SUMMARY FOR EQT271 Semester /2015 Maz Jamilah Masnan, Inst. of Engineering Mathematics, Univ. Malaysia Perlis

EQT 271 ENGINEERING STATISTICS 1. Basic Statistics 2. Statistics Inference 4. Simple Linear Regression 3. ANOVA 5. Nonparametric Statistics maz jamilah masnan/sem /2015 2

Chapter 1. Basic Statistics  Statistics in Engineering  Collecting Engineering Data (data type, group vs ungroup)  Data Summary and Presentation – 1. graphically (table, charts, graph etc) and 2. numerically (MCT – mean, mode, median, MOD – range, variance, std. dev., MOP – quartile, z-score, percentile, outlier, boxplot ≈ 5-number-summary ) maz jamilah masnan/sem /2015 3

1. Basic Statistics  Probability Distributions - Discrete Probability Distribution (Binomial & Poisson, Poisson Approximation of Binomial Probabilities – [ ]) - Continuous Probability Distribution (Normal & Normal approximation of Binomial – [ + continuous correction factor] & Poisson – [ 10 + continuous correction factor]) maz jamilah masnan/sem /2015 4

Population 1 Population 2 S2 S3 S4 Sn S2 S3 S4 Sn S n n S2 S3 S4 Sn S2 S3 S4 Sn Concept of Sampling Distribution of the Sample Mean maz jamilah masnan/sem /2015 5

n maz jamilah masnan/sem /2015 6

1. Basic Statistics  Uses Central of Limit Theorem, std. dev. = std. error i.e.  If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with and. ,,  Z-value for the sampling distribution of is Sampling Distribution of the Sample Mean maz jamilah masnan/sem /2015 7

Properties and Shape of the Sampling Distribution of the Sample Mean.  If n ≥30, is normally distributed, where  Note: If the unknown then it is estimated by.  If n<30 and variance is known. is normally distributed  If n<30 and variance is unknown. t distribution with n-1 degree of freedom is used maz jamilah masnan/sem /2015 8

Population 1Population 2 S S2 S3 S4 Sn S2 S3 S4 Sn n n maz jamilah masnan/sem /2015 9

Sampling Distribution of the Sample Proportion  The population and sample proportion are denoted by p and, respectively, are calculated as, and  For the large values of n (n ≥ 30), the sampling distribution is very closely normally distributed.  Mean and Standard Deviation of Sample Proportion ~ maz jamilah masnan/sem /

Statistical inference is a process of drawing an inference about the data statistically. It concerned in making conclusion about the characteristics of a population based on information contained in a sample. Since populations are characterized by numerical descriptive measures called parameters, therefore, statistical inference is concerned in making inferences about population parameters. Chapter 2. Statistical Inference maz jamilah masnan/sem /

1. Estimation (point & interval estimate [ a < < b] ) - Confidence interval estimation for mean ( μ ) and proportion (p) - Determining sample size 2. Hypothesis Testing - Test for one and two means - Test for one and two proportions Estimation & Hypothesis Testing maz jamilah masnan/sem /

μ Observed μμ 95% of the s lie in this interval F( ) μ n n interval estimates computed by using ± 1.96 maz jamilah masnan/sem /

Confidence Interval (Mean) maz jamilah masnan/sem /

Confidence Interval Estimates for the differences between two population mean, i) Variance and are known ii) If the population variances, and are unknown, then the following tables shows the different formulas that may be used depending on the sample sizes and the assumption on the population variances. Estimation (Confidence Interval – Difference in Means) maz jamilah masnan/sem /

Equality of variances, when are unknown Sample size maz jamilah masnan/sem /

Confidence Interval (Proportion) maz jamilah masnan/sem /

Hypothesis Testing 1. Test for one and two population means 2. Test for one and two population proportions Hypothesis Testing Require understanding of:- Definition of hypothesis test, null and alternative hypothesis, tests statistics, critical region (rejection region), critical value, p-value. maz jamilah masnan/sem /

Procedure for hypothesis testing 1. Define the question to be tested and formulate a hypothesis for a stating the problem. 2. Choose the appropriate test statistic and calculate the sample statistic value. The choice of test statistics is dependent upon the probability distribution of the random variable involved in the hypothesis. 3. Establish the test criterion by determining the critical value and critical region. 4. Draw conclusions, whether to accept or to reject the null hypothesis. maz jamilah masnan/sem /

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Hypothesis testing for the differences between two population mean, Test hypothesis Test statistics i) Variance and are known, and both and are samples of any sizes. ii) If the population variances, and are unknown, then the following tables shows the different formulas that may be used depending on the sample sizes and the assumption on the population variances. Hypothesis testing for the differences between two population mean, Test hypothesis Test statistics i) Variance and are known, and both and are samples of any sizes. ii) If the population variances, and are unknown, then the following tables shows the different formulas that may be used depending on the sample sizes and the assumption on the population variances. maz jamilah masnan/sem /

Equality of variances, when are unknown Sample size Equality of variances, when are unknown Sample size maz jamilah masnan/sem /

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For the single mean & proportion Confidence Interval vs Hypothesis Testing At the same level in confidence interval and hypothesis testing, when the null hypothesis is rejected, the confidence interval for the mean and proportion will not contain the hypothesized mean/proportion. Likewise, when we fail to reject null hypothesis the confidence interval will contain the hypothesized mean/ proportion. ** Applies only for two-tailed test. Allan Bluman, pg. 458 maz jamilah masnan/sem /

For the difference of means & proportions Confidence Interval vs Hypothesis Testing [-8.5, 8.5] Contains zero = If the CI contains zero, we fail to reject H 0 (Means that the there is NO DIFFERENCE in population means or proportions) [5.45, 12.45] No zero = If the CI does not contain zero, we reject H 0 (Mean/proportion for population 1 is GREATER than the mean/proportion for population 2) [-7.3, -3.3] No zero = If the CI does not contain zero, we reject H 0 (Mean/proportion for population 1 is LESS than the mean/proportion for population 2) maz jamilah masnan/sem /

maz jamilah masnan/sem / Outcomes for hypothesis result Ho (Claim) Reject Ho There is sufficient evidence to reject the claim. Fail to Reject Ho There is insufficient evidence to reject the claim. H 1 (Claim) Reject H There is sufficient evidence to support the claim. Fail to Reject H There is insufficient evidence to support the claim.

Chapter 3. ANALYSIS OF VARIANCE (ANOVA) 1. 1-way-ANOVA [Completely Randomized Design] 2. 2-way-ANOVA (without replication) [Randomized Completely Block Design] 3. 2-way-ANOVA (with replication) [Factorial Design] * Testing 3 or more population means maz jamilah masnan/sem /

1. 1-way-ANOVA [Completely Randomized Design] Hypothesis: H 0 : µ 1 = µ 2 =... = µ t * H 1 : µ i  µ j for at least one pair (i,j) (At least one of the treatment group means differs from the rest. OR At least two of the population means are not equal) The populations from which the samples were obtained must be normally or approximately normal distributed The populations from which the samples were obtained must be normally or approximately normal distributed The variance of the response variable, denoted  2, is the same for all of the populations. The variance of the response variable, denoted  2, is the same for all of the populations. The observations (samples) must be independent of each other The observations (samples) must be independent of each other nAssumptions for Analysis of Variance maz jamilah masnan/sem /

1. 1-way-ANOVA [Completely Randomized Design] Source of Variation Sum of Squares DFMean Square Fp-Value Treatments (between group var.) k-1 Error (within group var.) N-k TotalN-1 maz jamilah masnan/sem /

Source of Variation Sum of Squares Degrees of Freedom Mean Square Fp -Value Treatments SSTRt-1 ErrorSSEN-t TotalSSTN way-ANOVA [Completely Randomized Design] maz jamilah masnan/sem /

CONCLUSION Fail to Reject H 0 No difference in mean Between- group variance estimate approximately equal to the within-group variance F test value approximately equal to 1 Reject H 0 Difference in mean Between- group variance estimate will be larger than within-group variance F test value = greater than 1 * All treatments are equal * Treatments are not equal 1. 1-way-ANOVA [Completely Randomized Design] maz jamilah masnan/sem /

nSampling Distribution of MSTR/MSE MSTR/MSE Sampling Distribution of MSTR/MSE  Do Not Reject H 0 Reject H 0 Critical Value FF FF Comparing the Variance Estimates: The F Test If F_ratio < F_(critical value), FAIL to REJECT Ho If F_ratio > F_(critical value), REJECT Ho maz jamilah masnan/sem /

2-way-ANOVA (without replication) [Randomized Completely Block Design] Sample Mean Sample Variance Observation Wax Type 1 Wax Type 2 Wax Type Sample Mean Sample Variance Batch Wax Type 1 Wax Type 2 Wax Type Treatment Block *Treatment can be in column or row (in 1-way-ANOVA) *Treatment and block can either be in column or row (in 2-way-ANOVA) maz jamilah masnan/sem /

First Hypothesis: Treatment Effect: H 0 :  1 =  2 =... =  t =0 H 1 :  j  0 at least one j OR 2-way-ANOVA (without replication) [Randomized Completely Block Design] Second Hypothesis: Block Effect: H 0 :  i = 0 for each value of i through n H 1 :  i ≠ 0 at least one i OR H 0 : µ 1 = µ 2 =... = µ t * H 1 : µ i  µ j for at least one pair (i,j) H 0 : µ 1 = µ 2 =... = µ t * H 1 : µ i  µ j for at least one pair (i,j) maz jamilah masnan/sem /

Source of Variation Sum of Squares DF Mean Square F p -Value Treatmentsk-1 Blocks n-1 Error (k-1) *(n-1) Totalkn-1 2-way-ANOVA (without replication) [Randomized Completely Block Design] maz jamilah masnan/sem /

Source of Variation Sum of Squares Degrees of Freedom Mean Square Fp -Value Treatments SSTRt-1 Blocks SSBLn-1 ErrorSSE(t-1)(n-1) TotalSSTtn-1 Source of Variation Sum of Squares Degrees of Freedom Mean Square Fp -Value Treatments SSTRt-1 Blocks SSBLn-1 ErrorSSE(t-1)(n-1) TotalSSTtn-1 2-way-ANOVA (without replication) [Randomized Completely Block Design] maz jamilah masnan/sem /

Do Not Reject H 0 Reject H 0 (Critical Value) FF FF If F_ratio < F_(critical value), FAIL to REJECT Ho If F_ratio > F_(critical value), REJECT Ho 2-way-ANOVA (without replication) [Randomized Completely Block Design] DECISION TO MAKE maz jamilah masnan/sem /

Three Sets of Hypothesis: i. Factor A Effect: H 0 :  1 =  2 =... =  a =0 H 1 : at least one  i  0 ii. Factor B Effect: H 0 :  1 =  2 =... =  b =0 H 1 : at least one  j ≠ 0 iii. Interaction Effect: H 0 : (   ) ij = 0 for all i,j H 1 : at least one (   ) ij  0 H 0 : µ 1 = µ 2 =... = µ a * H 1 : µ i  µ k for at least one pair (i,k) H 0 : µ 1 = µ 2 =... = µ b * H 1 : µ i  µ k for at least one pair (i,k) H 0 : µ AB1 = µ AB2 =... = µ ABb * H 1 : µ ABi  µ ABk for at least one pair (i,k) REMEMBER THIS 2-way-ANOVA (with replication) [Factorial Design] maz jamilah masnan/sem /

2-way-ANOVA (with replication) [Factorial Design] H 0 : There is no difference in means of factor A H 1 : There is a difference in means of factor A H 0 : There is no difference in means of factor B H 1 : There is a difference in means of factor B H 0 : There is no interaction effect between factor A and B for/on ……… H 1 : There is an interaction effect between factor A and B for/on ……… Three Sets of Hypothesis: i. Factor A Effect: ii. Factor B Effect: iii. Interaction Effect: OR USE THESE maz jamilah masnan/sem /

maz jamilah masnan/sem / way-ANOVA (with replication) [Factorial Design] FIRST : Run test to check INTERACTION [Plot the interaction and test the hypothesis] If there is NO INTERACTION, then run a test to know which factor effect is significance If there EXIST INTERACTION, no need to run tests for each factor.

Source of Variation Sum of Squares DF Mean Square Fp -Value Factor A a-1 Factor B b-1 Interaction(a-1)(b-1) Errorab(r-1) Totalabr-1 2-way-ANOVA (with replication) [Factorial Design] maz jamilah masnan/sem /

2-way-ANOVA (with replication) [Factorial Design] Source of Variation Sum of Squares Degrees of Freedom Mean Square Fp -Value Factor A SSAa-1 Factor B SSBb-1 InteractionSSAB(a-1)(b-1) ErrorSSEab(r-1) TotalSSTabr-1 maz jamilah masnan/sem /

Do Not Reject H 0 Reject H 0 (Critical Value) FF FF If F_ratio < F_(critical value), FAIL to REJECT Ho If F_ratio > F_(critical value), REJECT Ho 2-way-ANOVA (with replication) [Factorial Design] DECISION TO MAKE maz jamilah masnan/sem /

maz jamilah masnan/sem / Disordinal Interaction There is a SIGNIFICANCE interaction between …… Ordinal Interaction There is an interaction but not significant. The main effect can be interpreted independently No Interaction (parallel) There is no significant interaction. The main effect can be interpreted independently

4. Simple Linear Regression Estimate the model using LEAST SQUARE METHOD maz jamilah masnan/sem /

ADEQUACY OF THE MODEL COEFFICIENT OF DETERMINATION ( OR ) PEARSON PRODUCT MOMENT CORRELATION COEFFICIENT (r) measure of the variation (%) of the dependent variable (Y) that is explained by the regression line and the independent variable (X) measures the strength of a linear relationship between the two variables X and Y. r +1 WeakStrong maz jamilah masnan/sem /

maz jamilah masnan/sem / TEST FOR LINEARITY OF REGRESSION 1. Determine the hypotheses. 2. Compute Critical Value/ level of significance. 3. Compute the test statistic. ( no linear r/ship) (exist linear r/ship) ( no linear r/ship) 4. Determine the Rejection Rule. Reject H 0 if : or p-value < a 5.Conclusion. t -Test F -Test 1. Determine the hypotheses. 3. Compute the test statistic. F = MSR/MSE * this value can get from ANOVA table 4. Determine the Rejection Rule. Reject H 0 if : p-value < a F test > ( no linear r/ship) (exist linear r/ship) 2. Specify the level of significance. There is a significant relationship between variable X and Y. 5.Conclusion. There is a significant relationship between variable X and Y.

maz jamilah masnan/sem / ANOVA APPROACH FOR TESTING LINEARITY OF REGRESSION 1) Hypothesis: 2) F-distribution table: 3) Test Statistic: F = MSR/MSE = or using p-value approach: 4) Rejection region: If F statistic > F table, we reject H 0 or if p-value < alpha, we reject H 0 5) Thus, there is a linear relationship between the variables X and Y.

maz jamilah masnan/sem / Nonparametric Statistics Sign Test (ST)Wilcoxon Signed Rank Test (WSRT) Man Whitney Test (MWT) (i.e. Wilcoxon Rank Sum Test) Kruskal Wallis Test (KWT)  Test for 1 sample (use median)  Can be performed for paired sample but not covered in EQT271  Test for 1 sample (use median)  Can be performed for paired sample but not covered in EQT271  The parametric version is t-test  Test for 2 samples (use medians)  The parametric version is Z test and t-test  Test for 3 or more samples (use medians)  The parametric version is ANOVA

maz jamilah masnan/sem / * Please double check the summary with the notes. Some of the complete descriptions and formulae might not be available in the summary. * * Please do more exercises for the final exam preparation *