Questions: Preform the following calculations on the sample below. Round to two decimals. (8,10,5,3,-2,-1,4,0,3,1) a) What is the sample size of the dataset? Ans: [Select] b) What is the mean of the dataset? Ans: [Select ] c) What is the median of the dataset? Ans: [Select] d) What is the variance of the dataset? Ans: [Select] e) What is the mode of the dataset? Ans: [Select] f) Find the 10th percentile of the dataset. Ans: g) Find the 75th percentile of the dataset. Ans: [Select]

Solution Steps

The sample size \( N \) of the dataset is calculated as follows: \[ N = 10 \]

The mean \( \mu \) of the dataset is calculated using the formula: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{31}{10} = 3.1 \]

To find the median, we first sort the data: \[ \text{Sorted data} = [-2, -1, 0, 1, 3, 3, 4, 5, 8, 10] \] The rank for the median is calculated as: \[ \text{Rank} = Q \times (N + 1) = 0.5 \times (10 + 1) = 5.5 \] Since the rank is not an integer, we average the values at positions 5 and 6: \[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{3 + 3}{2} = 3.0 \]

The variance \( \sigma^2 \) is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = 14.77 \]

The rank for the 10th percentile is calculated as: \[ \text{Rank} = Q \times (N + 1) = 0.1 \times (10 + 1) = 1.1 \] We average the values at positions 1 and 2: \[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{-2 + -1}{2} = -1.5 \]

The rank for the 75th percentile is calculated as: \[ \text{Rank} = Q \times (N + 1) = 0.75 \times (10 + 1) = 8.25 \] We average the values at positions 8 and 9: \[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{5 + 8}{2} = 6.5 \]

Final Answer