. the _________t falls in a specific range ( 1 2,t t ) equals the associated_____ under the t-curve. 4. Each t-curve is associated with its degree of freedom. When the degree of freedom is getting large, t-curve is close to z-curve. Example1: Given z value, try to find specific area under standard normal curve (p-value). Example2: Given a specific area under standard normal curve, try to find the corresponding z value (critical value). 5 Example3. Gun Control, In a survey conducted by Louis Harris of LH research, 1250 U.S. adults were polled regarding their views on banning handgun sales, 650 favored a ban. 1. Use a 95% confidence interval to estimate the proportion of all U.S. adults in favor of banning handgun sales. 2. Based on the confidence interval, can we infer that the majority of U.S. adults in favor of banning handgun sales? (check the sample size is large enough) . 8.1 The Elements of a test of hypothesis Introductory Example. Calcium: The level of calcium in the blood of healthy young adults is considered to be 10 milligrams per deciliter. A clinic measures the blood calcium of 64 healthy pregnant young women at their first visit for prenatal care. The mean of these 64 measurements is 9.8 and the standard deviation is 0.4. Is this evidence that the mean calcium level in the population of healthy pregnant young woman is less than 10? Use 01.0=α . 6 • Elements of a test of hypothesis: 1. _________________( 0H ): A theory about the values of one or more population parameters. The theory generally represents the status quo, which we adopt until it is proven false. 0H :___________________________ 2. ___________________( aH ): A theory that contradicts the null hypothesis. The theory generally represents that which we will accept only when sufficient evidence exists to establish its truth. 3. Test statistic: a ______________used to decide whether to reject the null hypothesis. 4. Rejection region: The numerical values of the test statistic for which the null hypothesis will be__________. The rejection region is chosen so that the probability of rejecting a true null hypothesis (making a Type I error) isα . α is also called _________________________. 5. Assumptions; 6. Calculation of test statistic based on the sample data; 7. Conclusion: a. If the numerical value of the test statistic _____________the rejection region , we _________the null hypothesis and conclude that the alternative hypothesis is true. b. If the numerical value of the test statistic _______________the rejection region, we _______________the null hypothesis and conclude that there is insufficient evidence to conclude that the alternative hypothesis is true. Note: Conclusions and consequences for a test of hypothesis 0H in fact is Conclusion true false Do not reject 0H Reject 0H Basic logic of hypothesis testing: 1. Take a random sample from the population of interest; 2. If the sample data is consistent with the_______________, _____________ 0H ; 3. If the sample data is consistent with the_________________, ______ 0H and in favor of aH . 7 8.2 Large-sample test of hypothesis about one population mean • Condition required for a valid large-sample hypothesis test for population meanμ : 1. A _____________is selected from the target population; 2. The sample size n is _______(n >___). Under these conditions, by Central Limit Theorem, The Sampling distribution of x is approximately ___________with: Mean: xμ = Standard error: xσ = • Large sample Z-test of hypothesis for a population meanμ : 1. set up null hypothesis and alternative hypothesis 0 :H __________________________ : ( ) : ( ) : ( ) a a a H two tailed or H lower tailed or H upper tailed − − − 2. significance level α 3. test statistic: xz n μ σ − = ≈ 4. rejection region: ____________when 0:aH μ μ≠ ___________ when 0:aH μ μ< _____________ when 0:aH μ μ> 5. conclusion: if the value of test statistic falls in rejection region, ______ 0H ，and conclude that at α level, there is sufficient evidence to conclude aH is true. if the value of test statistic does not fall in rejection region, ____________ 0H , and conclude that at α level, there is insufficient evidence to conclude aH is true. 10 8.3 Observed significance levels: p-values • p-value(OSL): the ___________(assuming 0H is true) of observing a value of the test statistic that is at least contradictory to the null hypothesis, and supportive of the alternative hypothesis, as the actual one computed from the sample data. P-value is the __________ significance level for which we would reject H0. • How to calculate the p-value: (the value of test statistic and aH ) 1. Determine _______________________ based on the sample data; 2. p-value = ____________ when 0:aH μ μ≠ p-value =______________ when 0:aH μ μ< p-value =______________ when 0:aH μ μ> Examples for p-value calculation: Ex.8.41, 42, p389, • How to make test conclusion based on p-value: If p-value is _____________the significance levelα , ________ 0H ; If p-value is _____________the significance levelα , _________________ 0H . 11 Three examples in section8.2 (using p-value to make conclusion). Example1, SSHA: A survey based on 81 incoming freshmen result in mean SSHA score is 116.2 and standard deviation is 25. Do the data provide evidence that the mean SSHA score is different from 115? Use significance level 05.0=α . Example2. Nicotine content: The nicotine content in cigarettes of a certain brand is normally distributed with standard deviation 1.0=σ milligrams. The brand advertises that the mean nicotine content of their cigarettes is 1.5 milligram, but measurements on a random sample of 100 cigarettes of this brand gave a mean of 1.53. Is this evidence that the mean nicotine content is actually higher than advertised? Use significance level 0.10α = . 12 Example3. HOSPLOS, P386, The length of stay (in days) in hospital for 100 randomly selected patients are presented in the table. Suppose we want to test the hypothesis that the true mean length of stay (LOS) at the hospital is less than 5 days. Use 05.0=α . LOS for 100 hospital patients 2, 3, 8, 6, 4, 4, 6, ………, 10, 2, 4, 2 SPSS output for HOSPLOS, P386: One-Sample Statistics N Mean Std. Deviation Std. Error Mean LOS 100 4.53 3.678 .368 One-Sample Test Test Value = 5 95% Confidence Interval of the Difference t df Sig. (2-tailed) Mean Difference Lower Upper LOS -1.278 99 .204 -.470 -1.20 .26 Example4. HOSPLOS, P386, The length of stay (in days) in hospital for 100 randomly selected patients are presented in the table. Do the data provide evidence that the true mean length of stay (LOS) at the hospital is different from 5 days. Use 05.0=α . 15 Example2. Ex.8.68, Mongolian desert ants, p396: To study the ants in Mongolia, the botanists placed seed baits at 11 sites and observed the number of ant species attracted to each site. Do the data indicate that the average number of ant species at Mongolian desert sites is greater than 5 species? Use α = 0.10. # of ant species 3, 3, 52, 7, 5, 49, 5, 4, 4, 5, 4 SPSS output for Example2. Ex.8.68, Mongolian desert ants, p396: One-Sample Statistics N Mean Std. Deviation Std. Error Mean ANTS 11 12.82 18.675 5.631 One-Sample Test Test Value = 5 95% Confidence Interval of the Difference t df Sig. (2-tailed) Mean Difference Lower Upper ANTS 1.388 10 .195 7.818 -4.73 20.36 16 Example3. A bottling company needs to produce bottles that will hold 12 ounces of liquid for a local brewery. To test this claim, the company randomly samples 14 bottles and finds the average amount of liquid held by the 14 bottles is 11.9 ounces with a standard deviation of 0.23 ounces. Do the data provide evidence that the mean liquid content differs from 12 ounces? Use α = 0.01. Section 8.4homework—(p394) 59-64, 69 17 8.5 Large-sample Test of hypotheses about a population proportion p • Condition required for a valid large-sample hypothesis test for p : 1. A____________ sample is elected from a binomial population; 2. The sample size n is _____(This condition will be satisfied if ˆ0 3 pp σ± falls between 0 and 1.) • Large-sample test for a population proportion p : 1. set up null hypothesis and alternative hypothesis 0 :H : ( ) : ( ) : ( ) a a a H two tailed or H lower tailed or H upper tailed − − − 2. significance level α 3. test statistic: 0 0ˆ 0 0, 1p p qz where q p n σ= = = − 4. rejection region: ______________when 0:aH p p≠ _____________ when 0:aH p p< ______________when 0:aH p p> 5. conclusion: if the value of test statistic falls in R.R, reject 0H ，and conclude that at α level, there is sufficient evidence to conclude aH is true. if the value of test statistic does not fall in R.R, do not reject 0H , and conclude that at α level, there is insufficient evidence to conclude aH is true.