Chapter 1 Introduction to Statistics Review

Statistics Exam Review Chapter 1: Introduction to Statistics Statistics: A set of mathematical procedures for organizing, summarizing, and interpreting information. Population: The entire set of individuals of interest for a particular research question. Sample: A set of individuals selected from a population, usually intended to represent the population in a research study. Parameter: A value, usually numerical, that describes a population. Statistic: A value, usually numerical, that describes a sample. Descriptive statistics: Statistical procedures used to summarize, organize, and simplify data. Inferential statistics: Techniques that allow us to study samples and them make generalizations about the populations from which they were selected. Sampling error: The naturally occurring discrepancy, or error, that exists between a sample statistic and the corresponding population parameter. Research methods:   Correlational method - Two different variables are observed to determine whether there is a relationship between them. - Cannot demonstrate a cause-and-effect relationship. - Quasi-independent variable: The nonexperimental form of an independent variable. Experimental method - One variable is manipulated (independent variable) while another variable is observed and measured (dependent variable). - Can establish a cause-and-effect relationship. - Control condition: Individuals receive either placebo treatment or no treatment to provide a baseline for comparison with the experimental condition. - Experimental condition: Individuals do receive the experimental treatment. Variables:   Discrete variable: Consists of separate, indivisible categories. No values can exist between two neighboring categories. Continuous variable: There are an infinite number of possible values that fall between any two observed values. A continuous variable is divisible into an infinite number of fractional parts. Scales:     Nominal scale: Consists of a set of categories that have different names. Measurements on a nominal scale label and categorize observations, but do not make any quantitative distinctions between observations. Ordinal scale: Consists of a set of categories that are organized in an ordered sequence. Measurements on an ordinal scale rank observations in terms of size or magnitude. Interval scale: Consists of ordered categories that are all intervals of exactly the same size. Equal differences between numbers on the scale reflect equal differences in magnitude. However, the zero point on an interval scale is arbitrary and does not indicate a zero amount of the variable being measured. Ratio scale: An interval scale with the additional feature of an absolute zero point. With a ratio scale, ratios of numbers do reflect ratios of magnitude. Chapter 2: Frequency Distributions Frequency distribution: An organized tabulation of the number of individuals located in each category on the scale of measurements. Obtaining ΣX from a frequency distribution table X 5 f 1 4 3 2 1 2 3 3 1 ΣX = 5 + 4 + 4 + 3 + 3 + 3 + 2 + 2 + 2 + 1 = 29 ΣX2 = 52 + 42 + 42 + 32 + 32 + 32 + 22 + 22 + 22 + 12 = 97 Proportion: The fraction of the total group that is associated with each score Proportion = p = f N Percentage = p(100) Interval width: A simple number that separates each interval (ex. 2, 5, 10, 20). Ex. An interval listed as 20-29 would have an interval width of 10. Grouped frequency distribution table:   Presents groups of scores rather than individual scores Does not provide enough information to obtain a complete listing of the original set of scores. Frequency distribution graphs:    Histograms and polygons: Used when the data consists of numerical scores that have been measured on an interval or a ratio scale. Bar graphs: Used when scores are measured on a nominal or ordinal scale (usually nonnumerical values). Graphs for population distribution: - Smooth curves: Used when a population distribution consists of numerical scores from an interval or a ratio scale. - Bar graphs: Used to graph relative frequencies of a population distribution. The shape of a frequency distribution: Note: The normal distribution is an example of a symmetrical distribution. Chapter 3: Measures of Central Tendency Central tendency: A statistical measure that attempts to determine the single value, usually located in the center of a distribution, that is most typical or most representative of the entire set of scores. The mean: The arithmetic average. Population mean is represented by μ. Sample mean is represented by M. Population mean: μ = ΣX N Sample mean: M = ΣX N The median: The midpoint of the distribution. Ex. 2 2 3 3 3 4 7 median = 3 Ex. 1 1 4 5 7 8 median = 4+5 = 4.5 2 Note: For a distribution with one or two extreme scores, the median is usually a more representative value than the mean. The mode: The score or category in a frequency distribution that has the greatest frequency. Note: It is possible for a distribution to have more than one mode. Ex. Selecting a measure of central tendency:   The median - Extreme scores or skewed distributions - Undetermined values - Open-ended distributions - Ordinal scale The mode - Nominal scales Discrete variables Describing shape Central tendency and the shape of the distribution: Positive skew Negative skew Chapter 4: Measures of Variability Variability: Provides a quantitative measure of the differences between scores in a distribution and describes the degree to which the scores are spread out or clustered together. Range: The distance covered by the scores in a distribution. Deviation: The distance from the mean. Variance: The average squared distance from the mean. Standard deviation: The square root of the variance and provides a measure of the standard, or average, distance from the mean. Standard Deviation and Variance for a Population Deviation score = X – μ SS or sum of squares: The sum of the squared deviation scores.   Definitional formula: SS = Σ(X – μ)2 - Steps to find SS 1. Find each deviation score (X – μ) 2. Square each deviation score (X – μ)2 3. Add the squared deviations Computational formula: SS = ΣX2 – ¿ ¿ Population variance = σ2 = SS N Population standard deviation = σ = √ σ 2 = √ SS N Standard Deviation and Variance for a Sample Deviation score = X – M SS or sum of squares:   Definitional formula: SS = Σ(X – M)2 Computational formula: SS = ΣX2 – ¿ ¿ Sample variance = s2 = SS n−1 Note: It is easier to see the mean difference between two samples if the sample variances are small. Sample standard deviation = s = √ s 2 = √ SS n−1 Degrees of freedom: df = n – 1 Chapter 5: z-scores: Location of Scores and Standardized Distribution z-score: Specifies the precise location of each X value within a distribution.   The sign tells whether the score is located above (+) or below (–) the mean. The number tells the distance between the score and the mean in terms of the number of standard deviations. For population: z = X = μ + zσ X−μ σ For sample: z = X−M s X = M + zs Notes:       A z-score of z = +1.00 always indicates a location exactly 1 standard deviation above the mean. If two individuals in the same population have identical X scores, they will also have identical z-scores. Transforming an entire distribution of scores into z-scores will not change the shape of the distribution. For any distribution of scores, the location identified by z = +1.00 and the location identified by z = –1.00 are exactly the same distance from the mean. The z-score distribution will always have a mean of zero. The distribution of z-scores will always have a standard deviation of 1. Standardized distribution: Composed of scores that have been transformed to create predetermined values for μ and σ. Standardized distributions are used to make dissimilar distributions comparable. Chapter 6: Probability The probability for any specific outcome is defined as a fraction or a proportion of all the possible outcomes. Ex. probability of A = ¿ of outcomes classified as A of possible outcomes ¿ total ¿ Note: All probabilities can be expressed as decimal values ranging from 0 to 1.00. A simple random sample: Requires that each individual in the population has an equal chance of being selected. Random sampling: Requires that each individual has an equal chance of being selected and that the probability of being selected stays constant from one selection to the next if more than one individual is selected. A sample produced by this technique is known as a random sample. Normal distribution: A symmetrical distribution, with the highest frequency in the middle and frequencies tapering off as you move toward either extreme. Ex. For a normal distribution with μ = 400 and σ = 80, find the following values: a. What X value separates the highest 10% of the distribution from the rest of the scores? z = 1.28  from column C = 0.1003 (closest value to 0.1 or 10%) z= 1.28 = X−μ σ X−400 80 102.4 = X – 400 X = 502.4 b. What X values form the boundaries from the middle 60% of the distribution? .3000 .3000 zupper = +0.84  from column D = 0.2995 (closest value to 0.3 or 30%) Xupper = zσ + μ Xlower = (-0.84)(80) + 400 = (0.84)(80) + 400 = 332.8 = 467.2 c. What is the probability of randomly selecting a score greater than X = 375? z= z= X−μ σ 375− 400 80 = -0.3125 p(X > 375) = p(z > -0.3125) = 0.6217 or 62.17% (from column B) zlower = -0.84 Chapter 7: Probability and Samples: The Distribution of Sample Means Sampling error: The natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter. Distribution of sample means: The collection of sample means for all of the possible random samples of a particular size (n) that can be obtained from a population. Notes:   The mean for the distribution of sample means is always equal to the mean for the population from which the samples are obtained. If samples are selected from a normal population, the distribution of sample means will also be normal. Sampling distribution: A distribution of statistics obtained by selecting all of the possible samples of a specific size from a population. Standard error (σM): The standard deviation of the distribution of sample means. The standard error provides a measure of how much distance is expected on average between a sample mean (M) and the population mean (μ). σ σM = n √ For a single score: σM = σ (standard deviation) Notes:   According to the central limit theorem, the standard error for a sample mean becomes smaller as the sample size increases. As the population standard deviation increases, the standard error will also increase. z-score for sample means: z = M −μ σM Ex. A population has a mean of μ = 80 with σ = 20. a. If a single score is randomly selected from this population, how much distance, on average, should you find between the score and the population mean? σM = σ = 20 b. If a sample of n = 6 scores is randomly selected from this population, how much distance, on average, should you find between the sample mean and the population mean? σM = = σ √n 20 √6 = 8.16 Ex. A sample of n = 16 scores is selected from a normal population with μ = 70 and σ = 20. a. What is the probability that the sample mean will be greater than 60? σM = σ √n z= M −μ σM p(M > 60) = p(z > -2.00) = 0.9772 or 97.72% 20 = √16 =5 = 60−70 5 = -2.00 b. What is the probability that the sample mean will be less than 66? z= = M −μ σM 66−70 5 = -0.80 p(M < 66) = p(z < -0.80) = 0.2119 or 21.19% c. What is the probability that the sample mean will be within 5 points of the population mean? That is, what is p(65 < X < 75)? z1 = = M −μ σM 65−70 5 z2 = = M −μ σM 75−70 5 p = p1 + p2 = 0.3413 + 0.3413 = -1.00  p1 = 0.3413 = 1.00  p1 = 0.3413 = 0.6826 or 68.26% Chapter 8: Introduction to Hypothesis Testing Hypothesis test: A statistical method that uses sample data to evaluate a hypothesis about a population. Null hypothesis (H0): States that in the general population there is no change, no difference, or no relationship. H0 predicts that the treatment (independent variable) has no effect on the scores (dependent variable) for the population. Notes:   In most situations, researchers would like the hypothesis test to reject the null hypothesis. If other factors are held constant, then increasing the sample size will increase the likelihood of rejecting the null hypothesis. Alternative hypothesis (H1): Opposite to null hypothesis. H1 predicts that the treatment has an effect on the scores for the population. Alpha level or level of significance: A probability value that is used to define the concept of “very unlikely” in a hypothesis test. Critical region: Composed of the extreme sample values that are very unlikely to be obtained if the null hypothesis is true. The boundaries from the critical region are determined by the alpha level. If sample data fall in the critical region, the null hypothesis is rejected. Note: If the sample data are in the critical region with  = 0.01, then the same sample data would still be in the critical region if  were changed to 0.05. Type Ι error: Occurs when a researcher rejects a null hypothesis that is actually true. The researcher concludes that a treatment has an effect when it actually doesn’t. Note: In a research report, p < 0.05 indicates that the probability of a Type Ι error is less than 0.05. Type ΙΙ error: Occurs when a researcher fails to reject a null hypothesis that is incorrect. Often occurs when a treatment actually does have an effect on the scores but the effect was not large enough to reject the null hypothesis. One-tailed test or directional hypothesis test: The critical region is contained in one tail of the distribution. In this test, the statistical hypotheses specify either an increase or a decrease in the population mean. Two-tailed test (standard format): The critical region is divided between the two tails of the distribution. Cohen’s d: Measures the distance between two means and is typically reported as a positive number even when the formula produces negative value. Cohen’s d = M −μ mean difference = standard deviation σ Ex. Some researchers claim that herbal supplements such as ginseng or ginkgo biloba enhance human memory. To test this claim, a researcher selects a sample of n = 25 college students. Each student is given a ginkgo biloba supplement daily for six weeks and then all the participants are given a standardized memory test. For the population, scores on the test are normally distributed with μ = 70 and σ = 15. The sample of n = 25 students had a mean score of M = 75. a. Are the data sufficient to demonstrate that the herb has a significant effect on memory? Use a two-tailed test with  = .05.  = 0.05  z = ± 1.96 (critical region) 15 σM = √25 =3 z= 75−70 3 = 1.67 Since the z-score is not in the critical region, we fail to reject the null hypothesis. Therefore, the data is not sufficient to demonstrate that the herb has a significant effect on memory. b. Compute Cohen’s d for this study. Cohen’s d = M −μ σ = 75−70 15 = 1 3 The power of a statistical test: The probability that the test will correctly reject a false null hypothesis. That is, power is the probability that the test will identify a treatment effect if one really exists. Note: If all other factors are held constant, increasing the sample size from n = 25 to n = 100 will increase the power of a statistical test. Chapter 14: Correlation Positive correlation: Both variables increase. Negative correlation: One variable increases as the other decreases. Pearson correlation: Measures the degree and the direction of the linear relationship between two variables. SP r = SS SS √ X Y Notes:   If the value of the Pearson correlation is r = +1.00 or -1.00, then all data points in the scatter plot fit perfectly on a straight line. Assuming that other factors are constant, a correlation of r = -0.95 will result in more accurate predictions than a correlation of r = +0.70. Ex. Compute the Pearson correlation for the following data. ΣX = 24 X 4 5 6 Y 3 3 5 4 4 5 1 ΣY = 16 ΣXY = 78 MX = 24 = 4.8 5 MY = SP = ΣXY - ΣXΣY n = 78 - 16 = 3.2 5 (24 )( 16 ) 5 = 1.2 SSX = Σ(X – MX)2 X 4 5 6 4 (X – MX) -0.8 0.2 1.2 -0.8 SSY = Σ(X – MY)2 (X – MX)2 0.64 0.04 1.44 0.64 X 3 3 5 4 1 (X – MY) -0.2 -0.2 1.8 0.8 -2.2 (X – MY)2 0.04 0.04 3.24 0.64 4.84 5 0.2 0.04 Σ(X – MX)2 = 2.8 r= SP √ SSxSSy 1.2 = √(2.8 )(8.8 ) = 0.24  The Pearson correlation is 0.24. Σ(X – MY)2 = 8.8