9 For an empirical investigation into the amount of rent paid by tenants in a town, data on income X and rent Y have been collected. Data for a total of 300 tenants of one- bedroom flats have been recorded. Assume that X and Y are both Normally distributed with expectations w, and Hy, and variances o; and o. Sy and Sy are the sample standard deviation for random samples of X and Y, respectively. The random variable Zx is defined as 2x=299 2993 (i) State the distribution of Zy and all of its parameters. [2] (ii) Write down the expectation and variance of Zx. [2] (iii) Explain why the distribution of Zx is approximately Normal. [2] (iv) Calculate values of an approximate 2.5% quantile and 97.5% quantile of the distribution of Zy using your answers to parts (ii) and (iii). [3] In the collected sample, the mean income is $1,838 with a realised sample standard deviation of $211, the mean rent is $608 with a realised sample standard deviation of $275 and Exy; = 348 10%. (v) Calculate a 95% confidence interval for the mean income. [2] (vi) Calculate a 95% confidence interval for the mean rent. [2] (vii) Calculate an approximate 95% confidence interval for the variance of income using your answer to part (iv). [2] (viii) Identify which one of the following options gives the correct form of the equation for the simple linear regression model of rent on income, including any assumptions required for statistical inference. [2] A1 A2 A3 Y;= a + bx; y; = a + bx, +z; with E[:] =0 y = a + bx;+z, with z; -X?, 299 df y; = a + bx, +z; with z;- N(0,0) A4 (ix) Calculate estimates of the slope and the intercept of the model in part (viii) based on the above data for the 300 tenants. [4] [Total 21] 9 For an empirical investigation into the amount of rent paid by tenants in a town, data on income X and rent Y have been collected. Data for a total of 300 tenants of one- bedroom flats have been recorded. Assume that X and Y are both Normally distributed with expectations w, and Hy, and variances o; and o. Sy and Sy are the sample standard deviation for random samples of X and Y, respectively. The random variable Zx is defined as 2x=299 2993 (i) State the distribution of Zy and all of its parameters. [2] (ii) Write down the expectation and variance of Zx. [2] (iii) Explain why the distribution of Zx is approximately Normal. [2] (iv) Calculate values of an approximate 2.5% quantile and 97.5% quantile of the distribution of Zy using your answers to parts (ii) and (iii). [3] In the collected sample, the mean income is $1,838 with a realised sample standard deviation of $211, the mean rent is $608 with a realised sample standard deviation of $275 and Exy; = 348 10%. (v) Calculate a 95% confidence interval for the mean income. [2] (vi) Calculate a 95% confidence interval for the mean rent. [2] (vii) Calculate an approximate 95% confidence interval for the variance of income using your answer to part (iv). [2] (viii) Identify which one of the following options gives the correct form of the equation for the simple linear regression model of rent on income, including any assumptions required for statistical inference. [2] A1 A2 A3 Y;= a + bx; y; = a + bx, +z; with E[:] =0 y = a + bx;+z, with z; -X?, 299 df y; = a + bx, +z; with z;- N(0,0) A4 (ix) Calculate estimates of the slope and the intercept of the model in part (viii) based on the above data for the 300 tenants. [4] [Total 21]