Final answer:
The mean of the original set of observations is 1137 and the variance is 322. This was determined by reversing the coding operation used on the observations and applying the appropriate transformations to the mean and variance of the coded data, X.
Step-by-step explanation:
To find the mean and variance of the original set of observations based on the coded data, we need to reverse the coding operation. The coding operation is given by X = (x - 1055) / 10. The mean of X is 8.2, which we can use to find the original mean, and the variance of X is 3.22 which will help us find the original variance.
To find the original mean (x-bar), we use the formula derived from the coding operation:
Substitute the given mean of X (8.2) into this equation to find x-bar:
- x-bar = 10 * 8.2 + 1055
- x-bar = 82 + 1055
- x-bar = 1137
The original variance (Var(x)) is found by squaring the multiplier used in the coding operation (since variance is based on squared differences). The coding operation divides by 10, so when reversing this, we multiply the coded variance by 102 (or 100).
- Var(x) = Var(X) * 102
- Var(x) = 3.22 * 100
- Var(x) = 322
The mean of the original observations is 1137 and the variance is 322.