Econometrics and Business Statistics - Solved Assignments
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This text provides solved assignments for Econometrics and Business Statistics course covering topics like probability, hypothesis testing, regression analysis and more. It includes descriptive statistics, F-test, two-sample t-test, scatter plot, regression model and significance level analysis.
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ECONOMETRICS AND BUSINESS STATISTICS STUDENT NAME/ID [Pick the date]
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Question 1 (a)Average numeracy score achieved by year 5 classes = 493 Standard deviation = 60 Randomly selected sample size = 30 Probability that a randomly selected sample would have average numeracy score less than 500 =? Standard error (S.E.) = Standard deviation/ sqrt (Randomly selected sample size) S.E.=60 √30=10.95 Assumption:Ithasbeenassumedthatthemeannumeracyscoreshasfollowednormal distribution. Standard Normal Table 1
Probability that a randomly selected sample would have average numeracy score less than 500 is 0.7389. (b)Assuming that 50% of student scored higher than 493 in numeracy test and hence, P(o) = 0.50 Randomly selected sample size N = 200 Total number of students scored higher than 493 = 115 Relevant proportionp=115 200=0.575 Thezscore=p−p(o) √p(o)∗1−p(o) N Hence, requisite probability P(z>2.1213) = 0.0169 (Standard Normal Table) (c)Average numeracy score achieved by year 5 classes (2016) = 493 Randomly selected sample size = 59 Standard deviation (2015) = 36.76 Average numeracy score (2015) = 506.63 Null and alternative hypotheses NullhypothesisH0:μ2015=μ2016 AlternativehypothesisH1:μ2015≠μ2016 2
The t stat tstat=493−506.63 36.76 √59 =−2.848 Degree of freedom (Dof) =Randomly selected sample size-1 =58 The p value It is apparent from the sign of alternative hypothesis that it is a two tailed hypothesis test and hence, the two tailed p value would be taken into account. The p value = 0.0060 Level of significance Alpha = 5% Result It is apparent that p value is lesser than level of significance and therefore, sufficient statistically evidence is available for the rejection of null hypothesis and for the acceptance of alternative hypothesis. Hence, the claim is correct that average numeracy scores in 2015 in three major capital cities differs from the average numeracy scores in 2016. (d)95% confidence interval Randomly selected sample size = 59 Standard deviation (2015) = 36.76 Average numeracy score (2015) = 506.6 The t critical value (95% confidence interval and 59 dof) = 2.0017 Lower limit = Average – (t value * Standard deviation/ sqrt(sample size)) = 506.6 – (2.0017* 36.76/sqrt(59)) = 497.05 3
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Upper limit = Average – (t value * Standard deviation/ sqrt(sample size)) = 506.6 +(2.0017* 36.76/sqrt(59)) = 516.20 95% confidence interval = [497.05516.20] There is 95% confidence that mean numeracy score for grade 5 class of three major capital citiies in the year 2015 will lie in the range of 497.05 and 516.20. (e)Assuming that 50% of year 5 classes Melbourne student achieved higher than or same as 500 in numeracy test and hence, P(o) = 0.50 Randomly selected sample size N = 20 Total number of students scored higher than or same as 500 = 15 Relevant proportionp=15 20=0.75 Null and alternative hypotheses NullhypothesisH0:μyear5,Melbourne<500 AlternativehypothesisH1:μyear5,Melbourne≥500 The t stat tscore=p−p(o) √p(o)∗1−p(o) N =0.75−0.5 √0.5(1−0.5) 20 =2.24 Degree of freedom (Dof) =Randomly selected sample size-1 =19 The p value It is apparent from the sign of alternative hypothesis that it is a one tailed hypothesis test and hence, the two tailed p value would be taken into account. The p value = 0.0186 4
(f)Level of significance Alpha = 5% (g)Result It is apparent that p value is lesser than level of significance and therefore, sufficient statistically evidence is available for the rejection of null hypothesis and for the acceptance of alternative hypothesis. Hence, the claim is correctof year 5 classes Melbourne student achieved higher than or same as 500 in numeracy test. (f)95% confidence interval 95% confidence interval = [0.56020.9398] There is 95% confidence that mean numeracy score of year5 classes Melbourne students achieved higher than or same as 500 in numeracy testwill lie in the range of 0.5602 and 0.9398. (h)It is apparent that proportion of year 5 classes Melbourne students at 65% fall within the range of 95% confidence interval and hence, null hypothesis will not be rejected and hence, it cannot reject the claim that 65% of year 5 classesMelbourne students achieved higher than or same as 500 in numeracy test. 5
Question 2 (a)Descriptive stat (year of 3 classes students from Sydney and Melbourne) (b)Hypothesis testing : Null and alternative hypothesis Ftest–twosampleforvariance 6
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Two tailed p value = 2*one tailed p value = 0.85 Level of significance (Alpha) = 5% It is apparent that p value is greater than level of significance and therefore, sufficient statistically evidence is not available for the rejection of null hypothesis. Hence, there is no significant differences is observed in the variance of average writing scores of the year 3 classes children from Sydney school and Melbourne school in the year 2015. (c)Hypothesis testing Null and alternative hypothesis Population standard deviation is not given which indicates that z value cannot be used and hence, relevant test would be two sample t test. (d)T he p value (two tailed) comes out to be 0.9209. It can be seen that p value is greater than alpha (5%) and therefore, sufficient statistically evidence is not available to reject the null hypothesis. Thus, no statistically significant different is present in average writing scores of the year 3 class children from Sydney and Melbourne school in the year 2015. 7
(e)Histogram It can be seen from the above that histogram does not show normal distribution because there is a long leftward tail which is indicative of presence of negative skew. The histogram also indicates the variation in the data points. (f)Hypothesis testing Population standard deviation is not given which indicates that z value cannot be used and hence, relevant test would be two sample t test. 8
One tailed p value = 0.0173 Level of significance (Alpha) = 5% It is apparent that p value is lesser than level of significance and therefore, sufficient statistically evidenceisavailablefortherejectionofnullhypothesisandacceptanceofalternative hypothesis. Hence, the mean reading scores is higher than mean writing scores of Melbourne school children of year 3 (g)The 95% confidence interval for difference of ready and writing scores of Melbourne school children of year 3 is shown below. 95% confidence interval = [5.54733.553] 9
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There is 95% confidence that difference of ready and writing scores of Melbourne school children of year 3 will lie in the range of 5.547 and 33.553. Question 3 (a)Scatter plot X: Independent variable= Numeracy scores Y: Dependent variable= Grammar scores The association between the variables can be termed as linear because numeracy and grammar scores are fall on a straight line. (b)Regression model 10
Least square regression equation Grammar scores = (-4.66) + (1.04*Numeracy scores) + (14.264) (c)Requisite sample regression model (d)The slope coefficient coms out to be 1.04 which indicates that for a unit variation in the numeracy scores would result corresponding change in the grammar scores by 1.04 times. (e)Significance level = 5% 11
Null Hypothesis: slope (βnumeracy)is significant. Alternative Hypothesis: slope (βnumeracy) is insignificant. The t stat from the regression for slope coefficient = 7.82 Corresponding p value = 0.00 Level of significance (Alpha) = 5% It is apparent that p value is lesser than level of significance and therefore, sufficient statistically evidenceisavailablefortherejectionofnullhypothesisandacceptanceofalternative hypothesis. Hence, it can be said that slope coefficient is significant which implies that numeracy scores is a statistically significant predictor of dependent variable grammar score. 12
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