Question

At King Kong High School, the average weight of senior boys is 168 lbs. with a standard deviation of 7 lbs. The average weight for senior girls is 117 libs. with a standard deviation of 5 lbs. Suppose a new distribution is created by adding the boys' weights and the girls' weights together. Assuming that the weights for boys and girls are independent, which of the following is true?

I. The mean of this new distribution is 168+117.
II. The standard deviation of this new distribution is 5+7.
III. The variance of this new distribution is 5^2+7^2

Answer 1

A. I and III only
B. II only
C. I and II only
D. I, II, and III
E. I only

Answer 2

If random variables X and Y are each independent and normally distributed, then their sum is also normally distributed.
$$\Large \rm {
If ~ X \sim N(\mu_X, \sigma_X^2)\\
Y \sim N(\mu_Y, \sigma_Y^2)\\
Z=X+Y\\

then\\

Z \sim N(\mu_X + \mu_Y, \sigma_X^2 + \sigma_Y^2)
}
$$
This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations).

Answer 3

would it include I,II, and III then? because thats the answer i was thinking it was i just wanted to double check

Answer 4

The new mean is the sum of the two old means, and its new variance is the of the two old variances. Remember that variance means standard deviation squared. Therefore II is incorrect.
The standard deviation of the new random variable is \( \sqrt{5^2 + 7^2} \)