The average mass of a man in an office is 85 kg with standard
deviation 12 kg. The average mass of a woman in the office is
68 kg with standard deviation 8 kg. The empty lift has a mass of
500 kg. What is the expectation and standard deviation of the
total mass of the lift when 3 women and 4 men are inside?
To find the expectation and standard deviation of the total mass of the lift when 3 women and 4 men are inside, we need to calculate the total mass of the people in the lift and add it to the mass of the empty lift.
First, let's calculate the total mass of the women:
Average mass of a woman = 68 kg
Total mass of 3 women = 3 * 68 kg = 204 kg
Next, let's calculate the total mass of the men:
Average mass of a man = 85 kg
Total mass of 4 men = 4 * 85 kg = 340 kg
Now, let's calculate the total mass of the people in the lift:
Total mass of people in the lift = Total mass of women + Total mass of men
= 204 kg + 340 kg
= 544 kg
Finally, let's calculate the total mass of the lift:
Total mass of the lift = Total mass of people in the lift + Mass of the empty lift
= 544 kg + 500 kg
= 1044 kg
The expectation of the total mass of the lift when 3 women and 4 men are inside is 1044 kg.
To calculate the standard deviation, we need to consider the variances of the women's and men's masses.
Variance of the women's mass = (standard deviation of women's mass)^2
= 8^2
= 64 kg^2
Variance of the men's mass = (standard deviation of men's mass)^2
= 12^2
= 144 kg^2
Next, we need to multiply the variances by the number of women and men, respectively, and sum them up:
Total variance = (Number of women * Variance of women's mass) + (Number of men * Variance of men's mass)
= (3 * 64 kg^2) + (4 * 144 kg^2)
= 192 kg^2 + 576 kg^2
= 768 kg^2
Finally, the standard deviation is the square root of the total variance:
Standard deviation = sqrt(Total variance)
= sqrt(768 kg^2)
= 27.71 kg
Therefore, the standard deviation of the total mass of the lift when 3 women and 4 men are inside is 27.71 kg.
To find the expectation and standard deviation of the total mass of the lift when 3 women and 4 men are inside, we need to calculate the sum of the masses of the men, the women, and the empty lift.
First, let's calculate the sum of the masses of the men. Since there are 4 men and the average mass of a man is 85 kg, the total mass of the men is 4 * 85 kg = 340 kg.
Next, let's calculate the sum of the masses of the women. Since there are 3 women and the average mass of a woman is 68 kg, the total mass of the women is 3 * 68 kg = 204 kg.
The empty lift has a mass of 500 kg.
Now, let's calculate the total mass of the lift when 3 women and 4 men are inside. The total mass of the lift is the sum of the masses of the men, the women, and the empty lift. So, the total mass is 340 kg (men's mass) + 204 kg (women's mass) + 500 kg (empty lift's mass) = 1044 kg.
To find the expectation of the total mass, we simply take the average of all the possible outcomes. In this case, we only have one outcome, which is 1044 kg. So, the expectation of the total mass is 1044 kg.
To find the standard deviation, we need to calculate the standard deviations of the men's mass, women's mass, and empty lift's mass, and then add them up.
The standard deviation of the total mass is calculated as the square root of the sum of the variances of the men's mass, women's mass, and empty lift's mass. The variance is equal to the square of the standard deviation.
The standard deviation of the men's mass = standard deviation of 12 kg.
The standard deviation of the women's mass = standard deviation of 8 kg.
The standard deviation of the empty lift's mass = 0 (since it is a constant).
So, the variance of the men's mass = (standard deviation of 12 kg) ^ 2 = 144 kg^2
The variance of the women's mass = (standard deviation of 8 kg) ^ 2 = 64 kg^2
The variance of the empty lift's mass = 0
Now, let's calculate the variance of the total mass. The variance of the total mass is the sum of the variances of the men's mass, women's mass, and empty lift's mass. So, the variance is 144 kg^2 (men's variance) + 64 kg^2 (women's variance) + 0 kg^2 (empty lift's variance) = 208 kg^2.
Finally, the standard deviation of the total mass is the square root of the variance. So, the standard deviation is the square root of 208 kg^2 ≈ 14.4 kg.
Therefore, the expectation of the total mass of the lift when 3 women and 4 men are inside is 1044 kg, and the standard deviation is approximately 14.4 kg.