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.