Econometrics and Business Statistics (ECON634) Semester 2, 2018 Take-home Test
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13 Econometrics and Business Statistics (ECON634) Semester 2, 2018 Take-home Test
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13 ANS: The distribution of numeracy scores was assumed to be normally distributed using central limit theorem, as the sample size was greater or equal to 30. Here,and The required probability was calculated as below. Figure1: Probability Area for
13 ANS: The distribution of numeracy score was assumed to be normally distributed using central limit theorem, as the sample size was greater than 30. According to the problem, population parameter ratio for numeracy score was Now, sample size Required probability was Figure2: Probability Area for ANS: Null hypothesis: H0:
13 Alternate Hypothesis: HA:wherewas the average numeracy score achieved by Year 5 classes. Level of significance was considered at 5%, and the test was a two tail test. Sample statistic values were Due to absence of population standard deviation, t-test was the appropriate choice. The t-statistic =with 58 degrees of freedom. The p-value was< 0.05, at 5% level. The95%confidenceintervalforpopulationmeanwasestimatedas ,andthepopulationaveragewas outside the interval. Hence, the null hypothesis was rejected at 5% level of significance, concluding thataveragenumeracyscoresinthethreemajorcapitalcitieswere significantly different from the average numeracy score of 493. ANS:The 95% confidence interval foraverage numeracy score of all grade 5 classesinthemajorcapitalcitiesin2015wasestimatedas ,andthepopulationaveragewas foundtobe outside theinterval. Hence,the nullhypothesiswascorrectly rejected at 5% level of significance. ANS: Null hypothesis: H0: Alternate Hypothesis: HA:
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13 Level of significance was considered at 5%, and the test was a two tail test. Sample statistic values were Z-test was the appropriate choice. The Z-statistic = The p-value was< 0.05, at 5% level. Figure3: Probability Area for The 95% confidence interval for population mean was estimated as, and the population parameterwas outside the interval. Hence, the null hypothesis was rejected at 5% level of significance, concluding thatproportionofnumeracyscoresinthethreemajorcapitalcitieswere significantly different from the population proportion of numeracy score. ANS:The95%confidenceintervalforpopulationmeanwasestimatedas , and the population parameterwas outside the interval. Hence, the null hypothesis was correctly rejected at 5% level of significance,
13 ANS:Sample statistic values were The 95% confidence interval for population mean was estimated as, and the population parameterwas still outside the interval. Hence, the null hypothesis would be still rejected at 5% level of significance, ANS: Descriptive statistics for the average Writing scores of Year 3 children from Sydney schools and Melbourne schools in 2015 was as follows. Table1 Descriptive Statistics for Writing Scores for Sydney Schools and Melbourne Schools DescriptivesWriting_SydneyWriting_Melbourne Mean439.16439.9 Standard Error5.195.29 Median442444.5 Mode438455 Standard Deviation22.6423.66 Sample Variance512.36559.57 Kurtosis0.460.12 Skewness-0.95-0.15 Range8593 Minimum384394 Maximum469487 Count1920 ANS: Null hypothesis: H0: Alternate Hypothesis: HA:whereandwere the variances of Writing scores of Year 3 children from Sydney schools and Melbourne schools in 2015. Level of significance was considered at 5%, and the test was a two tail test.
13 Sample statistic values were The F-test was the appropriate choice. The F-test was performed in Excel to test the hypothesis at 5% level. Table2: F-test for Two Sample Variances F-Test Two-Sample for Variances Writing_SydneyWriting_Melbourne Mean439.16439.90 Variance512.36559.57 Observations1920 df1819 F0.92 P(F<=f) one-tail0.428 F Critical one-tail0.45 The F-calculated was 0.92, and p-value = 0.428 > 0.05, at 5% level. Hence, the null hypothesis failed to get rejected at 5% level of significance, concluding that there was no statistically significant difference in variances of writing scores of Year 3 children from Sydney schools and Melbourne schools in 2015. . ANS: Null hypothesis: H0: Alternate Hypothesis: HA: Level of significance was considered at 5%, and the test was a two tail test. Sample statistic values were Due to absence of population standard deviations, t-test with equal variances (from 2.b) was the appropriate choice.
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13 Table3: T-test for Two Sample t-test t-Test: Two-Sample Assuming Equal Variances Writing_SydneyWriting_Melbourne Mean439.16439.90 Variance512.36559.57 Observations1920 Pooled Variance536.60 Hypothesized Mean Difference0 df37 t Stat-0.10 P(T<=t) one-tail0.460 t Critical one-tail1.69 P(T<=t) two-tail0.921 t Critical two-tail2.03 The t-statistic =with 37 degrees of freedom. The p-value was> 0.05, at 5% level. Hence, the null hypothesis failed to get rejected at 5% level of significance, concluding that there was no statistically significant difference in the average writing scores of Year 3 children from Melbourne schools and Sydney schools in 2015. ANS: The p-value was> 0.05, for two tail test at 5% level. Hence, the null hypothesis failed to get rejected at 5% level of significance, concluding that there was no statistically significant difference in the average writing scores of Year 3 children from Melbourne schools and Sydney schools in 2015. ANS: Histogram for the differences in the average Reading scores and the average Writing scores of Year 3 classes in 2015 in Melbourne schools is as follows.
13 Figure4: Histogram fordifferences in the average Reading scores and the average Writing scores The histogram was symmetric in nature, and the shape almost resembled to that of the normal distribution (bell shaped curve). ANS: Null hypothesis: H0: AlternateHypothesis:HA:whereandwereaverageReading and Writing scores. Level of significance was considered at 5%, and the test was a right tail test. Sample statistic values were Duetoabsenceofpopulationstandarddeviations,t-testwithunequal variances was the appropriate choice.
13 Table4: T-test for Two Sample t-Test: Two-Sample Assuming Unequal Variances ReadingWriting Mean459.45439.9 Variance1023.42559.57 Observations2020 Hypothesized Mean Difference0 df35 t Stat2.197 P(T<=t) one-tail0.017 t Critical one-tail1.690 P(T<=t) two-tail0.035 t Critical two-tail2.030 The t-statistic = 2.197 with 35 degrees of freedom. The p-value was< 0.05, at 5% level. Hence, the null hypothesis was rejected at 5% level of significance, concluding that the average Reading scores were statistically significantly more than the average writing scores of Year 3 classes (Fang, 2017). ANS:The95%confidenceintervalforaveragenumeracyscoreofmean difference in average Reading scores and average Writing scores of Year 3 classeswasestimatedas, and the difference in average Reading scores and average Writing scores was foundtobe outside theinterval. Hence,the nullhypothesiswascorrectly rejected at 5% level of significance.
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13 ANS: Scatter diagram between Grammar scores and Numeracy scores of Year 5 classes in Sydney in 2015 is constructed as below. Figure5:diagram between Grammar scores and Numeracy scores
13 ItappearedthatGrammarandNumeracyscoreswerelinearlypositively related.Grammarscoreswerenotedtoincreaselinearlywithincreasein Numeracy scores. ANS: Excel was used to estimate the regression line of Grammar on Numeracy scores. The estimated regression line was, wherewas the residual of the estimated equation Table5: regression Model ofGrammar on Numeracy scores Regression Statistics Multiple R0.885 R Square0.782 Adjusted R Square0.770 Standard Error14.264 Observations19 ANOVA dfSSMSFSignificance F Regression112443.0220312443.0220361.1520.000 Residual173459.083231203.475 Total1815902.10526 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept-4.65669.813-0.0670.948-151.949142.637 Numeracy1.0380.1337.8200.0000.7581.318 ANS:Theestimatedregressionlinewas, wherewas the residual of the estimated equation The standard error for intercept was SE = 69.813, and that of the Numeracy scores was SE = 0.133. The intercept (t = -0.067, p = 0.948) was statistically insignificant, as negative Grammar score due to zero in Numeracy scores does not make any statistical sense. Numeracywasstatisticallysignificant(t=7.82,p<0.05)predictorof Grammar scores at 5% level of significance.
13 The sample size was 19, and the coefficient of determination (R-square) = 0.782.Hence,Numeracywasabletoexplain78.2%variationinGrammar scores. ANS:Theslopecoefficientwas1.038,whichsignifiedasignificantstrong positive linear relation between the two variables. The angle of the regression line was noted to be greater than 45 degrees, as the slope was greater than 1. ANS: Null hypothesis: H0:(No linear relation exists) Alternate Hypothesis: HA: Level of significance was considered at 5%, and the test was a two tail test. Sample statistic values were Due to absence of population standard deviation, t-test was the appropriate choice. The t-statistic =with 18 degrees of freedom. The p-value was< 0.05, at 5% level. The95%confidenceintervalforpopulationmeanwasestimatedas ,andthepopulationparameter was outside the interval. Hence, the null hypothesis was rejected at 5% level of significance, concluding that a statistically significant linear relation exists between Grammar scores and Numeracy scores of Year 5 classes in Sydney in 2015 (Chatterjee, & Hadi, 2015). References
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13 Chatterjee, S., & Hadi, A. S. (2015).Regression analysis by example. John Wiley & Sons. Fang,K.W.(2017).SymmetricMultivariateandRelatedDistributions:0. Chapman and Hall/CRC.