COM 201_Inferential Statistics_18032022.pptx
- 1. Introduction to Inferential Statistics Sampling, Probability, Normal distribution, and Hypothesis Testing by Adekunle Fakunle (PhD)
- 2. Outline • Definitions of terms • Probability • Random Variables. • Probability Distributions • Binominal and Normal distribution • Inferential statistics • Types of inferential statistic
- 3. Important Definitions • Probability – the chance that an uncertain event will occur (always between 0 and 1) • Impossible Event – an event that has no chance of occurring (probability = 0) • Certain Event – an event that is sure to occur (probability = 1) •.
- 4. Probability •Outcomes: any possible result of probability event •Favorable outcomes: a successful result in a probability event e.g rolling the #1 on a die •Possible outcomes: All the result that could occur during probability event 4
- 5. Random Variables A random variable is a numerical description of the outcome of a statistical experiment A random variable that may assume only a finite number or an infinite sequence of values is said to be discrete; one that may assume any value in some interval on the real number line is said to be continuous
- 6. The sample Space, S The sample space, S, for a random phenomena is the set of all possible outcomes.
- 7. Examples 1. Tossing a coin – outcomes S ={Head, Tail} 2. Rolling a die – outcomes S ={ , , , , , } ={1, 2, 3, 4, 5, 6}
- 8. An Event , E The event, E, is any subset of the sample space, S. i.e. any set of outcomes (not necessarily all outcomes) of the random phenomena S E Venn diagram
- 9. The event, E, is said to have occurred if after the outcome has been observed the outcome lies in E. S E
- 10. Examples 1. Rolling a die – outcomes S ={ , , , , , } ={1, 2, 3, 4, 5, 6} E = the event that an even number is rolled = {2, 4, 6} ={ , , }
- 12. Histogramshowing the heights of 10000 males 0 200 400 600 800 1000 1200 1400 140 148 156 164 172 180 188 More Height (cm) Frequency A sample of heights of 10,000 adult males gave rise to the following histogram: Notice that this histogram is symmetrical and bell-shaped. This is the characteristic shape of a normal distribution.
- 13. The normal distribution is an appropriate model for many common continuous distributions, for example: If we were to draw a smooth curve through the mid-points of the bars in the histogram of these heights, it would have the following shape: This is called the normal curve. The masses of new-born babies; The IQs of school students; The hand span of adult females; The heights of plants growing in a field; etc.
- 14. Area Under the Curve •A normal distribution should be viewed almost like a histogram •Top Figure: The darker bars of the histogram correspond to ages ≤ 9 (~40% of distribution) •Bottom Figure: shaded area under the curve (AUC) corresponds to ages ≤ 9 (~40% of area) 2 2 1 2 1 ) ( x e x f
- 15. Parameters μ and σ •Normal distribution have two parameters μ - expected value (mean “mu”) σ - standard deviation (sigma) 7: Normal Probability Distributions 15 σ controls spread μ controls location
- 16. Mean and Standard Deviation of Normal Density 7: Normal Probability Distributions 16 μ σ
- 17. Standard Deviation σ •Points of inflections one σ below and above μ •Practice sketching Normal curves •Feel inflection points (where slopes change) •Label horizontal axis with σ landmarks 7: Normal Probability Distributions 17
- 18. Symmetry in the Tails 7: Normal Probability Distributions … we can easily determine the AUC in tails 95% Because the Normal curve is symmetrical and the total AUC is exactly 1…
- 19. Assessing Departures from Normality Same distribution on Normal “Q-Q” Plot Approximately Normal histogram Normal distributions adhere to diagonal line on Quantile-Quantile plot
- 20. Negative Skew Negative skew shows upward curve on Q-Q plot
- 21. Positive Skew Positive skew shows downward curve on Q-Q plot
- 23. What is inferential statistics? •Inferential statistics is a technique used to draw conclusions about a population by testing the data taken from the sample of that population •It is the process of how generalization from sample to population can be made. It is assumed that the characteristics of a sample is similar to the population’s characteristics. •It includes testing hypothesis and deriving estimates
- 24. Inferential Statistic Descriptive Statistics •are used to organize and/or summarize the parameters associated with data collection (e.g., mean, median, mode, variance, standard deviation) Inferential Statistics •are used to infer information about the relationship between multiple samples or between a sample and a population (e.g., t- test, ANOVA, Chi Square).
- 25. The process of inferential analysis Raw Data • It comprises of all the data collected from the sample. • Depending on the sample size, this data can be large or small set of measurements. Sample Statistics • It summarizes the raw data gathered from the sample of population • These are the descriptive statistics (e.g. measures of central tendency) Inferential Statistics • These statistics then generate conclusions about the population based on the sample statistics.
- 26. Inferential Statistics • Inferential statistics are used to draw conclusions about a population by examining the sample We want to learn about population parameter s… …but we can only calculate sample statistics
- 27. Parameters and Statistics We are going to illustrate inferential concept by considering how well a given sample mean “x-bar” reflects an underling population mean µ x µ
- 28. Inferential Statistic •Accuracy of inference depends on representativeness of sample from population •Random selection •equal chance for anyone to be selected makes sample more representative
- 29. Inferential Statistic •Inferential statistics help researchers test hypotheses and answer research questions, and derive meaning from the results •a result found to be statistically significant by testing the sample is assumed to also hold for the population from which the sample was drawn •the ability to make such an inference is based on the principle of probability
- 30. Inferential Statistic •Researchers set the significance level for each statistical test they conduct •by using probability theory as a basis for their tests, researchers can assess how likely it is that the difference they find is real and not due to chance
- 31. Inferential Statistics Provide Two Environments: •Test for Difference – To test whether a significant difference exists between groups •Tests for relationship – To test whether a significant relationship exist between a dependent (Y) and independent (X) variable/s •Relationship may also be predictive
- 32. Hypothesis Testing Using Basic Statistics •Univariate Statistical Analysis •Tests of hypotheses involving only one variable •Bivariate Statistical Analysis •Tests of hypotheses involving two variables •Multivariate Statistical Analysis •Statistical analysis involving three or more variables or sets of variables.
- 33. Hypothesis Testing Procedure • H0 – Null Hypothesis • “There is no significant difference/relationship between groups” • Ha – Alternative Hypothesis • “There is a significant difference/relationship between groups” • Always state your Hypothesis/es in the Null form • The object of the research is to either reject or accept the Null Hypothesis/es
- 34. Examples •Example 1: Three unrelated groups of people choose what they believe to be the best color scheme for a given website. • The null hypothesis is: There is no difference between color scheme choice and type of group •Example 2: Males and Females rate their level of satisfaction to a magazine using a 1-5 scale • The null hypothesis is: There is no difference between satisfaction level and gender
- 35. We can make two types of errors in hypothesis testing: In the population, Ho actually is: Not reject Ho Reject Ho True Correct decision made Type 1 error Researcher thinks there is an actual relationship between the variables when there is not False Type II error There is an actual relationship between variables although researcher has accepted null hypothesis Correct decision made
- 36. Concepts related to Sampling Error • Sampling Error: The degree to which a sample differs on a key variable from the population. • Confidence Level: The number of times out of 100 that the true value will fall within the confidence interval. • Confidence Interval: A calculated range for the true value, based on the relative sizes of the sample and the population. • Why is Confidence Level Important? Confidence levels, which indicate the level of error we are willing to accept, are based on the concept of the normal curve and probabilities. Generally, we set this level of confidence at either 90%, 95% or 99%. At a 95% confidence level, 95 times out of 100 the true value will fall within the confidence interval.
- 37. We can theoretically draw numerous samples from a population that examine the value of one variable. The more samples we draw from the population, the more likely it is that the frequency distribution of that variable will resemble a normal distribution
- 38. Important concepts about sampling distributions: •If a sample is representative of the population, the mean (on a variable of interest) for the sample and the population should be the same. •However, there will be some variation in the value of sample means due to random or sampling error. This refers to things you can’t necessarily control in a study or when you collect a sample. •The amount of variation that exists among sample means from a population is called the standard error of the mean. •Standard error decreases as sample size increases.
- 39. Significance Levels and p-values • Significance Level • A critical probability associated with a statistical hypothesis test that indicates how likely an inference supporting a difference between an observed value and some statistical expectation is true. • The acceptable level of Type I error. • p-value • Probability value, or the observed or computed significance level. • p-values are compared to significance levels to test hypotheses
- 40. Testing for Significant Difference •Testing for significant difference is a type of inferential statistic •One may test difference based on any type of data •Determining what type of test to use is based on what type of data are to be tested.
- 41. Example: Types of Relationships Positive Negative No Relationship Income ($) Education (yrs) Income ($) Education (yrs) Income ($) Education (yrs) 20,000 10 20,000 18 20,000 14 30,000 12 30,000 16 30,000 18 40,000 14 40,000 14 40,000 10 50,000 16 50,000 12 50,000 12 75,000 18 75,000 10 75,000 16
- 42. Income and Education 0 20000 40000 60000 80000 1 2 3 4 5 Education Income Income Education
- 43. In inferential statistics, the hypothesis that is actually tested in the null hypothesis. Therefore, what we must do is disprove that a relationship between the variables does not exist.
- 44. Different types of inferential statistics
- 45. Chi Square A chi square (X2) statistic is used to investigate whether distributions of categorical (i.e. nominal/ordinal) variables differ from one another.
- 46. General Notation for a chi square 2x2 Contingency Table Variable 2 Data Type 1 Data Type 2 Totals Category 1 a b a+b Category 2 c d c+d Total a+c b+d a+b+c+d 𝑥2 = 𝑎𝑑 − 𝑏𝑐 2 𝑎 + 𝑏 + 𝑐 + 𝑑 𝑎 + 𝑏 𝑐 + 𝑑 𝑏 + 𝑑 𝑎 + 𝑐 Variable 1
- 47. T test 2 1 2 1 x x S x x t Mean for group 1 Mean for group 2 Pooled, or combined, standard error of difference between means The pooled estimate of the standard error is a better estimate of the standard error than one based of independent samples. 1 x 2 x 2 1 x x S
- 48. Uses of the t test •Assesses whether the mean of a group of scores is statistically different from the population (One sample t test) •Assesses whether the means of two groups of scores are statistically different from each other (Two sample t test) •Cannot be used with more than two samples (ANOVA)
- 49. ANOVA • In statistics, analysis of variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation. • In its simplest form ANOVA provides a statistical test of whether or not the means of several groups are all equal, and therefore generalizes t-test to more than two groups. • Doing multiple two-sample t-tests would result in an increased chance of committing a type I error. For this reason, ANOVAs are useful in comparing two, three or more means.
- 50. ANOVA Hypothesis Testing •Tests hypotheses that involve comparisons of two or more populations •The overall ANOVA test will indicate if a difference exists between any of the groups •However, the test will not specify which groups are different •Therefore, the research hypothesis will state that there are no significant difference between any of the groups 𝐻0: 𝜇1 = 𝜇2 = 𝜇3
- 51. Regression Analysis •The description of the nature of the relationship between two or more variables •It is concerned with the problem of describing or estimating the value of the dependent variable on the basis of one or more independent variables.
- 52. Predictive Versus Explanatory Regression Analysis •Prediction – to develop a model to predict future values of a response variable (Y) based on its relationships with predictor variables (X’s) •Explanatory Analysis – to develop an understanding of the relationships between response variable and predictor variables
- 53. Correlation Analysis Spearman's correlation: It is the nonparametric version of the Pearson product- moment correlation. Spearman correlation is often used to evaluate relationships involving ordinal variables. For example, you might use a Spearman correlation to evaluate whether the order in which employees complete a test exercise is related to the number of months they have been employed.
- 54. Correlation analysis Pearson correlation: It is the test statistics that measures the statistical relationship, or association, between two continuous variables. It is known as the best method of measuring the association between variables of interest because it is based on the method of covariance. Pearson's correlation is used when you want to see if their is a linear relationship between two quantitative variables.
- 55. Simple Regression Model 𝑦 = 𝑎 + 𝑏𝑥 𝑺𝒍𝒐𝒑𝒆 𝒃 = (𝑁Σ𝑋𝑌 − Σ𝑋 Σ𝑌 ))/(𝑁Σ𝑋2 − Σ𝑋 2) 𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 𝒂 = (Σ𝑌 − 𝑏 Σ𝑋 )/𝑁 Where: y = Dependent Variable x = Independent Variable b = Slope of Regression Line a = Intercept point of line N = Number of values X = First Score Y = Second Score ΣXY = Sum of the product of 1st & 2nd scores ΣX = Sum of First Scores ΣY = Sum of Second Scores ΣX2 = Sum of squared First Scores
- 56. y x Predicted Values Actual Values i Y i Y i r ˆ Residuals Slope (b) Intercept (a) Simple regression model
- 58. THANK YOU