Show that If a Random Variable X has the Binomial Distribution with Parameters n,p: Quantitative Methods Coursework, SMU, Singapore

Assignment Details:

1. Show that if a random variable X has the binomial distribution with parameters n,p, then the probability of successive values are related by the formula:
2. During 2017, a proportion of 0.225 of road accident casualties aged 16 and over in England and Wales who were fatally injured had a blood alcohol concentration above the legal limit for driving (80mg/100m1 of blood). Supposing that the same proportion applies to cyclists as does to all fatally injured casualties, use the binomial distribution to calculate the probability that 3 (the observed number) or fewer out of 40 cyclists (the sample for which data are available) had blood alcohol concentrations above this limit.
3. Use the normal distribution to estimate this probability and compare your results.
4. During 2017, 67 road accident pedestrian casualties aged 16 and over in England and Wales who were fatally injured and who were tested had concentrations above the legal limit for driving (80mg1100m1 of blood). Supposing that the same proportion 0.225 applies to pedestrians as does to all fatally injured casualties. use the normal distribution to estimate the probability that 67 (the observed number) or more out of 184 pedestrians (the sample for which data are available) had blood alcohol concentrations above this limit and discuss your results.

Question 2:

The linear regression model can be written

1. Tables Al and A2 give the results of two linear regression models estimated from two independent samples of bus trips in a single city. The dependent variable yi is the bus journey time (in minutes). and the single independent variable xi is trip length (Trip length in lm). Explain what is meant by the terms unbiased and efficient in relation to the estimator β‾ of the population parameter β. Given that the true value of β is 1.50, comment on the bias and efficiency of the parameter estimates of trip length.
2. In Table B1 below. the estimate of the parameter β from Model 1 is given along with the standard error of the estimate. the total sum of squares (St²). the model sum of squares (Sm²), and the residual sum of squares (Se²). Using the information given:

• Determine whether there is a statistically significant association between trip length and bus journey time; and,
• Assess the overall model performance.

Include all relevant formulas to support your answers and comment on your findings.

3. A multivariate linear regression model is estimated with the number of bus trips per household per month as the dependent variable, and three independent variables: the number of persons per household (iv’_person HEI). monthly household income (N income), and the distance to the nearest bus stop (Iv distance). Table Dl gives the results of the model estimated using the original data set, while Table D2 gives results from a model where the independent variables have been aggregated into groups as described. Explain how and why the parameter estimates and overall model performance have been affected by grouping. Include relevant formulas to support your explanation.