Inferential statistics
- 1. INFERENTIAL STATISTICS DesmondAyim-Aboagye, Ph.D SAM PLING DISTRIBUTIONS AND HYPOTHESIS TESTING
- 2. OVERVIEW We will understand the logic underlying the testing of hypothesis Understanding of sampling, and probability will be brought to bear in the exploration of hypothesis Testable questions or focused predictions
- 3. Steps and Motivation 1. A hypothesis is identified 2. Research is executed 3. Data are collected 4. Inferential statistics are used 5. To test the viability of the hypothesis 6. Was an anticipated relationship found within the data?
- 4. EXAMPLE 1 Thisresearcherwantsto knowifthetwo-prong treatmentismore beneficialthanbehavior therapyalone A clinical psychologist believes that the link between obsessive thoughts and behavior can be disrupted with a combination of behavioral and cognitive therapies. Control group of (0bsessive compulsive disorder) OC patients receives standard behavior therapy Experimental group is exposed to the same therapy coupled with work on reducing obsessive thoughts.
- 5. EXAMPLE 2 Thisresearchwantsto determineiffourth-years studentsaremorelikelyto voteforliberalcandidates, whilefirst-yearstudentswill tendtoendorseconservative candidates. A political psychologist studies how higher education affects college students' voting behavior in national elections. He hypothesizes that students generally become more liberal and politically aware across their four college years. He sets to compare the voting behavior of first-year students with that of fourth-year students in a mock national election. Students from both classes read a series of mock candidate profiles and then answer questions about their beliefs concerning a variety of public and social policy issues.
- 6. EXAMPLE 3 A Health Psychologist Purpose Hebelievesthatmiddle-aged individualswho carefor elderlyparentsareatgreater riskforillnessthan similarly agedpersonswith no caregiverresponsibilities. Study The researcher interviews the two sets of adults and then gains permission to examine their medical records at the end of a 1- year period. Hypothesis The care giver group will show more frequent illnesses, visits to the doctor, hospitalizations, and medicine prescriptions than the noncaregiver group.
- 7. Important! None of these researchers can ever hope to measure the responses of every possible respondent in their population of interest, so such data are usually collected in the form of some random sample.
- 8. Random sample is divided into 2 distinct groups Group 1 Control group Group 2 Experiment al group
- 9. Each group is then exposed to one level of the independent variable—the variable manipulated by or under the control of the investigator.
- 10. HYPOTHESIS TESTING Hypothesis testing entails comparing the groups' reactions to the dependent measure following the introduction of the independent variable.
- 11. PRACTICAL MATTER Did the independent measure create an observed and systematic change in the dependent measure? Did the experimental group behave or respond differently than the control group after both were exposed to the independent variable?
- 12. THEORETICAL MATTER Following exposure to the independent variable, is the 𝝁 of the experimental group verifiably different from the 𝝁 of the control group? In other words, Can we attribute the differential and measured between group differences to the fact that the control group and experimental group now effectively represent different populations with different parameters?
- 13. We now turn to: 1. How to draw samples from population? 2. How do we make estimation? 3. How do we conduct experiment?
- 14. Population Random Sample Representative EXTIMATION OF POPULATION PARAMETERS
- 15. Things we must be concerned with: Representative of the population from which it was drawn The average behavior witnessed in the sample reflects what is usually true of the population The sample statistics N (i.e., sample mean ˉχ and standard deviation, s) are similar to those of population's parameters (i.e., population mean 𝝁 and standard deviation σ)
- 16. Point estimation Point estimation is the process of using a sample statistic (e.g., ˉχ, s) to estimate the value of some population parameter (e.g.,𝝁, σ)
- 17. ˉχ ≅ 𝝁 ˉχ ≅ 𝝁 sample mean is close or equal in value to population mean
- 18. Sampling error Any given sample statistic is apt to contain some degree of error (i.e., the difference between estimated and actual reality)—sampling error
- 19. Overcoming point estimation's limitations can be achieved through a somewhat more laborious process known as Interval estimations Interval estimation involves careful examination and estimation of the variability noted among sample statistics based on the repeated sampling of the same population.
- 20. STATISTIC INFERENCE AND HYPOTHESIS TESTING Random Sample Representativeness Divide into two a. Control group b. Experimental group Present each group with a distinct level of some independent variable and then measures subsequent reactions based on changes to the dependent variable.
- 21. MEAN DIFFERENCES Whether the sample data—now divided into two groups—show detectable differences due to the influence of some independent variable
- 22. This process of detection centers on the role inferential statistics play in hypothesis testing, which is generally to demonstrate mean differences. How does the average behavior in one group of participants differ from the average behavior found in other groups? Did the manipulation of an independent variable lead to a different average level of behavior in the dependent measure for the experimental mental group than the control group? Control group and Experimental group (manipulated)
- 23. Mean Sample Mean, χ, average ˉχ = 𝜒/N Population mean 𝝁 𝝁 = 𝜒/N
- 25. Importance of Hypothesis testing Whether a sample statistic fits one or another population Independent variable impact on a dependent variable Researchers are able to make educated guesses when limited information is available to guide judgment. Involving every person or animal is impossible, impractical, and nonsensical. Statistical inference enables researchers to evaluate the veracity of a hypothesis as if a whole population of participants were available instead of merely a small representative sampling.
- 26. Random sample drawn from Population Control Group Experime ntal Group Ctr level Indep Var Exp level Inde Var Does a diff exist between sample means? Random Ass to one of 2 Groups Χc ,Sc Χe ,Se Is the sample mean of the control Is the sample mean of the experim gr Group ≅population μ? ≠to the original population μ? Figure 9.1. The process of sampling and inferring whether a statistic is from one or another population. Is the sample mean of the control Is the sample mean of the experim gr Group ≅population μ? ≠to the original population μ?
- 27. Distribution of Sample Means A distribution of sample means is a group or collection of sample means based on random samples of a fixed size N from some population (Several Samples chosen to compare them. USA Voting as example, ˉχ1, ˉχ2, ˉχ3… ˉχ N).
- 28. X Y X1 X2 X3 Xn Unknown P 𝜇 Repeated Samples Drawn Sampling Distribution of X A Sampling DistributionCreated by Repeated Sampling (Fixed Sample Size N) of a Population
- 29. Sampling Distribution A sampling distribution is a distribution comprised of statistic (e.g., ˉχ, S) based on samples of some fixed size N drawn from a larger population.
- 30. Important!! A sampling distribution of means of some fixed size N will take on the shape of the standard normal distribution.
- 31. CENTRAL LIMIT THEOREM The Central LimitTheorem proposes that as the size of any sample, N, becomes infinitely large in size, the shape of the sampling distribution of the mean approaches normality—that is, it takes on the appearance of the familiar bell-shaped curve – with a mean equal to μ, the population's mean, and the standard deviation equal to σ/ 𝑵, which is known as the standard error of the mean. As N increases in size, the standard error of the mean or 𝝈ˉχ will decrease in magnitude, indicating that the sample will be close in value to the actual population μ. Thus, it will also be true that μ ˉχ ≅ 𝛍 and that 𝝈ˉχ ≅ 𝛔/ 𝑵.
- 32. Y X 40 55 70 85 100 115 130 145 160 𝝁 Figure 5.1. Hypothetical distribution of scores on an IQ test Third Second First First Second Third 𝜎 𝜎 𝜎 𝜎 𝜎 𝜎 Below 𝜇 Below𝜇 Below 𝜇 Above 𝜇 Above 𝜇 Above 𝜇 Pop mean (𝜇) of 100 Standard deviation 𝜎15 110 1st 𝜎 around the mean NORMAL DISTRIBUTION
- 33. The mean of any sampling distribution of a sample statistic is referred to as the sampling distribution's expected value. Symbolically μ ˉχ Formula for calculating μ ˉχ = ˉ𝜒/Nk
- 34. Distribution's Standard Error The standard deviation of any sampling distribution of a sample statistic is referred to as the sampling distribution's standard error. (standard error of the mean)
- 35. Standard Error Symbolically 𝝈ˉχ Formula for calculating it 𝝈ˉχ = √𝝈 2 /N = 𝜎/ 𝑁
- 36. These must be known N is sample size, 𝛔 2 is population variance, σ population standard deviation.
- 37. LAWOF LARGE NUMBERS The law of large numbers proposes that the larger the size of a sample (N), the greater the probability that the sample mean (ˉχ) will be close to the value of the population mean (μ).
- 38. STANDARD ERRORAND SAMPLING ERROR Sampling error is the difference between a given sample mean and a population mean (ˉχ - μ). Theoretically, if we have a distribution of sufficient size, we could readily show that the (ˉχ - μ) = 0. (That is, the sum of the sampling errors—the positive as well as negative differences between sample means and a population mean—would be equal to 0. In theory, these random deviations would cancel one another out.)
- 39. ESTIMATING THE STANDARD ERROROF THE MEAN The standard error of the mean provides a researcher with a very important advantage: The sample mean estimates the value of a population mean. A smaller standard error specifies a close match between a sample mean and a population mean. On the other hand, a larger error points to considerable disparity between the two indices.
- 40. To determine standard error: 1. Estimating population variance, and 2. Standard deviation from known values of a. sample variance and b. standard deviation. Sample variance formula S² = ⅀(χ -ˉχ)² N
- 41. Standard deviation formula S = √S² = √⅀(χ -ˉχ)² N
- 42. A biased estimate can be corrected, that is, converted into an unbiased estimate by reducing the value of the denominator by one observation— in formula terms, N becomes N – 1. σ² = ŝ² = ⅀(χ -ˉχ)² N- 1 The caret (ˆ) over ŝ indicates that the statistic is an unbiased estimate.
- 43. Population Standard Deviation (σ) The population standard deviation (σ) can be determined by the formula σ = ŝ =√ŝ² =√⅀(χ -ˉχ)² N- 1
- 44. By relying on the formula and logic behind these unbiased estimators, we can now approximate the variance of the population (σ²-χ) To do so, of course, we need to rely on estimated variance of the population (s²-χ) and the estimated error of the mean (sˉχ). Given that we do not know the true values of σ² and σ, we must use estimates provided by ŝ² and ŝ, respectively.
- 45. Estimated σ²-χ = s²-χ = ŝ²/N. It follows that the standard error mean can be estimated by using : σˉχ = sˉχ = √ŝ²/N = ŝ/√N when ŝ² and ŝ may be unavailable, the standard error of the mean can be estimated using this formula: Estimate σˉχ = sˉχ = s/√N-1 = √⅀ (x -ˉχ)²/N(N -1) = √ss/N(N- 1)