Unit Three: Null Hypothesis Statistical Testing Chapter 4.4: Probability & Normal Distribution 1. Determining % of scores that fall above or below a certain z score in a normal distribution: Turn the raw score into a zscore • 𝑟𝑎𝑤− 𝜇 𝜎 Interest in the proportion above or below the mean? • Depends on whether the question is asking you for something above or below. Base it off of question wording. =Z State the % that fall above/below the mean using the table. Draw histogram and shade area of interest. •Choose correct value in your book & look up bsolute value. 2. Random sample, is it true? Why? Random sample Every case in a population has an = chance of selection. Occurs everytime you make a selection. a) Selecting people with the highest test scores. Not a random bc chance isn’t equal. b) DPU students interested in going to grad school. Randomly pick 30/50. No because people volunteer, then you choose – since not everyone sees the ad, they don’t know about it and therefore don’t have an equal chance. c) Random digit dialing of homes with phones. Yes, all numbers have an = chance of being dialed. 3. Application of the law of large numbers to evaluate what scenario is more likely, given known probabilities. For each event, determine the probability of the outcome for option A & for B Event(s) •# of outcomes qualified as A/total # of outcomes. • probabilities = more accurate w/ random sample. The more events completed, the closer you will be to the true probabilities. • Larger = closer accuracy. Chapter 5: Probability & Normal Distribution and how they allow us to make inferences about populations based on samples 1. Define the following… Term/concept Definition Sampling Distribution Frequency distribution generated by taking repeated, random samples from a population and generating some value, like a mean, for each sample. Behaves on simulations & math – not actually something you do in the study. Dist. Of sample o Distribution of means taken from possible random samples of a certain size from means a population. o Demostrates how much sampeling error exists in the samples. Tells the f of different means of the samples that will occur when taking all samples of a certain size from a sample. Standard Error o The Standard deviation of a sampling distribution of the mean. on average, how far each random sample mean is from the true population mean. 2. Identify factors that affect the standard error & how they affect it. Population Standard deviation Sample Size = = Standard error sample mean (i.e. its further from the population value). Standard error sample mean (i.e. closer to true value so, more accurate). 3. What is the Central limit theorem? What does it tell us about the distribution of sample means? Central Limit Theorem Definition: 3 predictions about sampling distribution of the mean What does it tell us about the distribution of sample means? Statement about the shape that a sampling distribution of the mean takes if the size of the sample is large and every possible sample were obtained. If we know the population, the mean, the standard deviation, & the sample size, we can predict the behavior of sampling distribution of means. 1. Mean of the distribution of sample means = true population value 2. Standard error of the distribution of the mean = the standard deviation of the distribution of sample means. 3. Regardless of the shape of the original distribution, of scores, if you take a large enough sample, distribution of samples will be normally distributed. a. Only applies to samples of 30+ 1. It can predict the behavior of the sampling distribution means. 2. If sample is 30+, the shape will be a normal distribution. Chapter 6: Null hypothesis testing and the logic of inferential statistics Concept: Definition Hypothesis Null/Alternative Hypothesis One-tailed test Directional Two-tailed test Non-directional Alpha (or alpha level) Critical Value P-value A proposed explanation for observed facts. Statement OR prediction about a population value. Statement about population, not a sample. They must cover all possible outcomes. They must be all-inclusive/mutually exclusive. 1. Null (H0): Negative The population of the explanatory variable impact on the outcome variable. Specific prediction 2. Alternative (H1): Positive Explanatory effects the outcome in the population. Usually statement of what researcher believes to be true Predicts that explanatory has an impact on the outcome variable in a SPECIFIC DIRECTION. Says explanatory has positive or negative affect. Hypothesis predicting the explanatory variable has an impact on the outcome variable, but DOESN’T PREDICT DIRECTION. Doesn’t say whether explanatory has positive or negative affect. probability that a result will fall in the rare zone. Null true = reject null Set at .05 or 50% Value of test stat that forms the boundary btw the rare zone and the common zone of the sampling distribution of the test statistic. Probability of type I error: Error that occurs when the null is true but rejected. The same as alpha or significance level. Example Researcher is testing a technique to improve intelligence… 1. (H0): “The technique does not improve intelligence.” a. Since it makes a specific prediction, it would say, “the technique has 0 impact and doesn’t improve intelligence at all. 2. (H1): “The technique has some impact on intelligence” a. No specific prediction. “Cardiac patients who receive support from former patients have less anxiety and higher efficiency than other patients.” “There is a difference in anxiety and self-efficiency btw cardiac patients who receive support from former patients and those who do not.” 2. Given the results, determine the outcome of hypothesis test. 3. Interpret… 4. Appropriate vs. inappropriate interpretations of the results of hypothesis test. Explain. Day 13: get chapter/title (example of z-test is on page 195) 1. Z-Tests Z Test 𝒛= 𝑴−𝝁 𝝈𝒎 Definition/Application - Compares a sample mean to a population mean. - Standard deviation of the population is known. M = Sample mean 𝛍 = 𝐏𝐨𝐩 𝐦𝐞𝐚𝐧 𝛔𝐦 = 𝐬𝐭𝐝. 𝐞𝐫. 𝐦𝐞𝐚𝐧 Example “The test used to see whether adopted children differ in intelligence from the general population.” 2. Rejecting the Null or Failing to reject the Null Decision Appropriateness Reject the Null Z 1.96 -1.96 Z Z = - 1.96 or 1.96 Fail to Reject the Null -1.96 < Z < 1.96 Interpretation Results fall in rare zone so, the doctor rejects the null – therefore, he accepts the alternative and concludes that the population mean is something other than 100.