Descriptive Probability Distributions

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timer Asked: Jan 15th, 2021

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Republic of the Philippines CAVITE STATE UNIVERSITY Don Severino de las Alas Campus Indang, Cavite  (046) 4150-010 / (046) 4150-13 loc 253 www.cvsu.edu.ph OFFICE OF THE GRADUATE SCHOOL AND OPEN LEARNING COLLEGE LEARNING ACTIVITY SPEARMAN’S RANK ORDER CORRELATION Name:___________________________ Section:_____________________ Score: _________ Contact Number: ___________ Objectives: At the end of this exercise, the student is expected to: 1. know when and how to estimate the Spearman’s correlation coefficient; 2. interpret the estimated correlation coefficient. Part I. Problem Solving 1. The marks obtained by twelve students in Math and Physics are given below: MATH PHYSICS 1 2 3 4 5 6 7 8 9 10 11 12 75 89 84 95 79 87 88 91 84 81 92 78 85 94 87 94 86 88 85 96 86 84 92 83 Assume that the variables are not correlation coefficient. normal, Compute and interpret the You may use this example as your guide. Dr. Cristina Hobbs, a Medical Curriculum Specialist, wants to determine if the score of nursing students in a Creativity Test is correlated to their score in Board Examination for Nursing. She randomly selected seven (7) graduates of nursing program from different university. The result is given below: Student Creativity 1 2 3 4 5 6 7 97 94 90 89 87 85 83 Score in the Board Exam 95 90 85 93 84 80 96 Is the score of students in a Creativity Test correlated to their performance in the Board Exam? Make a simple table like this one: Student Creativity Score (y) (x) 1 97 95 2 94 90 3 90 85 4 89 93 5 87 84 6 85 80 7 83 96 Rx (Rank of x) 7 6 5 4 3 2 1 Ry (Rank of y) 6 4 3 5 2 1 7 D = Rx Ry 1 2 2 -1 1 1 -6 D2 1 4 4 1 1 1 36 ΣD2 = 48 SOLUTION: where: N = number of observations = 7 rs = 0.14 Interpretation: The obtained rs value is 0.14 hence, we can say that there is a very weak correlation between the score of students in creativity test and their performance in the Board Exam. Part II. Problem Formulation Using the data that you have as a teacher, create a realistic problem situation that will require study of the relationship between two variables using non-parametric correlation analysis. Use SPSS Package to facilitate the analysis( Please provide screenshots/picture/raw outputs as proof) Expected output: 1. Background of the problem including specific objective(s). 2. Estimation of the coefficient. 3. Interpretation of the obtained correlation coefficient in the context of the problem. 4. Test of hypothesis on the significance of the correlation coefficient. PREPARED BY: ANALYN A. MOJICA Associate Professor January 9, 2021 Republic of the Philippines CAVITE STATE UNIVERSITY Don Severino de las Alas Campus Indang, Cavite  (046) 4150-010 / (046) 4150-13 loc 253 www.cvsu.edu.ph OFFICE OF THE GRADUATE SCHOOL AND OPEN LEARNING COLLEGE Name: __________________________ Program: ________________________ Date: ______________ Learning Activity 3 DISCRETE PROBABILITY DISTRIBUTIONS OBJECTIVES: At the end of this exercise the student must be able: 1. to determine the different discrete probability functions. 2. to construct the distribution of a random variable. METHODS: 1. For each of the following, determine whether it can serve as the probability distribution of some random variable: a. f(x) = 1/10 for x = 1, 2, 3, … , 10; b. f(x) = 1/8 for x = 0, 1, 2, …, 8; c. f(x) = for x = 1, 2, 3, 4, 5; d. f(x) = for x = 0, 1, 2, 3. 2. From a group of 24 students, 15 are boys and 9 are girls. Let X equal the number of boys in a sample of five students selected at random and without replacement from the group. Find the following probabilities: a. P(X = 2) b. P(X≤ 2) 3. Say there are 3 principals in a school of 50 teachers. A sample of size 10 is taken at random and without replacement. Let X denote the number of principals in the sample. Find the probability that the sample contains: a. exactly 1 principal. b. at most 1 principal. 4. On a ten-question multiple-choice test there are five possible answers, of which one is correct (C) and four are incorrect (I). If a student guesses randomly and independently, find the probability of: a. being correct on two questions. b. getting at least 1 correct answers. c. getting between 4 and 8 correct answers inclusive. 5. Let p equal the proportion of all college and university students who would say yes to the question, “Would you drink from the same glass as your friend if you suspected that this friend was an AIDS virus carrier?” Assume that p=0.10. Let X equal the number of students out of a random sample of size n=9 who would say yes to this question. a. How is X distributed b. Find P(X= 2) c. Find P(X ≥ 2) Note: To be submitted of November 28, 2020 Republic of the Philippines CAVITE STATE UNIVERSITY Don Severino de las Alas Campus Indang, Cavite  (046) 4150-010 / (046) 4150-13 loc 253 www.cvsu.edu.ph OFFICE OF THE GRADUATE SCHOOL AND OPEN LEARNING COLLEGE Name: __________________________ Program: ________________________ Date: ______________ Score:____________ Learning Activity COMPARISON OF 2 OR MORE POPULATIONS (NONPARAMETRIC) OBJECTIVES: At the end of the exercise, the student must be able to: 1. implement Signed Rank test, Wilcoxon Signed Rank test and Mann-Whitney U test for the analysis of 2 populations; 2. implement Kruskal Wallis H test for the analysis of k populations; and 3. formulate a realistic problem situation for comparing k populations. MATERIALS: Calculator, Class Record, Stat Software PART 1: PROBLEM SOLVING Directions: 1. For each problem, follow the steps in Hypothesis Testing. 2. Use SPSS or Excel. 3. Copy paste Outputs. 1. The following are the weights (in pounds) before and after, of 16 persons who stayed on a certain weight-reducing diet for 2 weeks: Before After 169.0 159.9 188.6 181.3 222.1 209.0 160.1 162.3 187.5 183.5 202.5 197.6 167.8 171.4 214.3 202.1 143.8 145.1 198.2 185.5 166.9 142.9 160.5 198.7 149.7 181.6 158.6 145.4 159.5 190.6 149.0 183.1 Use the large sample Signed Rank test at 0.05 level of significance to test whether the weight-reducing diet is effective. 2. The weights of five people before they stopped smoking and 5 weeks after they stopped smoking, in kilograms, are as follows: 1 66 71 Before After INDIVIDUAL 3 69 68 2 80 82 4 52 56 5 75 73 Use the Wilcoxon signed-rank test to test the hypothesis, at the 0.05 level of significance, that giving up smoking has no effect on a person’s weight against the alternative that one’s weight increases if he quits smoking. 3. An experiment measures the intelligence quotients (IQs) of adult male students of tall, And short stature. The results are given below. Assuming that the data are not normally distributed, determine at the 0.01 significance level whether there is any difference in the IQ scores relative to the height differences using Mann Whitney U Test. Tall Short 110 95 105 103 118 115 112 107 90 3. A teacher wishes to test three different teaching methods: I, II, and III. To do this, three groups of five students are chosen at random, and each group is taught by a different method. The same examination is then given to all students, and the grades obtained are shown below. Assuming that assumption of normality is not satisfied, determine whether there is a difference between the teaching methods at 0.05 level of significance using Kruskal Wallis H Test. Method I 75 62 71 58 73 Method II 81 85 68 92 90 Method III 73 PART II: Problem Formulation 79 60 75 81 Using the any data you have, construct a problem situation that would require the comparison of k independent samples that can be analyzed by Kruskal Wallis Analysis of Variance. Guidelines: 1. Provide sufficient background of the problem. 2. State the necessary assumptions needed to perform the test. 3. Perform test of hypothesis at  = 5 % to answer the objective of the study. PREPARED BY: ANALYN A. MOJICA ASSO. PROF. January 9, 2021 Republic of the Philippines CAVITE STATE UNIVERSITY Don Severino de las Alas Campus Indang, Cavite  (046) 4150-010 / (046) 4150-13 loc 253 www.cvsu.edu.ph OFFICE OF THE GRADUATE SCHOOL AND OPEN LEARNING COLLEGE LEARNING ACTIVITY PEARSON PRODUCT MOMENT CORRELATION Name:___________________________ Section:_____________________ Score: ___________________ Contact Number: ___________ Objectives: At the end of this exercise, the student is expected to: 1. know when and how to estimate the Pearson’s correlation coefficient; 2. interpret the estimated correlation coefficient; and 3. perform test of hypothesis on the Pearson correlation coefficient. Part I. Problem Solving 1. Princess wants to determine if there is a significant linear relationship between the expenditures (Y) of UPLB students in a month measured in pesos are related to their age in years (X). A random sample of 10 UPLB students was obtained and the following data were gathered: Student 1 2 3 4 5 6 7 8 9 10 Age 18 17 16 18 19 20 19 21 22 17 Expenditure 5000 5200 5300 4800 4900 4500 5000 4600 4500 5300 a. Compute and interpret the correlation coefficient between the expenditures (Y) and age (X) of UPLB students. Assume that variables are bivariate normal. SSx = _________________________ SSy = _________________________ SPxy = ________________________ r = ___________________________ Interpretation: ___________________________________________________________ _______________________________________________________________________. b. Is there evidence to conclude that as the UPLB students get older, they become thriftier? Use a 0.05 level of significance. Ho: _______________________________________________________________________ Ha: _______________________________________________________________________ Test Statistic: _______________________________________________________________ Decision Rule: Reject Ho if _____________________________, otherwise fail to reject Ho. Computations: Decision: __________________________________________________________________ Conclusion: ________________________________________________________________ __________________________________________________________________________. Part II. Problem Formulation Using the data that you have as a teacher, create a realistic problem situation that will require study of the relationship between two variables (at least interval in scale) using parametric correlation analysis. Use SPSS Package to facilitate the analysis ( PLEASE provide screenshots, picture or raw outputs as proof). Expected output: 1. 2. 3. 4. Background of the problem including specific objective(s). Estimation of the Pearson Product Moment Correlation Coefficient. Interpretation of the obtained correlation coefficient in the context of the problem. Test of hypothesis on the significance of the correlation coefficient. For manual computation, you may use this example as your guide. 1. It is believed that there is a linear relationship between an infant’s length and weight at birth. A pediatrician obtained the following measurements on the length and weight of newly born infants in a certain district hospital. Infant 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Length (cm) Weight (kg) 57.5 2.75 52.8 2.15 61.3 4.41 67.0 5.52 53.5 3.21 62.7 4.32 56.2 2.31 68.5 4.30 69.2 3.71 68.7 4.40 67.1 4.89 56.7 2.50 58.3 2.38 51.0 2.14 59.9 3.72 Assume that the variables are bivariate normal. 1. Compute and interpret the correlation coefficient. 2. Is there evidence to say that the length and weight of newly born infants are positively linearly related? Test at  = 0.05. 1. Computations: X Y 1 57.5 2.75 X*X 3306.25 Y*Y 7.5625 X*Y 158.125 2 52.8 2.15 2787.84 4.6225 113.520 3 61.3 4.41 3757.69 19.4481 270.333 4 67 5.52 4489.00 30.4704 369.840 5 53.5 3.21 2862.25 10.3041 171.735 6 62.7 4.32 3931.29 18.6624 270.864 7 56.2 2.31 3158.44 5.3361 129.822 8 68.5 4.3 4692.25 18.4900 294.550 9 69.2 3.71 4788.64 13.7641 256.732 10 68.7 4.4 4719.69 19.3600 302.280 11 67.1 4.89 4502.41 23.9121 328.119 12 56.7 2.5 3214.89 6.2500 141.750 13 58.3 2.38 3398.89 5.6644 138.754 14 51 2.14 2601.00 4.5796 109.140 15 59.9 Totals 910.4 3.72 3588.01 13.8384 222.828 52.71 55798.54 202.2647 3278.392 (910.4)2 SS x = 55798.54 − = 543.3293333 [1 pt] 15 (52.71) 2 SS y = 202.2647 − = 17.04176 [1 pt] 15 (910.4)*(52.71) SPxy = 3278.392 − = 79.2464 [1 pt] 15 79.2464 r= = 0.823551464 [2 pts] 543.3293333*17.04176 Interpretation: There is a very strong positive linear relationship between length and weight of newly born infants. 2. Test of hypothesis Ho: There is no linear relationship between length and weight of newly born infants. ( = 0) Ha: There is a positive linear relationship between length and weight of newly born infants. ( > 0) Test Procedure: t-test at  = 0.05. Decision Rule: Reject Ho if |tc| > t(n-2), otherwise fail to reject HO. Computation: t(n-2) = t0.05(15-2) = t0.05(13) = 1.771 tc = 0.823551464* 15 − 2 1 − (0.823551464) 2 = 5.2347 Decision: Since 5.2347 > ttab, we reject Ho. Conclusion: At  = 0.05, we have evidence to say that there is a positive linear relationship between length and weight of newly born infants. PREPARED BY: ANALYN A. MOJICA Associate Professor January 9, 2021 Republic of the Philippines CAVITE STATE UNIVERSITY Don Severino de las Alas Campus Indang, Cavite  (046) 4150-010 / (046) 4150-13 loc 253 www.cvsu.edu.ph OFFICE OF THE GRADUATE SCHOOL AND OPEN LEARNING COLLEGE Name: __________________________ Program: ________________________ Date: ______________ Score: _____________ Learning Activity COMPARISON OF TWO OR MORE POPULATIONS (PARAMETRIC) OBJECTIVES: At the end of the exercise, the student must be able to: 1. implement t-test for the analysis of 2 populations; 2. implement ANOVA for the analysis of k populations; and 3. formulate a realistic problem situation for the ONE-WAY ANOVA. MATERIALS: Calculator, Class Record, Stat Software PART I: Computation Direction: For each problem, show your complete solution and follow the steps in Hypothesis Testing. Verify your answer using SPSS or Excel (Copy paste SPSS or Excel Outputs after each manual computation) 1. A Psychology professor, teaching at large university, wants to know whether there is a difference between the IQs of male and female students in attendance. She randomly and independently selects 20 female students and 20 male students and has them take the IQ test. The resulting IQ data are as follows: Female 130 117 124 120 125 109 100 118 127 130 126 Male 106 114 134 120 107 131 133 134 122 111 109 118 131 104 130 116 122 115 129 117 101 144 122 124 116 120 119 114 110 What conclusion can be drawn out of the given data? Test your hypothesis using 0.01 level of significance. Use t-test for two independent samples, assuming unequal population variances. 2. As part of the study to determine the effects of a feeding program on weight gain, 12 healthy students were weighed at the beginning of the said program. They were reweighed after three months. Do the results suggest evidence of weight gain? Use ttest for dependent/related samples with alpha . Subject Initial Weight (lb) 3-Month Weight (lb) 3. 1 120 2 141 3 130 4 162 5 150 6 148 7 135 8 140 9 129 10 120 11 140 12 130 123 143 140 162 145 150 140 143 130 118 141 132 A school district supervisor wishes to compare four programs for training teachers to perform a certain manual task. Twenty new teachers are randomly assigned to the training programs, with 5 in each program. At the end of the training period, a test is conducted to see how quickly teachers can perform the task. The number of times the task is performed per minute is recorded for each trainee, with the following results: Program 1: 9, 12, 14, 11, 13 Program 2: 10, 6, 9, 9, 10 Program 3: 12, 14, 11, 13, 11 Program 4: 9, 8, 11, 7, 8 (a) Construct the ANOVA table (b) Using α = .05, determine whether the treatments differ in their effectiveness. For ANOVA, use this example as your guide. The following data represent the scores of a random sample of students in each section during the first long examination: Section Samples A 87 45 75 65 82 B 78 56 66 49 56 45 C 53 76 59 73 43 32 62 Is there a significant difference between the performances of students in the three sections? Use 0.01 level of significance. Solution: 1. Ho : There is no significant difference between means Ha : At least one mean is significantly different 2. Specify the level of significance and the sample sizes 3. Test Statistics : F-test (One-way ANOVA) 4. Critical Region : Reject Ho if Fc > Ftab at 0.05 level of significance 5. Computations: Construct an ANOVA Table Section ΣX ΣX2 354 26168 350 21138 398 24112 Samples A 87 45 75 65 82 B 78 56 66 49 56 45 C 53 76 59 73 43 32 62 ΣX2 = 71418 GT = 354+350+398 = 1102 SST = 71418 - (11022/18) = 3951.11 SSB = [(3542/5) + (3502/6)+ (3982/7)] - (11022/18) = 642.12 SSW = SST – SSB =3951.11 – 642.12 = 3308.99 MSB = SSB/df = 642.12 / 2 = 321.06 MSW = SSW/df = 3308.99 / 15 = 220.60 Fc = MSB/MSW = 321.06/220.60 = 1.455 ANOVA TABLE F tab df Sum of Square Mean Square Fcomp 0.05 0.01 Between(TREATMENT) 2 642.12 321.06 1.455 3.68 6.36 Within(ERROR) 15 3308.99 220.60 Total 17 3951.11 Source of Variation 6. Decision : Since Fc < Ftab at 0.05 level of significance, therefore accept Ho and conclude that there is no significant differences between the mean scores of three sections. Note: Use F table for the critical value of F, (Ftab) Part II. Problem Formulation Construct a problem situation that would require the comparison of k independent samples that can be analyzed by ONE-WAY ANOVA (parametric). Guidelines: 1. Provide sufficient background of the problem. 2. State the necessary assumptions needed to perform the test. 3. Perform test of hypothesis at  = 5 % to answer the objective of the study. PREPARED BY: ANALYN A. MOJICA ASSO. PROF. January 9, 2021 ...
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