Discussion: Sociology Hypothesis Testing

Discussion: Sociology Hypothesis Testing ORDER NOW FOR CUSTOMIZED AND ORIGINAL ESSAY PAPERS ON Discussion: Sociology Hypothesis Testing 1. Hypothesis testing: how to form hypotheses (null and alternative); what is the meaning of reject the null or fail to reject the null; how to compare the p-value to the significant level (suchlike alpha = 0.05), and what a smaller p-value means. Discussion: Sociology Hypothesis Testing 2. How to interpret the one-sample t-test results: what are Ho and Ha; the standard for determining statistical significance, i.e., t statistic and p-value; what are the steps for the one-sample t test; what a normal distribution looks like. 3. How to interpret the one-way ANOVA results: what are Ho and Ha; the standard for determining statistical significance, i.e., F statistic and p-value; what an F distribution looks like. 4. How to interpret the simple linear regression results: what are Ho and Ha; the standard for determining statistical significance, i.e., t statistic and p-value of the slope; what is the slope and what it means; what is the R-square (not R, it is R-square!) and what it means; what are independent variables and dependent variable, and what their relationships are; how would you plot the relationship between a dependent variable and an independent variable; from a given independent variable, how would you predict the value of a dependent variable. 5. How to interpret the multiple regression results: how to interpret the slope of an independent variable (i.e., the impact of this independent variable, holding other independent variables constance). Discussion: Sociology Hypothesis Testing _midterm_review.docx Form: 20 questions in total. 10 multiple choice or filling the blanks; 10 short responses, related to the statistical tables provided (suchlike those tables in HW assignments). Key points are summarized below: Level of measurement: understand what are continuous and discrete variables, and examples of different types (discrete, continuous, and the 4 types below) Hypothesis testing: how to form hypotheses (null and alternative); what is the meaning of reject the null or fail to reject the null; how to compare the p-value to the significant level (suchlike alpha = 0.05), and what a smaller p-value means. How to interpret the one-sample t-test results: what are Ho and Ha; the standard for determining statistical significance, i.e., t statistic and p-value; what are the steps for the one-sample t test; what a normal distribution looks like. How to interpret the one-way ANOVA results: what are Ho and Ha; the standard for determining statistical significance, i.e., F statistic and p-value; what an F distribution looks like. How to interpret the simple linear regression results: what are Ho and Ha; the standard for determining statistical significance, i.e., t statistic and p-value of the slope; what is the slope and what it means; what is the R-square (not R, it is R-square!) and what it means; what are independent variables and dependent variable, and what their relationships are; how would you plot the relationship between a dependent variable and an independent variable; from a given independent variable, how would you predict the value of a dependent variable. How to interpret the multiple regression results: how to interpret the slope of an in dependent variable (i.e., the impact of this independent variable, holding other independent variables constance). Discussion: Sociology Hypothesis Testing Understand how to use SPSS or Stata to produce all of the tables that you have had to handle so far. Homework 1: Tables used: Homework 2: Tables used: Homework 3: Tables used: Be familiar with the variables housed in the GSS dataset. Limited because it doesn’t have a lot of the best kind of variables, but the variables still work. Limitations: level of measurement / going to be a lot of times you have to overlook the problems HAPMAR (happiness in marriage), RINCOME (income), PAPRES10 (father’s prestige score) `How are they coded? HAPMAR ? 1 = very happy, 2 = pretty happy, 3 = not too happy, 8 = don’t know, 9 = no answer, 0 = Not applicable RINCOME ? 1 = Lt $1000, 2 = $1000 – $2999, […], 12 = $25000 or more, 13 = Refused, 98 = Don’t know, 99 = No answer, 0 = applicable PAPRES ? F“or the 3 different ‘papres’ variables on GSS, there are no labels associated with the codes Levels of measurement? HAPMAR – nominal RINCOME – ordinal PAPRES – interval Be able to distinguish among various levels of measurement for variables. Nominal Data cannot be ordered nor can it be used in calculations Republican, democrat, green, libertarian Not useful in calculations – Data is qualitative, can’t be used in a meaningful way such as means and standard deviations. Discussion: Sociology Hypothesis Testing Ordinal Data that can be ordered, differences cannot be measured Small – 8oz, medium – 12oz, large – 32oz Cities ranked 1-10, but differences between the cities don’t make sense/ can’t know how much better life is in city 1 vs city 2 Also shouldn’t be used in calculations Interval Data with a definite ordering but not starting point; the differences can be measured, but there is no such thing as a ratio Not only classifies and orders the measurements, but it also specifies that the distances between each interval on the scale are equivalent along the scale from low interval to high interval Can be ordered and differences between the data make sense Data at this level does not have a starting point 0 degrees doesn’t mean absence of temperature think temperature: 10?+10?=20? but 20? is not twice as hot as 10?. We can see this when we convert to Farenheit; 10?= 50?, but 20?= 68?. Ratio Data Data with a starting point that can be ordered; the differences have meaning and ratios can be calculated All features of interval data plus absolute zero Phrases such as “four times as likely” are actually meaningful Is defined as a quantitative data, having the same properties as interval data, with an equal and definitive ratio between each data and absolute “zero” being treated as a point of origin Tell us about the order, the exact value in between units Height, weight, duration Both descriptive and inferential statistics can be applied Discussion: Sociology Hypothesis Testing Your highest level, your most sophisticated Axis of whatever you are measuring There can be no negative numeric value in ratio data Amount of money in your pocket right now Understand the difference between continuous and discrete variables. Discrete data Very discrete spaces in between values / not going to have values in between whole numbers Certain number of values; positive, whole numbers (like number of people) Continuous data Fractional size spaces in between Capturing every moment of the process / any value between a given range Height, weight, etc. Not restricted to separate values Occupies any value over a continuous data value Age Why is it important to know #4 and #5 in performing statistical procedures. Not all variable types can have statistical procedures performed on them Affects what type of analytical techniques can be used on the data and what conclusions can be drawn Important to understand that they are just 2 different types of data which will explain the relationship of the data & create a better understanding for analysis Important because you always want to know the level of measurement before you start analysis – you want to choose the right way of doing analysis What do we mean by inference? Inference: causal Something caused/influenced another thing A caused by B Concerned primarily with understanding the quality of parameter estimates How sure are we that estimated xbar is near true population mean µ Reliability of statistical relationships, typically on the basis of random sampling Would you need to perform any work regarding inference with population data? No, inferential statistics allows you to make inferences about the population based on sample data. No inferences would need to be made if you had population data. What is the purpose of hypothesis testing, and on what kind of data? Hypothesis testing is the primary mechanism for making decisions based on observed sample statistics We want to know if there’s any relationship – causal or correlated Related to the conclusion we can get/ pre-score and post-score see if there’s a difference Must be done with continuous sample data The alpha level tells you that you’re operating at the possibility of being wrong Working cautiously and understanding limitations What are the important components of hypothesis testing? What are the essential elements? Read all the elements to understand what it’s about. Discussion: Sociology Hypothesis Testing Know sampling statistic – derive from own data Critical value – get off curve Compare critical value to the point you derive from your data Based on the level of significance, you draw a conclusion There’s a lot of components – you have to have a dataset, have to construct your own hypothesis, find mean & variance to construct analysis Null & alternative hypotheses Test statistic Sampling statistic Critical value Probability values and statistical significance Conclusions of hypothesis testing What are the steps in performing a hypothesis test? Specify the null hypothesis and alternative hypothesis assumptions / givens Random sampling, known parameters, levels of measurement, known statistics Set the significance level (alpha value) Calculate the test statistic and corresponding p-value Drawing a conclusion Be able to draw a “curve” and label that curve appropriately for a hypothesis test. Plot number line below curve and be able to do the math Make sure math matches curve If it’s a two tailed test make sure you break it up into two sides F is always one tail Question about greater than or equal to – it’s a one-sided test What alternative is there to a “curve”? You can walk through the equation without drawing a curve Ex: calculate p-value and compare that to the critical value You perform the test and afterwards and tell people how to determine if that’s significant or not How do tests of proportion differ from tests of means? A test of proportions seeks to find a statistically significant difference between the proportions of two groups. A test of means seeks to find a statistically significant difference between the means of two groups. What is a sampling distribution and how is it derived? A sampling distribution is a probability distribution of a statistic obtained through a large number of samples drawn from a specific population It tells us which outcomes we should expect for some sample statistics (mean, standard deviation, correlation, etc Discussion: Sociology Hypothesis Testing Represents the distribution of the point estimates based on samples of a fixed size from a certain population. It is useful to think of a particular point estimate as being drawn from such distribution. Understanding the concept of a sampling distribution is central to understanding statistical inference. Example below: unimodal and approximately symmetric. Centered exactly at true population mean µ=3.90. Sample means should tend to fall around population mean. What are sampling distributions used for? Knowledge of sampling distribution & making inferences about the overall population What is a significance level? How is it interpreted? (significance level = a) Probability of error / doing our best to get as close as we can. Restricting to 5%, 1%, etc. The significance level, also denoted as alpha or a is the probability of rejecting the null hypothesis when it is true. For example, a significance level of .05 indicates a 5% risk of concluding that a difference exists when there is no actual difference (95% confidence interval to evaluate hypothesis test). With this example, we will make an error whenever the point estimate is at least 1.96 standard errors away from population parameter (about 5% of the time, 2.5% on each tail) Can you set your level of significance anywhere? Yes you can – you’re essentially making an assumption at the beginning of your statistical experiment so you can adjust it to whatever you want Lower the alpha(significance level), more confident Coming in with an alpha of .01 – one would most likely assume that findings would be somewhat significant What do we mean by a “significant” finding? Differences that are being studied are real and not due to chance What are the basic things you need to perform a hypothesis test? Parameter & Statistic parameter: summary description of a fixed characteristic or measure of the target population. Denotes the true value that would be obtained if a census rather than a sample were undertaken Mean (µ), Variance ( o ˆ2), standard deviation ( o ), proportion (p) Statistic: summary description of a characteristic or measure of the sample. The sample statistic is used as an estimate of the population parameter Sample mean (xbar), sample variance (S^2), sample standard deviation (S), sample proportion (pbar) Sampling Distribution: probability distribution of a statistic obtained through a large number of samples drawn from a specific population Standard Error: similar to standard deviation – both are measures of spread. The higher the number, the more spread out your data is. Standard error uses statistics (sample data) and standard deviation uses parameters (population data) Tells you how far your sample statistic (such as sample mean) deviates from the actual population mean. Larger your sample size, the smaller the SE/closer your sample mean is to the actual population mean. Null hypothesis: a statement in which no difference or effect is expected Alternate hypothesis: a statement that some difference or effect is expected.Discussion: Sociology Hypothesis Testing Descriptive statistics Brief descriptive coefficients that summarize a given data set, which can be either a representation of the entire or a sample of a population/ summarizes or describes characteristics of a data set Broken down into measures of central tendency (mean, median, mode) and measures of variability (spread – standard deviation, variance, minimum and maximum variables, skewness) What do you run on the computer at the very start of a hypothesis test? (Varies with type of test) Run a frequency distribution to make sure your levels of measurement match the procedures you want to do What is a test statistic and how many test statistics have we worked with so far? Test statistic measures how close the sample has come to the null hypothesis. Its observed value changes randomly from one random sample to a different sample. A test statistic contains information about the data that is relevant for deciding whether to reject the null hypothesis or not Hypothesis test Test Statistic Z-Test Z-Statistic t-test t-statistic ANOVA F-statistic Chi-square tests Chi-square statistic What is a frequency distribution and a cross tabulation and how do you interpret them? Frequency distribution: shows you how common values are within the variable We can get an idea about whether something is a continuous or categorical variable/ snapshot view of the characteristics of a data set – allows you to see how scores are distributed across the whole set of scores (spread evenly, skew, etc.) SPSS steps: click on analyze —> descriptive statistics —> frequencies Move the variable of interest into the right-hand column Click on the chart button, select histograms, and press continue and OK to generate distribution table Cross tabulations: shows where the variables have something in common, seen at the intersec tion of the row and the column summarize the association between two categorical variables joint frequency distribution of cases based on two or more categorical variables SPSS steps: analyze —> descriptive statistics —> select cross tabulation Here you will see Rows and Columns. You can select one or more than one variable in each of these boxes, depending on what you have to compare, then click on OK. For percentages – analyze —> descriptive statistics —> crosstabs —> cells —> under percentage, select all 3 options Can you determine the level of measurement from a frequency distribution? Yes, the independent variable of a frequency distribution should indicate its level of measurement – which is typically categorical What is the purpose of an analysis of variance? Is it relevant for data that comes in proportions? Discussion: Sociology Hypothesis Testing ANOVA uses a single hypothesis test to check whether the means across many groups are equal: H0: The mean outcome is the same across all groups. In statistical notation, µ1 = µ2 =…… = µk where µi represents the mean of the outcome for observations in category i. HA: At least one mean is different. Generally we must check three conditions on the data before performing ANOVA: the observations are independent within and across groups, the data within each group are nearly normal, and the variability across the groups is about equal How do you calculate Eta 2 from ANOVA and how do you interpret it? (from the reading) A measure in ANOVA that tells you how much variance is in between each variable Is a measure in ANOVA (h^2) – proportion of the total variance that is attributed to an effect. It is calculated as the ratio of the effect variance (SSeffect) to the total variance (SStotal) We will be given value and just need to interpret it on test Example: Total SS: 62.29, Anxiety SS: 4.08 —> 4.08/62.29 = 6.6% 6% of variance is associated with anxiety What kind of data is needed for an analysis of variance? Dependent variable must be a continuous (interval or ratio) level of measurement Independent variable must be a categorical (nominal or ordinal variable) Two way ANOVA has 2 independent variables Females may have higher IQ scores compared to males, but this difference could be greater or less in European countries compared to North American countries ANOVA assumes: data is normally distributed, homogeneity of variance (variance among groups should be approx. equal), observations independent of each other How does ANOVA work with both means and variances? Inferences about means are made by analyzing variance What is the equation for ANOVA? F = MST/MSE where F = Anova coefficient, MST = mean sum of squares due to treatment, MSE = mean sum of squares due to error MST = SST/p-1 SST = ?n(x-xbar)^2 where SST = sum of squares due to treatment, p = total number of populations, n = total number of samples in a population MSE = SSE/N-p SSE = ?(n-1)S^2 Where SSE = sum of squares due to error, S = standard deviation of samples, and N = total number of observations F=MSbetween/MSwithin What kind of conclusion are we looking to draw from an ANOVA procedure? What is ALL that we can report? We are looking to see if the means between groups are statistically equal to one another, which is all we can report. P-value and Eta^2 What are we able to conclude from linear regression that we have not been able to conclude with other procedures? Based on what? The growth of dependent variable due to changing (can be positive or negative) of 1 unit of independent variable. Which group is significantly different from the others (coding each group as one binary independent variable). What level of variable measurement is ideal for regression? Why? Continuous variable Any time you’re working with means, you want to be working with ratios because you want to be able to have continuous data with an absolute zero Why are certain levels of measurement problematic? TA doesn’t think they are problematic, but – for some variables getting the mean doesn’t make sense If not continuous, maybe it’s not normally distributed.Discussion: Sociology Hypothesis Testing OTHER NOTES / READING NOTES Descriptive statistics: uses the data to provide descriptions of the population, either through numerical calculations or graphs or tables Inferential statistics: makes inferences and predictions about a population based on a sample of data taken from the population in question ANOVA Analysis of variance using a test statistic F/ uses single hypothesis test to check whether the means across many groups are equal Null: mean outcome is the same across all groups; Alternate: at least one mean is different Interval or ratio level data 3 conditions before performing ANOVA: the observations are independent within and across groups The data within each group are nearly normal The variability across the groups is about equal Example: consider a stats department that runs three lectures of an introductory stats course. We might like to determine whether there are statistically significant differences in first exam scores in these three classes (A,B, and C). Describe appropriate hypotheses to determine whether there are any differences between the 3 classes. H0= Average score is identical in all lectures, any observed difference is due to chance. HA: average score varies by class Mean square between groups( MSG) : Simultaneously consider many groups, and evaluate whether their sample means differ more than we would expect from natural variation Mean square between groups is quite useless so we compute a pooled variance estimate mean square error (MSE). MSE has an associated degrees of freedom value dfE= n – k It is helpful to think of MSE as a measure of variability within the groups. When the null hypothesis is true, any differences among the sample means are only due to chance and the MSG and MSE should be about equal. As a test statistic for ANOVA, we examine the fraction of MSG and MSE F = MSG/MSE The MSG represents a measure of the between-group variability, and MSE measures the variability within each of the groups ANOVA on SPSS One-way: Analyze > Compare means > One way ANOVA Dependent list: variable whose means will be compared between the samples Factor: the independent variable: categories will define which samples will be compared F test When to use F-test: F: represents a standardized ratio of variability in the sample means relative to the variability within groups. If null is true, F follows an F distribution. The upper tail of the F distribution is used to represent the p-value We can use the F statistic to evaluate the hypotheses in what is called an F test. A p-value can be computed from the F statistic using an F distribution, which has two associated parameters df1 and df2 The larger the observed variability in the sample means (MSG) relative to the within-group observations (MSE), the larger F will be and the strongest evidence against the null hypothesis. Because larger values of F represent stronger evidence against the null hypothesis, we use the upper tail of the distribution to compute a p-value.Discussion: Sociology Hypothesis Testing P-value is how significant your findings are Used to determine statistical significance in a hypothesis test; evaluate how well the sample data support the devil’s advocate argument that the null hypothesis is true. Measures how compatible your data are with the null hypothesis. The result you find from your z or t score after doing test Lower the better – more likely that you can reject your null For F—> tail is where significant values are Alpha is what you set beforehand to see if p-value is going to be below it a= 1 – confidence interval µ= population mean, xbar = sample mean Variance Trying to see how close together a data set is T test (steps and components) When to do T-Test: On SPSS: Analyze > Compare means > Independent Samples T Test Test variables: the dependent variable(s)/ continuous variable whose means will be compared between the two groups Grouping variable: independent variable/categories of the independent variable will define which samples will be compared in the t test Steps using calculator/walk through procedure: Discussion: Sociology Hypothesis Testing Z tests (steps and components) When to do Z-test Population is always a z test The formula for calculating a z -score is z =(x-?)/?, where ? is the population mean and ? is the population standard deviation (note: if you don’t know the population standard deviation or the sample size is below 6, you should use a t-score instead of a z -score). Giving frequency table and understanding how its coded What does the table/number represent What kind of data is that Happiness of marriage: categorical One-tailed test (steps and components) A statistical hypothesis test in which the critical area of a distribution is one-sided so that it is either greater than or less than a certain value, but not both. If sample being tested falls into the one sided critical area, the alternative hypothesis will be accepted instead of the null Two-tailed test (steps and components) Method in which the critical area of a distribution is two-sided and tests whether a sample is greater than or less than a certain range of values. If sample being tested falls into either of the critical areas, the alternative hypothesis is accepted instead of the null. Review Session/OH notes Know how to read the curve and table for a Z, T, and F test Probability, different parameters, different testing, etc. Don’t need to know all equations – but do need to know really straightforward equations e. F test = means squared/ another means squared Sum of squared/sum of squared Know what all of these mean Know how sampling distributions work Know something about z, t, f scores/ what they mean Z-tests are statistical calculations that can be used to compare population means to a samples A z-score is a measure of position that indicates the number of standard deviations a data value lies from the mean. Positive if above mean, negative if below. T-tests are calculations used to test a hypothesis, most useful when we need to determine if there is a statistically significant difference between two independent sample groups Comparing two related samples Population is infinite and normal, population variance is unknown and estimated from sample, mean is known, sample observations are random and independent, sample size is small, null may be one sided or two sided F-test is used to test the equality of two populations/ if data conforms to a regression model which is acquired through least square analysis/ determines whether any of the independent variables is having a linear relationship with the dependent variable A statistical test which determines the equality of the variances of the two normal datasets How much proportion of the variation is being contributed by this effect —> n^2=SSeffect/SStotal Might encounter a situation where we have so many groups, might not be a huge impact because of so many groups Hypothesis test with slope y=B0+B1X1 Testing if slope is significant (B1) Discussion: Sociology Hypothesis Testing Null hypothesis: B1=0 Sociology 113 Cumulative Final Exam Study Guide All the knowledge you need included below. Understand how to use SPSS or Stata to produce all of the tables that you have had to handle so far. Frequency Distribution, Cross tabulation, ANOVA Output, Two-Sample T-Test Frequency Distribution Analyze ? Descriptive Statistics ? Frequency Distribution Used in order to summarize categorical variables Cross Tabulation Analyze ? Descriptive Statistics ? Cross Tabulation Used in order to expose relationships between two separate variables ANOVA Output Analyze ? Compare Means ? One-Way ANOVA output Independent variable goes under ‘factor’ Dependent variable goes under ‘dependent list’ Discussion: Sociology Hypothesis Testing Post Hoc test at significance level 0.05 If P is = or < 0.05, then reject the null. If above, then fail to reject. Two-Sample T-Test Analyze ? Compare Means ? Independent Samples T-Test Input Test variable and grouping variable If sig (2-tailed) is below 0.05, reject the null hypothesis Be familiar with the variables housed in the GSS dataset. Be familiar particularly with the variables used in the homework happiness in marriage (HAPMAR), respondent’s income (RINCOME), father’s prestige score (PAPRES10) How are they coded? [a] Levels of measurement? HAPMAR – nominal [b] [c] [d] RINCOME – ordinal PAPRES – interval Be able to distinguish among various levels of measurement for variables. Nominal – name only; labels with no numerical significance cannot perform statistical procedures Ordinal – ordered levels or ranks; differences between each is unknown cannot perform statistical procedures Interval – numeric scales in which we know both the order and the exact differences between the values, like temperature can perform some statistical analysis, but the problem is that they don’t have a “true zero” (0 does not mean the absence of value; it is actually another number used on the scale, like 0? or 0?, there is no absence of temperature) which means we cannot calculate ratios think temperature: 10?+10?=20? but 20? is not twice as hot as 10?. We can see this when we convert to Farenheit; 10?= 50?, but 20?= 68?. Ratio – tell us about the order, the exact value between units, and they also have an absolute zero, like height, weight, durationDiscussion: Sociology Hypothesis Testing both descriptive and inferential statistics can be applied Understand the difference between continuous and discrete variables. [e] discrete variables refer to those that have a certain number of values; positive, whole numbers (like number of people) Whole values continuous variables refer to t hose that can take any value between a given range (like height, weight, etc) Any values like fractions of values Why is it important to know #4 and #5 in performing statistical procedures. it is important to know the level of measurement because not all variable types can have statistical procedures performed on them (see #3) You can do means test on ratio data, but not on nominal data What do we mean by inference? statistical inference is the theory, methods, and practice of forming judgments about the parameters of a population and the reliability of statistical relationships, typically on the basis of random sampling causal inference is finding the causal relationship between variables Would you need to perform any work regarding inference with population data? no, inferential statistics allows you to make inferences about the population based on sample data. no inferences would need to be made if you have the population data What is the purpose of hypothesis testing, and on what kind of data? hypothesis testing is the primary mechanism for making decisions based on statistical analysis in order to make inferences about population parameters based on observed sample statistics is there a causal relationship? must be done on continuous sample data What are the important components of hypothesis testing? What are the essential elements? the null and alternative hypotheses test statistic Discussion: Sociology Hypothesis Testing sampling statistic critical value (aka significance aka alpha value) probability values and statistical significance conclusions of hypothesis testing get data set 2. find variable 3. construct hypothesis 4. construct analysis. What are the steps in performing a hypothesis test? hypothesis: null and alternative assumptions/givens: random sampling, normal population distribution, level of measurement, known parameters/known statistics sampling distribution test statistic: use appropriate sampling distribution to calculate value for test statistic level of significance, the critical value/re

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