Question

The rainfalls for a six day period for Omarama and Twizel are as follows a₁​=7,a₂=10,a₃​=0,a₄​=0,a₅​=2, and a₆=5,t₁​=19,t₂​=18,t₃​=0,t4​=1,t₄₅=5, and t₆​=5,​ where ai​ and ti​ are the recorded rainfalls in Omarama and Twizel respectively on day i. The data are put into six vectors, where the first and second components of each vector are the days' rainfalls for Omarama and Twizel, viz.

z​i​=(ai
​ti​​) for i=1,…,6
(a) Calculate the mean vector and variance matrix for the six data points z​₁,…,z​₆.

Answer 1

To calculate the mean vector and variance matrix for the given data points, we need to find the mean and variance for the rainfalls in Omarama and Twizel separately. The mean vector is (4, 8) and the variance matrix is [[13.67, 0], [0, 58.67]].

Step-by-step explanation:

To calculate the mean vector and variance matrix for the six data points z₁,…,z₆, we need to find the mean and variance for the rainfalls in Omarama and Twizel separately.

Mean vector:

1. Find the mean rainfall for Omarama by summing up the rainfalls for each day and dividing by the total number of days (6): (7 + 10 + 0 + 0 + 2 + 5) / 6 = 24 / 6 = 4.
2. Find the mean rainfall for Twizel using the same process: (19 + 18 + 0 + 1 + 5 + 5) / 6 = 48 / 6 = 8.

The mean vector for the six data points is (4, 8).

Variance matrix:

1. Find the variance for Omarama by calculating the sum of squared differences between each rainfall value and the mean rainfall, and dividing by the total number of days (6): ((7 - 4)² + (10 - 4)² + (0 - 4)² + (0 - 4)² + (2 - 4)² + (5 - 4)²) / 6 = (9 + 36 + 16 + 16 + 4 + 1) / 6 = 82 / 6 ≈ 13.67.
2. Find the variance for Twizel using the same process: ((19 - 8)² + (18 - 8)² + (0 - 8)² + (1 - 8)² + (5 - 8)² + (5 - 8)²) / 6 = (121 + 100 + 64 + 49 + 9 + 9) / 6 = 352 / 6 ≈ 58.67.

The variance matrix for the six data points is:

[[13.67, 0], [0, 58.67]].