SIMPLE LINEAR REGRESSION ANALYSIS

STATISTICS FOR MASTERAL STUDENTS

REPORTER: NORMA M. MONISIT MPA 1

SIMPLE LINEAR REGRESSION ANALYSIS

Regression Analysis deals with the estimation of one variable based on the changes and movement of two variables.One of the main uses of regression is to make prediction.

Prior to executing the regression analysis, there should be a significant relationship between the X (independent variable) and the Y (dependent variable)

SituationThe data below show the study time and examination

scores of 10 students. Determine the regression equation and test the hypothesis that study time is a predictor of examination scores at 95% confidence level.

Student Hours of Study (X) Examination Score (Y)

123456789

10

1578

101114151519

105

53745943568496698483

701

1. Problem StatementIs study hour a predictor of

examination score?2. Hypotheses

Ho: The study hour is not a predictor of the examination score. Ha: The study hour is a predictor

of the examination score

3. Choice of test statistics & level of significance: Simple linear regression, ᾳ = 0.05

4. Computation:

A. Manual 𝑦= 𝑎+ 𝑏𝑋 Where: 𝑎 = (σ𝑥)(σ𝑥²)/N(σ𝑥²) − (σ𝑥) ² 𝑏= 𝑁(σ𝑥𝑦) −(σ𝑥)(𝛴𝑦)/ N(σ𝑥²) − (σ𝑥) ²

Substituting:

a = ሺ105ሻሺ1,367ሻ10ሺ1,367ሻ−(105)² = 49.4771

𝑏= 10(7,880) − (105)(701)/10(1,367) – (105)²

𝑏= 1.9641 𝑌= 49.4771+ 1.9641𝑋

Stu-dent

Hours of Study (X)

Exami-nation Score

(Y)

XY X² Y²

123456789

10

1578

101114151519

105

53745943568496698483

701

53370413344560924

13441035125015777880

1254964

100121196225225361

1367

2809547634811849313670569216476170566889

51729

B. Scatter Plot (using Excel)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

53

74

59

43

56

8496

69

84 83f(x) = 1.9640831758034 x + 49.4771266540643R² = 0.394121522588693

Series1Linear (Series1)

C. Using SPSS:

STEPS:OPEN SPSSON DATA VIEW, ENCODE DATAON VARIABLE VIEW, ENCODE THE VARIABLE LABELS

ANALYZEREGRESSIONLINEARTRANSFER STUDY HOUR TO INDEPENDENT VARIABLE BOXTRANSFER EXAM SCORE TO DEPENDENT VARIABLE BOXOPTION, α = .05CONTINUEOK

Using SPSS:

5. Decision Rule and FindingDecision Rule: Reject Ho if level of

significance, ᾳ > p-valueFinding: ᾳ (0.05) < p-value (0.052)

6. Decision: Do not reject the null hypothesis7. Interpretation & Analysis

The study hour is not a predictor of examination score. The regression equation Exam_Score (Y͡.) =49.48 + 1.96 (hrs of study) reflects that, on the average, each additional hr of study yields a little less than 2 additional exam points (the slope). A student who did not study (hrs_study=0) would expect a score of 49 (the intercept)

The scatter plot shows an imperfect fit since not all of the variation in exams scores reflects other factors e.g. previous night’s sleep, class attendance, text anxiety.

Using the fitted regression equation, y₁ =49.76 + 1.96x₁, find each student’s expected examination score.

Student & study hours (X)

Actual Score (Y)

Expected Exam Scorey₁ =49.76 + 1.96x₁

No. 4, 8 hrsNo. 7, 14 hrs

No. 10, 19 hrs

439683

y₁ =49.76 + 1.96(8) = 65.19y₁ =49.76 + 1.96(14) =76.98y₁ =49.76 + 1.96(19) = 86.79

8. ConclusionFor the given data, the number of spent study

hours cannot predict examination scores of the students.

9. ImplicationsThis could be due to small sample size of the

given data. To have a more fitted predictive model need large sample size. Say hundreds or more of students’ study hour and their examination performance included in the regression analysis

Source:Ferdinand T. Abocejo & Zosima A. Pañares, Statistics Handbook I for Graduate Students (Revised April 2012)