Psychological Statistics Exam 2 Actual Questions With 100% Correct Detailed Answers.

The Normal Distribution - -unimodal -symmetric -mathematically defined (shape has equation) -theoretical Big Five Traits ("OCEAN") - usually fit normal curve -Openness -Conscientiousness -Extraversion - high = very social; low = enjoys time alone -Agreeable -Neuroticism - high = stressed reactive, depression, anxiety - low = nothing bothers you - still a problem As your sample size ______, shape of distribution will more closely resemble the ______. As long as the underlying population distribution is ______ - 1. increases 2. the normal curve 3. normal Standardization - - Allows for comparisons across different measurement scales - Converts individual scores (from different normal distributions) to a shared normal distribution Z-score - Numerical value of a z-score specifies the distance (# of SD) that a value is above or below the mean Standard normal distribution - A normal distribution of z-scores Z- distribution - - The normal distribution of standardized scores mean (μ) = 0 SD (σ) = 1 If z = 1 then your value is 1 SD above the mean If z = -2 then your value is 2 SD below the mean What does the z-distribtuion allow us to do? - 1. Transform raw scores to z scores (standard scores) 2. Transfrom z scores into raw scores 3. Compare z scores to each other - even if the raw scores are measured on different scales 4. Transform z scores into percentiles Transforming raw scores -- z-scores - 1) subtract the population mean (μ) from the raw score (X) 2) Divide by the population SD (σ) z = (X-μ)/σ Transforming z-scores -- raw scores - 1) Multiple the z-score by the population SD (σ) 2) Add population mean (μ) to that product X = z(σ)+μ Transforming z-scores -- percentiles - 100% of the population is represented under the bell-shaped curve - Remember the Empirical Rule: +/- 1 SD = 68% of population +/1 2 SD = 95% +/- 3 SD = 99.7% The Central Limit Theorem - As sample size *increases*, the shape of a distribution of *sample means* approximates the normal curve, even when the population from which it was drawn does not CLT principles - 1. Repeated sampling approximates a normal curve, even when the original population is not normally distributed 2. A distribution of means is less variable than a distribution of individual scores Sampling Distribution (Distribution of means) - A distribution of *means* calculated on all samples of size n (same size), randomly drawn from a particular population - It is a good estimator of the value of the population mean Characteristics of Sampling Distribution - 1. *Unbiased estimator* = any sample statistic that equals the value of its respective population parameter 2. *Biased estimator* = any sample statistics that does not equal the value of its respective population parameter Example of estimators: - Pop of 3 (N=3) Person A: 8 Person B: 5 pop mean = 5 Person c: 2 sample mean = 5 -- unbiased Sampling Distribution - The Mean - The mean of the sampling distribution (μx) equals the mean of the population (μ) μx=μ - Sample mean = unbiased estimator Sampling Distribution - Variance - Will vary *minimally* from the value of the population Thus, the SD of the sampling distribution is smaller than the SD of the population SD sample means SD population Standard Error of the Mean (SEM) - Standard Error of a distribution of sample means sample mean = (σx) - This is the name for the SD of a distribution of means The SEM ____ as the population SD ____ and the sample size ____. - The SEM *decreases* as the population SD *decreases* and the sample size *increases* Sampling Error - The extent to which sample means selected from the same population differ from one another (by chance) raw scores -- z-scores -- percentages - 1. Convert a raw score into a z score 2. Look up given z score on the z table to find the percentage of scores between the mean and that z score If its a positive z-score - add it to 50% If its a negative z-score - subtract from 50% Hypothesis Testing - We test a hypothesis by determining the likelihood that a sample statistic would be selected if the hypothesis regarding the population parameter were true Parametric Tests - Inferential statistical analyses based on a set of assumptions about the population - requires assumptions about the population Assumptions - the criteria that are met, ideally, before a hypothesis is conducted Robust hypothesis test - One that produces valid results even when all assumptions are not met Assumptions or Conducting Analyses (Parametric Tests) - 1. The Dependent Variable is assessed using a scale measure (cannot be nominal or ordinal) 2. Participants are randomly selected 3. The distribution of the population of interest must be approximately normal Hypothesis testing with Z tests - 1. State the hypothesis 2. Set criteria for a decision 3. Compute the test statistics 4. Make a decision 1. Stating the Hypothesis - Null = Ho Alternative = H1 Null hypothesis - the hypothesis that there is no significant difference between specified populations, any observed difference being due to sampling or experimental error Alternative Hypothesis - the hypothesis that sample observations are influenced by some non-random cause 2. Setting the Criteria - 1. Critical Values 2. Critical Regions 3. Significance Testing Critical Values - test statistic values beyond which we reject the null hypothesis "cut off values" Critical Regions - the area in the tails of the distribution in which the null hypothesis will be rejected Significance Testing - significance level or alpha level (α level) = highest probable value taken as evidence against the null hypothesis - p-value: probability of obtaining sample values when the null hypothesis is true - low = good - high = bad 3. Computing the test statistic - Compute and then compare an inferential test statistics to the critical values - Z statistic can be used when the population mean and population SD are known - If a test gives a p-value lower than the significance level (α level), - the null hypothesis is rejected, and the results are "statistically significant". - The lower the α level chosen, the stronger the evidence required 4. Making a decision - - Does the s statistic fall in the critical regions? - Does is exceed the critical values? Non-directional (2-tailed) tests - Used to test hypotheses when we are interested in *any* alternative to the null hypothesis - The research hypothesis does not indicate a direction of the mean difference or change in the DV, but merely indicates that there will be a difference ex. Ho: μ=100 H1: μ =/= 100 Directional (1 tailed) tests - Used to test hypotheses when we are interested in a *specific* alternative to the null hypothesis - The research hypothesis is direction, posing either a mean decrease of a mean increase in the DV, but not both, as a result of the IV ex. Ho: μ=100 H1: μ 100 - These are rare in research Computing test statistics for a two-tailed test - In non-directional (two-tailed) tests, we divide the α level in half so that an equal proportion of area is placed in the upper and lower tail: .05/2 = .025 in each tail. Characteristics of a Hypothesis Test - Value being measured? *p-value* Type of distribution test is based on? *sampling distribution* What does it measure? *The probability of obtaining a measured sample mean* What can be inferred from the test? *Whether an effect exists in the population* Confidence intervals - - Centered around the mean of the sample - Most commonly used: 95% confidence level 1. Point estimate 2. Interval estimate Point estimate - a summary statistic from a sample that is just one number as an estimate of the population parameter Interval estimate - a sample statistic and provide a range of plausible values for the population parameter