STAT 210 1nd EditionExam # 5 Study Guide Lectures: 24-29Lecture 24 (October 20)- Population Parameterso In statistics, inferences calculated from data that was collected from a sample and used to make statements (inferences).- Two types of statistical inferences:o Estimation of parameters using confidence intervalso Statistical tests about parameters - State the practical question in need of answering in inference procedure o Involves specifying population of interest and specific parameter of which inferences need to be made about- Confidence Intervalso Procedures that allow one to estimate unknown population parameters. Involves calculation and interpretation of interval.- Interpretation=statistical inferenceo Select sample from population to estimate unknown population parameter  Random selection Data collected from sample used to compute a statistic which becomes the starting point for confidence interval thus statistical inferenceo Point estimate- value computed from the data collected from the sampleo Unlikely that real value of population parameter will equal point estimate value. So, subtract/add margin of error to create interval of values in which parameter should be contained. This interval is called a confidence interval.- Confidence interval for a parameter: point estimate ±margin of erroro Margin of error calculated form sampling distribution of point estimate in use- Most common values of confidence interval: 90%, 95%, 98%, and 99%.o However, any value above 0% and less than 100% can be used.o Amount of confidence is called the confidence level.- Confidence interval is interpreted by stating that there is 100*C% confidence that unknown population parameter falls between lower limit (L= point estimate – margin of error) and upper limit (U= point estimate + margin of error)- Width of confidence interval: width=2*margin of erroro Want margin of error/interval width to be as small as possible- Tests of Significance: hypothesize that unknown population parameter equals some value and use data in sample to test if this value is reasonable or not- Statistical hypothesis: Statement about some population parameter. o Use Greek letters to denote parameters. - Tests of Significance1. Null Hypothesisa. H0b. Hypothesis about population parameter presumed to be true. Usually a statement of no effect/changec. Will always have an equal to sign2. Alternative/Research Hypothesisa. Ha or H1b. Hypothesis about population parameter that researcher suspects/hopes is truei. Ha can have greater than sign; upper one-sided testii. Ha can have less than sign; lower one-sided testiii. Ha can have not equal to sign; two-sided testLecture 25 (October 22)- Tests of Significance continued…:3. Test Statistica. Quantity calculated from sample data collected. Used to determine strength of evidence against null hypothesis.b. If test statistic is close to hypothesized value, likely that null hypothesis is correct. If not, alternative hypothesis is correct.4. When assume null hypothesis is true, test will result in one of two decisions:a. Reject H0; conclusion that alternative hypothesis is true; this test is said to be significant b. Fail to reject H0; don’t conclude alternative hypothesis is true; this testis said to be insignificant i. Can have two kinds of errors involved in these decisions:1. Type I error where reject the null hypothesis when it is really true2. Type II error where fail to reject null hypothesis when it is really false- Type I error more serious than Type II. In a statistical test, attempt to control probability oftype I error. Leads to concept of a significance level of a test.- Significance level of a test is the max probability of type I error that researcher willing to risk. Denoted by Greek letter alpha () and usually set to be relatively small.o Significance level always stated at beginning of the process when null/alternative hypothesis are stated.- Can take two approaches to determine if reject/don’t reject null hypothesis:1. Compute p-value of test, compare it directly to significance level2. Create rejection region, compare test statistic to rejection region to formulate decision.- The p-value is the probability under the assumption that the null hypothesis is true. The test statistic takes this value as extreme/more extreme than the value observed.- Calculation of p-value requires:o Determination of the distribution of test statistic (either Z or T distribution)o Determination of test being conducted (depends on sign in alternative hypothesis)- After calculating p-value, compare directly to significance level ()o If p-value   then reject HO and conclude that alternative hypothesis is trueo If p-value >  then fail to reject HO and conclude that not enough evidence that alternative hypothesis is true. Small p-values favor alternative hypothesis- Rejection region gives range of values that test statistic can take that leads to the rejectionof the null hypothesis in favor of the alternative hypothesis. Dependent on the type of test, distribution of test statistic, and significance level chosen.- General Significance Testing Procedure1. State null & alternative hypotheses as well as the significance level  that will be used.2. Do the experiment, collect data, verify assumptions, and if appropriate compute value of test statistic3. Calculate the p-value4. Decide on significance of the test (reject or fail to reject HO)5. Create conclusion statement in words of original problem. This is the statisticalinference. Lecture 26 (October 24)- Sample proportion= ^p- Population proportion= - From sample data compute sample proportion which ten becomes the basis for inferencesthat will be made about unknown population proportiono Sample proportion is called the point estimate of population proportion- 0-1 random variableo Success is 1, failure is 0o Proportion of successes in population is , so proportion of failures is 1-o Will never be normal- Assumptions1. Data used to make inferences must be simple random sample from population2. Population distribution must be normal (sample size must be large enough for central limit theorem to apply)- Central Limit Theoremo If sample size is large enough then distribution of sample proportion is approximately normal Sample is large enough if both the expected number of successes n and expected number of failures 1- are greater