Problem 69E

Do social robots walk or roll? Refer to the International Conference on Social Robotics (Vol. 6414, 2010) study of the trend in the design of social robots, Exercise 4.25 (p.196). The study found that of 106 social robots, 63 were built with legs only, 20 with wheels only, 8 with both legs and wheels, and 15 with neither legs nor wheels. Suppose you randomly select 10 of the 106 social robots and count the number, x, with neither legs nor wheels.

a. Demonstrate why the probability distribution for x should not be approximated by the binomial distribution.

b. Show that the properties of the hypergeometric probability distribution are satisfied for this experiment.

c. Find μ and σ for the probability distribution for x.

d. Calculate the probability that x = 2.

4.25 Do social robots walk or roll? Refer to the International Conference on Social Robotics (Vol. 6414, 2010) study of the trend in the design of social robots, Exercise 2.3 (p. 48). Recall that in a random sample of 106 social (or service) robots designed to entertain, educate, and care for human users, 63 were built with legs only, 20 with wheels only, 8 with both legs and wheels, and 15 with neither legs nor wheels. Assume the following: Of the 63 robots with legs only, 50 have two legs, 5 have three legs, and 8 have four legs; of the 8 robots with both legs and wheels, all 8 have two legs. Suppose one of the 106 social robots is randomly selected. Let x equal the number of legs on the robot.

a. List the possible values of x.

b. Find the probability distribution of x.

c. Find E(x) and give a practical interpretation of its value.

2.3 Do social robots walk or roll? According to the United Nations, social robots now outnumber industrial robots worldwide. A social (or service) robot is designed to entertain, educate, and care for human users. In a paper published by the International Conference on Social Robotics (Vol. 6414, 2010), design engineers investigated the trend in the design of social robots. Using a random sample of 106 social robots obtained through a Web search, the engineers found that 63 were built with legs only, 20 with wheels only, 8 with both legs and wheels, and 15 with neither legs nor wheels. This information is portrayed in the accompanying graph.

a. What type of graph is used to describe the data?

b. Indentify the variable measured for each of the 106 robot designs.

c. Use the graph to identify the social robot design that is currently used the most.

d. Compute class relative frequencies for the different categories shown in the graph.

e. Use the results from, part d to construct a Pareto diagram for the data.

Solution 69E

Step1 of 5:

Total number of robots = 106.

The number of robots built with legs = 63.

The number of robots built with wheels = 20.

The number of robots built with both legs and wheels = 8.

The number of robots built with neither legs nor wheels = 15.

Here our goal is:

a). We need to demonstrate the probability distribution for x should not be approximated by the binomial distribution.

b). We need to show that the properties of the hypergeometric probability distribution are satisfied for this experiment.

c). We need to find μ and σ for the probability distribution for x.

d). We need to calculate the probability that x = 2.

Step2 of 5:

a).

We know that the characteristics of a binomial random variable and it is given below:

1). Identical trials: Suppose there are ‘n’ identical trials, in that we are selecting 10 robots from 106. On the first trial, we are selecting 1 robot out of 106. And second trial we are selecting 1 robot out of 105. Similarly, On the 10th trial, we are selecting 1 robot out of 97. Therefore these trials are not identical.

2). Two possible outcomes: in a above given problem a selected robot either has no legs or wheels or it has some legs or wheels. Here the events are:

S = Robot has no legs or wheels and

F = Robot has either legs and/or wheels.

Hence, this condition holds.

3).

The probability that the Robot has no legs or wheels on first trial is:

P(S) =

The probability that the Robot has no legs or wheels on second trial is:

P(S) =

If a robot with neither legs nor wheels is not selected on the first trial, then P(S) on the second trial is:

Therefore, The value of P(S) is not constant from trial to trial. This condition does not holds.

4). Trials are independent: in a given problem a robot selected on one trial affects the type of robot selected on the next trial. Therefore, these trials are not independent and hence, This condition does not holds.

5). Necessary condition: let a random variable x it presents number of robots selected that do not have legs or wheels in 10 trials. Therefore, the necessary conditions for a binomial random variable are do not holds.

Step3 of 5:

b).

We know that the characteristics of a hypergeometric random variable and it is given below:

1). The experiment consists of randomly drawing ‘n’ elements without replacement from a set of N elements, in that ‘r’ of which are successes and ‘(N – r)’ of which are failures.

Here in a given problem there are a total of N = 106 robots, of which r = 15 have neither legs nor wheels and N - r = 106 – 15 = 95 have some legs and/or wheels. We are selecting n = 10 robots.

2). Let a random variable x it presents the number of successes in the draw of ‘n’ elements. In a given example, x = number of robots selected with no legs or wheels in 20 selections.

Step4 of 5:

c).

Let,

Therefore,

And,

= 1.0539

Therefore,

Step5 of 5:

d).

Let,

=

= 0.2801

Therefore, P(X = 2) = 0.2801.