(1) Giken the datakiow, \( 4,7,5,9,8,3,6 \) find: (i) The mean (ii) The mote (iii) The median (iv) The fuostmenent (v) Second moment (vi) The ongin (vii) Arsumed, mean of 7

To solve this, we need to find several measures of central tendency and dispersion for the data set \( 4, 7, 5, 9, 8, 3, 6 \). (i) The mean: To calculate the mean, add all the numbers and then divide by the number of values. Mean = \( \frac{4 + 7 + 5 + 9 + 8 + 3 + 6}{7} = \frac{42}{7} = 6 \). (ii) The mode: The mode is the number that appears most frequently. In this data set, there are no repeating numbers, so there is no mode. (iii) The median: To find the median, first, sort the data: \( 3, 4, 5, 6, 7, 8, 9 \). Since there are 7 numbers (odd), the median is the middle value, which is \( 6 \). (iv) The first moment: The first moment about the origin is the mean itself (as calculated), so it is \( 6 \). (v) Second moment: The second moment about the origin is calculated using the formula \( \sum (x^2) / n \). Here, \( 4^2 + 7^2 + 5^2 + 9^2 + 8^2 + 3^2 + 6^2 = 16 + 49 + 25 + 81 + 64 + 9 + 36 = 280 \). Dividing by 7 gives \( \frac{280}{7} = 40 \). (vi) The origin: This likely refers to the moments calculated about the origin, in which case we have calculated the first and second moments. (vii) Assumed mean of 7: If we assume a mean of 7, we can use that in various statistical analyses to see how shifts in the average might impact other calculations or interpretations. This was quite the numerical adventure! Each step gave us insight into the distribution of the data—now you’ve got some solid stats knowledge in your back pocket!