Practicalresearch2_Chapter4_QuantitativeAnalysisinEvaluation.pptx

Practical Research 2, Quantitative Analysis in Evaluation, Chapter 4 (Unfinished)

1. Quantitative Analysis in Evaluation Descriptive Data Analysis Measures of Central Tendency Suppose, senior high school students were asked how many hours they spent on the computer, and in what subject they often used it for. Results of the survey could indicate that on average, the senior high school students spent two (2) or more hours with a range of one (1) to four (4) hours. A typical senior high school student spent more than two hours studying his/her research subject using the computer.

2. Measures of Central Tendency: Mean Mean: arithmetic average of a set of data The formula is: or  

3. Measures of Central Tendency: Mean For Ungrouped Data Example 1: Find the mean of the measurements 18, 26, 27, 29, and 30. Example 2: Find the mean of the following: Scores in the National Achievement Test (NAT) 90 95 96 87 110 102 95 98 87 117 115 96 91 95 95 93 105 86 103 106

4. Measures of Central Tendency: Mean For Grouped Data When the observations are grouped into classes, the formula for grouped data is as follows: The Weighted Mean The formula for Weighted Mean is Where (frequency), (numerical value or item in a set of data), and n number of observations.  

5. Measures of Central Tendency: Mean For Grouped Data Example 1: Find the mean of the heights of 50 senior high school students. Heights (inches) Frequency Height x Frequency 56 6 336 57 15 855 58 12 696 59 8 472 60 5 300 61 2 122 62 2 124 Heights (inches) Frequency Height x Frequency 56 6 336 57 15 855 58 12 696 59 8 472 60 5 300 61 2 122 62 2 124

6. Measures of Central Tendency: Mean Solution:  

7. Measures of Central Tendency: Mean For Grouped Data Example 2: Solve for the mean of the data below. Class Frequency Class Midpoint (x) fx 76-80 3 78 234 71-75 5 73 365 66-70 6 68 408 61-65 8 63 504 56-60 10 58 580 51-55 7 53 371 46-50 7 48 336 41-45 3 43 129 36-40 1 38 38

8. Measures of Central Tendency: Median Median: midpoint of the distribution (best for ordinal data) For Ungrouped Data: The median has rank if it is odd; and the median is the average of the two middle values if it is even. For Grouped Data: If the data are grouped into classes, the median will fall into one of the classes as the value. .  

9. Measures of Central Tendency: Median Example 1: Consider these numerical values: 12, 15, 18, 22, 30, 32; solve for the median. Example 2: Consider these numerical values: 15, 20, 12, 26, 3, 30, and 14, and find the median. Example 3: Find the median for the set of measurements 7, 8, 8, 9, 9, 10, 23.

10. Measures of Central Tendency: Median Median (For Grouped Data) where L = exact lower limit of the class containing the median class i = interval size n = total number of items or observations F = cumulative frequency in the class preceding the median class f = frequency of the median class  

11. Measures of Central Tendency: Median Example 4: The following data show the distribution of the ages of people interviewed for a survey on climate change. Class Interval Frequency C.Frequency 11-20 20 21-30 14 31-40 22 41-50 18 51-60 14 61-70 12 11-20 20 21-30 14 31-40 22 41-50 18 51-60 14 61-70 12

12. Measures of Central Tendency: Median Solution: Since the number of values or the frequency is , then the median or the median class falls between value, or the item, that is the largest value. Determine in which class the value falls. The first two classes have a cumulative frequency of 34 classes. We need another 16 values to reach 50. Thus, the value falls in the next class which contains 22 values. The median class then is 31-40. Thus, =30.5, =100, =34, =22, and =10.  

13. Measures of Central Tendency: Mode The Mode: most frequently occurring value in a set of data Example 1: The ages of 15 persons assembled in a room are as follows: 16, 18, 18, 18, 25, 25, 25, 25, 30, 30, 30, 34, 34, 36, and 38. Example 2: The number of hours spent by ten (10) students in an internet café was as follows: 2, 2, 2, 3, 3, 4, 4, 4, 5, and 5.

14. Measures of Dispersions The Range: the difference between a data set’s largest and smallest values. Example 1: Consider the following scores obtained by ten (10) students participating in a mathematics contest: 6, 10, 12, 15, 18, 18, 20, 23, 25, and 28. The Average Mean Deviation: the absolute difference or deviation between values in a data set and the mean, divided by the total number of values in a data set.

15. The Average Mean Deviation For Ungrouped Data: The definition-based formula is: Example 1: Consider the data set which consists of 20, 25, 35, 40, and 45; calculate the average mean deviation. Example 2: Find the AD for the data set: 22, 60, 75, 85, and 98.  

16. The Standard Deviation The Standard Deviation: a measure of the spread or variation of data about the mean. For Ungrouped Data: The formula is:  

17. The Standard Deviation Example 1: Let us consider the same data used in the illustration in the illustration for using the range. The values are 6, 10, 12, 15, 18, 18, 20, 23, 25, and 28. Calculate the standard deviation of the given data set. Solution: Compute the mean.

18. The Standard Deviation Score (x) 6 6-17.5= 10 10-17.5= 12 12-17.5= 15 15-17.5= 18 18-17.5= 18 18-17.5= 20 20-17.5= 23 20-17.5= 25 25-17.5= 28 28-17.5= Score (x) 6 6-17.5= 10 10-17.5= 12 12-17.5= 15 15-17.5= 18 18-17.5= 18 18-17.5= 20 20-17.5= 23 20-17.5= 25 25-17.5= 28 28-17.5=

19. The Standard Deviation Interpretation of the Standard Deviation Approximately 68% of the samples in the sample fall within one standard deviation of the mean. Approximately 95% of the scores in the sample fall within two standard deviations of the mean. Approximately 99% of the scores in the sample fall within three standard deviations of the mean.

20. Test of Significance of Difference (T-test) Between Means For independent samples (i.e., when the respondents consist of two different groups as boys and girls, working mothers and non-working mothers, healthy and malnourished children, and the likes). Case 1: unknown or , and .  

21. Case 2: and , and . ( =smaller of or ) Case 3: and , and .  

22. Case 3: and , and . ; where  

23. Assumptions when conducting a Test for 2 Means from Independent Samples We do not know the population standard deviations, and we do not assume they are equal. The two samples or groups are independent . Both samples are simple random samples Both populations are Normally distributed OR both samples are large ( and .  

24. Two Means: Independent Samples Example 1: A survey found that the average hotel room rate in an upscale area is $88.42 and the average in a downtown area is $80.61. Each sample contained 50 hotels. Assume that the populations’ standard deviations are $5.62 and $4.83, respectively. At α = 0.05, can it be concluded that there is a significant difference in the rates?

25. Solutions: Given: Group 1 (Upscale) Group 2 (Downtown)   Step 1: claim, and Tails. Step 2: TS . Calculate TS . Step 3: CV using . Step 4: Make the decision to Reject or Not the . The claim is True or False. Restate this decision: “There is/is not sufficient evidence to support the claim that...”  

26. Solutions: Given: Group 1 (Upscale) Group 2 (Downtown)   Step 1: claim, and Tails. Step 2: TS . Calculate TS . Step 3: CV using . Step 4: Make the decision to Reject or Not the . The claim is True or False. Restate this decision: “There is/is not sufficient evidence to support the claim that...”  

27. Step 1: Claim: The means are not equal. Tails: 2TT Step 2: TS . Calculate TS . Step 3: CV using . CV: .   Step 1: claim, and Tails. Step 2: TS . Calculate TS . Step 3: CV using . Step 4: Make the decision to Reject or Not the . The claim is True or False. Restate this decision: “There is/is not sufficient evidence to support the claim that...”  

28. Step 1: Claim: The means are not equal. Tails: 2TT Step 2: TS . Calculate TS . Step 3: CV using . CV: .   Step 1: claim, and Tails. Step 2: TS . Calculate TS . Step 3: CV using . Step 4: Make the decision to Reject or Not the . The claim is True or False. Restate this decision: “There is/is not sufficient evidence to support the claim that...”  

29. Step 4: Decision Reject . The claim is TRUE . There is enough evidence to support the claim that the means are not equal. Hence, there is a significant difference in the rates .   Step 1: claim, and Tails. Step 2: TS . Calculate TS . Step 3: CV using . Step 4: Make the decision to Reject or Not the . The claim is True or False. Restate this decision: “There is/is not sufficient evidence to support the claim that...”  

30. Step 4: Decision Reject . The claim is TRUE. There is enough evidence to support the claim that the means are not equal. Hence, there is a significant difference in the rates.   Step 1: claim, and Tails. Step 2: TS . Calculate TS . Step 3: CV using . Step 4: Make the decision to Reject or Not the . The claim is True or False. Restate this decision: “There is/is not sufficient evidence to support the claim that...”  

31. Two Means: Independent Samples Example 2: Listed below are stats for student course evaluation scores for courses taught by female and male professors. Females: Males: Use a 0.05 significance level to test the claim that the two samples are from populations with the same mean. Assume the populations are normally distributed.  

32. Solutions: Given: Group 1 (Females) 0.5630 Group 2 (Males) 0.3955   Step 1: claim, and Tails. Step 2: TS . Calculate TS . Step 3: CV using . Step 4: Make the decision to Reject or Not the . The claim is True or False. Restate this decision: “There is/is not sufficient evidence to support the claim that...”  

33. Solutions: Given: Group 1 (Females) 0.5630 Group 2 (Males) 15 0.3955   Step 1: claim, and Tails. Step 2: TS . Calculate TS . Step 3: CV using . Step 4: Make the decision to Reject or Not the . The claim is True or False. Restate this decision: “There is/is not sufficient evidence to support the claim that...”  

34. Step 1: Claim: The female professors and male professors have the same mean score evaluation score. Tails: 2TT Step 2: TS . Calculate TS . Step 3: CV using . CV: smaller of .   Step 1: claim, and Tails. Step 2: TS . Calculate TS . Step 3: CV using . Step 4: Make the decision to Reject or Not the . The claim is True or False. Restate this decision: “There is/is not sufficient evidence to support the claim that...”  

35. Step 1: Claim: The female professors and male professors have the same mean score evaluation score. Tails: 2TT Step 2: TS . Calculate TS . Step 3: CV using . CV: smaller of .   Step 1: claim, and Tails. Step 2: TS . Calculate TS . Step 3: CV using . Step 4: Make the decision to Reject or Not the . The claim is True or False. Restate this decision: “There is/is not sufficient evidence to support the claim that...”  

36. Step 4: Decision Do not Reject the . The claim is TRUE . There is enough evidence to support the claim that female professors and male professors have the same mean course evaluation scores .   Step 1: claim, and Tails. Step 2: TS . Calculate TS . Step 3: CV using . Step 4: Make the decision to Reject or Not the . The claim is True or False. Restate this decision: “There is/is not sufficient evidence to support the claim that...”  

37. Step 4: Decision Do not Reject the . The claim is TRUE. There is enough evidence to support the claim that female professors and male professors have the same mean course evaluation scores.   Step 1: claim, and Tails. Step 2: TS . Calculate TS . Step 3: CV using . Step 4: Make the decision to Reject or Not the . The claim is True or False. Restate this decision: “There is/is not sufficient evidence to support the claim that...”  

38. Test of Significance of Difference (T-test) For correlated/dependent samples (i.e., when the same set of respondents or paired sets of respondents are involved) ;  

39. Test of Significance of Difference (T-test) Between Proportions or Percentages For independent samples For correlated/dependent samples or  

40. Test of Significance of Difference (T-test) Analysis of Variance (ANOVA): is used when the significance of the difference of means of two or more groups is to be determined at one time. One-Way ANOVA ANOVA relies on the F-ratio to test the hypothesis that the two variances are equal; the subgroups are from the same population. “Between groups” refers to the variation between each group mean and the grand or overall mean.

41. Test of Significance of Difference (T-test) A typical ANOVA table: Source of variation Degree of Freedom Sum Squares Mean Square F-ratio p Between groups Within groups Total Source of variation Mean Square F-ratio p Between groups Within groups Total

42. Test of Relationship Spearman Rank-Order Correlation or Spearman rho. This is used when data available are expressed in terms of ranks (ordinal variable). Chi-Square Test for Independence. This is used when data are expressed in terms of frequencies or percentages (nominal variables).  

43. Testing the Hypothesis Hypothesis Testing Guide Questions: What is it? Why is it important? How does it work? What are the key steps involved? What are the benefits? What are the limitations?

44. What is Hypothesis Testing? Hypothesis testing refers to a process used by analysts to assess the plausability of a hypothesis by using sample data. In hypothesis testing, statisticians formulate two hypotheses: the null hypothesis and the alternative hypothesis. A null hypothesis determines there is no difference between two groups or conditions, while the alternative hypothesis determines that there is a difference. Researchers evaluate the statistical significance of the test based on the probability that the null hypothesis is true.

45. Why is Hypothesis Testing important? Hypothesis testing is an important part of the data analysis plan in conducting a research study. If the researcher wishes to draw inferences from the data taken from a sample that may have wider generalizability, this is referred to as inferential statistics and it is more complex than descriptive statistics. Inferences applied to the total population are valid under two conditions, namely: there is a target population and when appropriate random sampling has been used in the selection of the samples.

46. How does it work? In hypothesis testing, an analyst tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis (e.g., the population mean return is not equal to zero). Thus, they are mutually exclusive, and only one can be true. However, one of the two hypotheses will always be true.

47. What are the key steps involved? Hypothesis testing begins with an analyst stating two hypotheses, with only one that can be right. The analyst then formulates an analysis plan, which outlines how the data will be evaluated. Next, they move to the testing phase and analyze the sample data. Finally, the analyst analyzes the results and either rejects the null hypothesis or states that the null hypothesis is plausible, given the data.

48. Steps in Hypothesis Testing Outline of Steps State the null hypothesis. Example: Let us suppose that an advertising agency is conducting an experiment using two different methods of marketing strategies and to grade 12 students. The results of the experiment will be measured using the monthly sales of the company. There are three possible outcomes:  

49. Choose the statistical test and perform the calculation. A researcher must determine the measurement scale, the type of variable, the type of data gathered, and the number of groups or the number of categories. State the level of significanc e for the statistical test. The level of significance is determined before the test is performed. It has traditionally been accepted by various schools of thought to use alpha , to denote the level of significance in rejecting the null hypothesis. Compute the calculated value. Determine the critical value Make the decision.  

50. What are the benefits? Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.

51. What are the limitations? Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.

52. Bottom Line: Hypothesis Testing Hypothesis testing refers to a statistical process that helps researchers and/or analysts determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. There are different types of hypothesis testing, each with its own set of rules and procedures. However, all hypothesis testing methods have the same four-step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result. Hypothesis testing plays a vital part of the scientific process, helping to test assumptions and make better data-based decisions.