Question

The dataset "chickwts" you can access in R

Weight    Feed

1 179 horsebean

2 160 horsebean

3 136 horsebean

4 227 horsebean

5 217   horsebean

6 168   horsebean

Let X1, .... X6 be random variables for the 6 different feedtypes. Let

xj =∑ xij /n

and

s2 =∑ (xij -xj )^2 / n

be estimators for the mean and variance of the chickweights for each of the feedtypes. Find the values of thee estimators for each. Assume these random variables are independent. use your estimates for expected value and variance to assign constants c1..c6 such that

E(c1X1 + ...c6X6) = 300

Separately select constants such that, d1, ... d6

Var(d1X1 +...d6X6) = 2500

Are these constants the same? Could they ever be?

Answer 1

All R commands are shown in bold.

Determine the levels of feed.

levels(chickwts$feed)

[1] "casein" "horsebean" "linseed" "meatmeal" "soybean" "sunflower"

Mean and variance of the chickweights for each of the feedtypes are

> mean(chickwts$weight[chickwts$feed == "casein"])
[1] 323.5833333

> mean(chickwts$weight[chickwts$feed == "horsebean"])
[1] 160.2
> mean(chickwts$weight[chickwts$feed == "linseed"])
[1] 218.75
> mean(chickwts$weight[chickwts$feed == "meatmeal"])
[1] 276.9090909
> mean(chickwts$weight[chickwts$feed == "soybean"])
[1] 246.4285714
> mean(chickwts$weight[chickwts$feed == "sunflower"])
[1] 328.9166667

Variance of each feedtypes are

> var(chickwts$weight[chickwts$feed == "casein"])
[1] 4151.719697
> var(chickwts$weight[chickwts$feed == "horsebean"])
[1] 1491.955556
> var(chickwts$weight[chickwts$feed == "linseed"])
[1] 2728.568182
> var(chickwts$weight[chickwts$feed == "meatmeal"])
[1] 4212.090909
> var(chickwts$weight[chickwts$feed == "soybean"])
[1] 2929.956044
> var(chickwts$weight[chickwts$feed == "sunflower"])
[1] 2384.992424

We can assign any random values to c1..c5 and then calculate c6 such that E(c1X1 + ...c6X6) = 300

Let c1 = c2 = c3 = c4 = c5 = 0.2

then,

0.2 * (323.5833333 + 160.2 + 218.75 + 276.9090909 + 246.4285714) + c6 328.9166667 = 300

=> 245.1741991 + c6 328.9166667 = 300

=> c6 = (300 - 245.1741991) / 328.9166667 = 0.1667

Thus, c1 = c2 = c3 = c4 = c5 = 0.2 and c6 = 0.1667 makes

E(c1X1 + ...c6X6) = 300

Similarly, we can assign any random values to d1..d5 and then calculate d6 such that  Var(d1X1 +...d6X6) = 2500

Let d1 = d2 = d3 = d4 = d5 = 0.2

Var(d1X1 +...d6X6) = 2500

=> d1² Var(X1) + ... + d6² Var(X1) = 2500

then, 0.04 * (4151.719697 + 1491.955556 + 2728.568182 + 4212.090909 + 2929.956044) + d6 2384.992424 = 2500

=> 620.5716155 + d6 2384.992424 = 2500

=> c6 = (2500 - 620.5716155) / 2384.992424 = 0.788

Thus, d1 = d2 = d3 = d4 = d5 = 0.2 and d6 = 0.788 makes

Var(d1X1 +...d6X6) = 2500

No, these constants are not same.