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Statistics for the Behavioral and Social Sciences: A Brief Course Fifth Edition Arthur Aron, Elaine N. Aron, Elliot Coups Prepared by: Genna Hymowitz Stony Brook University This multimedia product and its contents are protected under copyright law. The following are prohibited by law: -any public performance or display, including transmission of any image over a network; -preparation of any derivative work, including the extraction, in whole or in part, of any images; -any rental, lease, or lending of the program. Copyright © 2011 by Pearson Education, Inc. All rights reserved Some Key Ingredients for Inferential Statistics Chapter 4 Copyright © 2011 by Pearson Education, Inc. All rights reserved Chapter Outline • • • • The Normal Curve Sample and Population Probability Normal Curves, Samples and Populations, and Probabilities in Research Articles Copyright © 2011 by Pearson Education, Inc. All rights reserved Inferential Statistics • Allow us to draw conclusions about theoretical principles that go beyond the group of participants in a particular study Copyright © 2011 by Pearson Education, Inc. All rights reserved The Normal Curve • Normal Distribution – histogram or frequency distribution that is a unimodal, symmetrical, and bell-shaped – a mathematical distribution – Researchers compare the distributions of their variables to see if they approximately follow the normal curve. Copyright © 2011 by Pearson Education, Inc. All rights reserved Why the Normal Curve Is Commonly Found in Nature • A person’s ratings on a variable or performance on a task is influenced by a number of random factors at each point in time. • These factors can make a person rate things like stress levels or mood as higher or lower than they actually are, or can make a person perform better or worse than they usually would. • Most of these positive and negative influences on performance or ratings cancel each other out. • Most scores will fall toward the middle, with few very low scores and few very high scores. – This results in an approximately normal distribution (unimodal, symmetrical, and bell-shaped). Copyright © 2011 by Pearson Education, Inc. All rights reserved The Normal Curve and the Percentage of Scores Between the Mean and 1 and 2 Standard Deviations from the Mean • There is a known percentage of scores that fall below any given point on a normal curve. – 50% of scores fall above the mean and 50% of scores fall below the mean. – 34% of scores fall between the mean and 1 standard deviation above the mean. – 34% of scores fall between the mean and 1 standard deviation below the mean. – 14% of scores fall between 1 standard deviation above the mean and 2 standard deviations above the mean. – 14% of scores fall between 1 standard deviation below the mean and 2 standard deviations below the mean. – 2% of scores fall between 2 and 3 standard deviations above the mean. – 2% of scores fall between 2 and 3 standard deviations below the mean. Copyright © 2011 by Pearson Education, Inc. All rights reserved The Normal Curve Table and Z Scores • A normal curve table shows the percentages of scores associated with the normal curve. – The first column of this table lists the Z score – The second column is labeled “% Mean to Z” and gives the percentage of scores between the mean and that Z score. – The third column is labeled “% in Tail.” . Z % Mean to Z % in Tail .09 3.59 46.41 .10 3.98 46.02 .11 4.38 45.62 Copyright © 2011 by Pearson Education, Inc. All rights reserved Using the Normal Curve Table to Figure a Percentage of Scores Above or Below a Raw Score • If you are beginning with a raw score, first change it to a Z Score. – Z = (X – M) / SD • • • Draw a picture of the normal curve, decide where the Z score falls on it, and shade in the area for which you are finding the percentage. Make a rough estimate of the shaded area’s percentage based on the 50%–34%–14% percentages. Find the exact percentages using the normal curve table. – Look up the Z score in the “Z” column of the table. – Find the percentage in the “% Mean to Z” column or the “% in Tail” column. • If the Z score is negative and you need to find the percentage of scores above this score, or if the Z score is positive and you need to find the percentage of scores below this score, you will need to add 50% to the percentage from the table. • Check that your exact percentage is within the range of your rough estimate. Copyright © 2011 by Pearson Education, Inc. All rights reserved Using the Normal Curve Table to Figure Z Scores and Raw Scores • • • • Draw a picture of the normal curve and shade in the approximate area of your percentage using the 50%–34%–14% percentages. Make a rough estimate of the Z score where the shaded area stops. Find the exact Z score using the normal curve table. Check that your Z score is within the range of your rough estimate. – From your picture, estimate the percentage of scores in the tail or between the mean and where the shading stops. • To figure the percentage between the mean and where the shading stops, you will sometimes need to subtract 50 from your percentage. – Look up the closest percentage in the appropriate column of the normal curve table. – Find the Z score for that percentage. • If you want to find a raw score, change it from the Z score. – X = (Z)(SD) + M Copyright © 2011 by Pearson Education, Inc. All rights reserved Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 1 • Draw a picture of the normal curve and shade in the approximate area of your percentage using the 50%– 34%–14% percentages. – We want the top 5%. – You would start shading slightly to the left of the 2 SD mark. Copyright © 2011 by Pearson Education, Inc. All rights reserved Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 2 • Make a rough estimate of the Z score where the shaded area stops. – The Z Score has to be between +1 and +2. Copyright © 2011 by Pearson Education, Inc. All rights reserved Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 3 • Find the exact Z score using the normal curve table. – We want the top 5% so we can use the “% in Tail” column of the normal curve table. – The closest percentage to 5% is 5.05%, which goes with a Z score of 1.64. Copyright © 2011 by Pearson Education, Inc. All rights reserved Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 4 • Check that your Z score is within the range of your rough estimate. – +1.64 is between +1 and +2. Copyright © 2011 by Pearson Education, Inc. All rights reserved Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores • Check that your Z score is within the range of your rough estimate. – From your picture, estimate the percentage of scores in the tail or between the mean and where the shading stops. • To figure the percentage between the mean and where the shading stops, you will sometimes need to subtract 50 from your percentage. – Look up the closest percentage in the appropriate column of the normal curve table. – Find the Z score for that percentage. • If you want to find a raw score, change it from the Z score. – X = (Z)(SD) + M Copyright © 2011 by Pearson Education, Inc. All rights reserved Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 5 • If you want to find a raw score, change it from the Z score. – X = (Z)(SD) + M – X = (1.64)(16) + 100 = 126.24 Copyright © 2011 by Pearson Education, Inc. All rights reserved How Are You Doing? • Use the partial normal curve table found below to answer the following question: • If the data from your study were normally distributed, what percentage of scores would fall between the mean and a Z score of .10? Z % Mean to Z % in Tail .09 3.59 46.41 .10 3.98 46.02 .11 4.38 45.62 Copyright © 2011 by Pearson Education, Inc. All rights reserved Sample and Population • Population – entire set of things of interest • e.g., the entire piggy bank of pennies • e.g., the entire population of individuals in the US • Sample – the part of the population about which you actually have information • e.g., a handful of pennies • e.g., 100 men and women who answered an online questionnaire about health care usage Copyright © 2011 by Pearson Education, Inc. All rights reserved Why Samples Instead of Populations Are Studied • It is usually more practical to obtain information from a sample than from the entire population. • The goal of research is to make generalizations or predictions about populations or events in general. • Much of social and behavioral research is conducted by evaluating a sample of individuals who are representative of a population of interest. Copyright © 2011 by Pearson Education, Inc. All rights reserved Methods of Sampling • Random Selection – method of choosing a sample in which each individual in the population has an equal chance of being selected • e.g., using a random number table • Haphazard Selection – method of selecting a sample of individuals to study by taking whoever is available or happens to be first on a list • This method of selection can result in a sample that is not representative of the population. Copyright © 2011 by Pearson Education, Inc. All rights reserved Statistical Terminology for Sample and Populations • Population Parameters – mean, variance, and standard deviation of a population – are usually unknown and can be estimated from information obtained from a sample of the population • Sample Statistics – mean, variance, and standard deviation you figure for the sample – calculated from known information Copyright © 2011 by Pearson Education, Inc. All rights reserved Probability • Expected relative frequency of a particular outcome – outcome • term used for discussing probability for the result of an experiment – expected relative frequency • number of successful outcomes divided by the number of total outcomes you would expect to get if you repeated an experiment a large number of times • long-run relative-frequency interpretation of probability – understanding of probability as the proportion of a particular outcome that you would get if the experiment were repeated many times Copyright © 2011 by Pearson Education, Inc. All rights reserved Steps for Figuring Probability • Determine the number of possible successful outcomes. • Determine the number of all possible outcomes. • Divide the number of possible successful outcomes by the number of all possible outcomes. Copyright © 2011 by Pearson Education, Inc. All rights reserved Figuring Probability • You have a jar that contains 100 jelly beans. • 9 of the jelly beans are green. • The probability of picking a green jelly bean would be 9 (# of successful outcomes) or 9% 100 (# of possible outcomes) Copyright © 2011 by Pearson Education, Inc. All rights reserved Range of Probabilities • Probability cannot be less than 0 or greater than 1. – Something with a probability of 0 has no chance of happening. – Something with a probability of 1 has a 100% chance of happening. Copyright © 2011 by Pearson Education, Inc. All rights reserved p • p is a symbol for probability. – Probability is usually written as a decimal, but can also be written as a fraction or percentage. – p < .05 • the probability is less than .05 Copyright © 2011 by Pearson Education, Inc. All rights reserved Probability, Z Scores, and the Normal Distribution • The normal distribution can also be thought of as a probability distribution. – The percentage of scores between two Z scores is the same as the probability of selecting a score between those two Z scores. Copyright © 2011 by Pearson Education, Inc. All rights reserved Normal Curves, Samples and Populations, and Probability in Research Articles • Normal curve is sometimes mentioned in the context of describing a pattern of scores on a particular variable. • Probability is discussed in the context of reporting statistical significance of study results. • Sample selection is usually mentioned in the methods section of a research article. Copyright © 2011 by Pearson Education, Inc. All rights reserved Key Points • • • • • • • In behavioral and social science research, scores on many variables approximately follow a normal curve which is a bell-shaped, symmetrical, and unimodal distribution. 50% of the scores on a normal curve are above the mean, 34% of the scores are between the mean and 1 standard deviation above the mean, and 14% of the scores are between 1 standard deviation above the mean and 2 standard deviations above the mean. A normal curve table is used to determine the percentage of scores between the mean and any particular Z score and the percentage of scores in the tail for any particular Z score. This table can also be used to find the percentage of scores above or below any Z score and to find the Z score for the point where a particular percentage of scores begins or ends. A population is a group of interest that cannot usually be studied in its entirety and a sample is a subgroup that is studied as representative of this larger group. Population parameters are the mean, variance, and standard deviation of a population, and sample statistics are the mean, variance, and standard deviation of a sample. Probability (p) is figured as the proportion of successful outcomes to total possible outcomes. It ranges from 0 (no chance of occurrence) to 1 (100% chance of occurrence). The normal curve can be used to determine the probability of scores falling within a particular range of values. Sample selection is sometimes discussed in research articles. Copyright © 2011 by Pearson Education, Inc. All rights reserved