FANR 3000 1nd EditionExam # 2 Study Guide Lectures: 7 – 10Lecture 7 (February 11)Data InterpretationI. Data Interpretation and T-tablea. T-distributioni. More area in the tails and less in the center than z-distributionii. As “n” increases though, a t-distribution will approach the standard normal distributioniii. T-values allows us to estimate uncertainty - Alpha= acceptable probability of erroro As we want to be more confident in our analyses, we decrease our alpha level o Alpha value is the acceptable probability of confidence (the inferencesyou make from the data you collect is wrong) Decrease in alpha levels= more confidence levels  Acceptable probability of error is the alpha value - Ex. For 95% confidence, we are saying that we are willing to accept a 5% chance our range of possible means is wrong, so alpha=.05 - Based on your degrees of freedom, you can calculate t-score from any alpha level interested o df= degrees of freedom= n-1 - within the distribution there is a true population mean- more samples results in less variability tighten cluster of t-distribution  less range of predicted values (more confident, more accurate numbers) - less samples more variability= bad II. Confidence intervals - A way to incorporate our sampling error in our population parameter estimate- Range of values around a sample mean that has a probability of containing the true population mean - Steps for determining confidence interval1. Calculate mean of sample2. Calculate variance, SD and SE3. Determine critical value from distribution table4. Plug values into CI formula- As CI increases, the CI get wider- Decrease variability, add more samples- Smaller sample sizes generate wider intervals Lecture 8 (February 16) P-Values III. Hypotheses - Inferential Statistics o To estimate a population parametero To test a hypothesis - The assumption made about the population parameters1. Determine null (H0) and alternative hypothesesa. H0- the belief that the mean of the population is <,>, or = a specific valueb. H1- opposite of the null hypothesis; holds true if H0 is found to be false c. Results of test will either bei. Reject null hypothesisii. Fail to reject null hypothesis IV. Type I and Type II Errors2. Specify level of significance a. Type I Error: a false positive; rejection of H0 when in reality it is true b. Type II Error: a false negative; failure to reject H0 when in reality it is not true - Increasing the sample size will reduce both types of error 3. Select the test statistic that will be used 4. Collect the sample data and compute the value of the test statistic V. P-values5. Use value of the test statistic to make a decision using a p-value a. P-values: the smallest level of significance at which the H0 will be rejected, assuming the H0 is true; the probability that the observed statistic occurred by chance alone b. To determine if the observed outcome is statistically significant, compare p-value to acceptable probability (α)i. If p-value <α, reject H0ii. If p-value >α, fail to reject H06. Interpret statistical results in “real world” termsLesson 9 (February 23) T-TestsI. Hypothesis Testing- Comparing 2 populations1) Determine null (H0) and alternative (H1) hypotheses 2) Specify level of significance (α, probability of Type I error) 3) Select the test statistic that will be used 4) Collect the sample data and compute the value of the test statistic 5) Use the value of the test statistic to make a decision using a rejection point (Zc, tc) or a p-value 6) Interpret statistical results in “real world” terms- α 1) Allowable error 2) The level of significance (p-value < α denotes significant results) 3) Acceptable probability of a Type I Error (0.01-0.10)4) the location of the rejection boundary is a function of α5) The smaller the value of α, the more difficult it is to reject H06) Test results are “stronger” with a lower αII. T-Tests - A 2 sample t-test is appropriate ONLY if the samples meet the following criteria:1. 2 samples (one from each population) are independent and random2. Both populations are approximately normally distributed - Compare 2 means using a new sampling distribution: the sampling distribution for the difference in means o To determine if the samples come from the same population - The larger the difference, the larger the calculated t-score, so the further out in the tail the calculated t-score will be 1. Determine Null and Alternative hypothesesa. When testing differences between means, the null hypothesis is that the difference between means is some specified value (usually zero)2. Choose a significance level (α )3. Select the test statistic that will be used (difference between sample means)4. Collect sample data from both populations5. Execute the test statistic III. Types of T-tests 1. One-sample t-test: trying to determine if there is a difference in the population mean calculated from your samples against the hypothesized population mean2. Two-sample t-test: to determine if there is a difference in the population mean between the two groups you have sampled. For equal or unequal variance3. Paired sample t-test: to determine differences between populations when you have paired samples. When you have similar sample/individuals each receiving a different treatment or when the same sample/individual is in the first treatment(control) and then is subjected to the second treatment - For t-values, if we calculate a t-value that is greater than our “critical t-value”, then the two populations are statistically different.- For p-values, if we calculate a p-value less than our alpha value, then we assume the two populations to be statistically different.Lecture 10 (March 2)Determining Sample Size I. Determining Sample Size- Mean+/- variance o Increase samples= decrease variance - Subject to time, budget, and ease of selection constraints o Task: determine how large of a sample a “good” estimate of population parameter would require - Determine how much error you are willing to accept in estimating the mean of a populationo Sample size should be statistically and practically efficient II. Calculate Sample Size 1. Using the bound on the sampling meana. Based on the absolute amount of error acceptable based on the standard error of the mean b. Bx= 2Sx 2. Using the standard deviation and desired closeness, or coefficient of variation (%,CV) and allowable/acceptable error (%, AE) - The formula used varies based