Question

Suppose the US plans to send "rovers" to Mars. Rovers are mobile robots, programmed to collect rock samples and bring them back to a return rocket at the landing site. The weight of rocks collected by rover i is Xi. Weights collected by rovers are approximately normally distributed, with mean u and standard deviation 5 pounds. Weights collected by different rovers are independent.

Hint: you are not given the value of u in the first questions of this set. you do not need it. also, the questions of this set do not depend on each other.

a) What is the probability Xi will be within 3 pounds of the mean. Four decimals

Hint: within 3 pounds of u means from3 below u to 3 above u. Write down a probability statement and then standardize

b) Calculate the length of the interva (a,b) such that the probability is 0.99 that Xi will be in that interval. Specifically, calculate the length of the interval that is symmetric around the mean u

Hint: you do not have enough information to calculate values of a and b. But you do have enough information to calculate b-a. write down conditions for the values of a and b (but don't try to solve for a and b). then take b-a, and you will get a numerical value. three decimals

c)X-bar is the mean weight of rocks collected by n rovers (the average of the rovers' rock samples in pounds). How many rovers would the US have to send in order for the probability to be at least 0.99 that X-bar is within 3 pounds of u? Integers

Hint: Now X-bar is the random variable you're interested in, not Xi.

d) The rocket that will return the rovers' rock samples to earth cannot lift off if the total weight of the samples exceeds 530 pounds. Suppose the US sends 10 rovers that are programmed with u=50 and standard deviation 5 pounds. What is the probability that the rocket will not be able to lift off? Four decimals.

Answer 1

a) Define the standard random variable Z as ______(1)

Then required probability is

b) WE know from normal table that ____(2)

WE have to find a and b such that

From (1) and (2) we get

Subtract the two gives

------------------------(3)

c) For the sample mean standard random variable Z is defined as n being sample size

Using b) part equation (3) we have

   As b-a=3-(-3)=6

d) The sum of 10 rovers weights is and standard deviation is

The using normal table we have the required probability as