Question

According to a certain golf​ association, the weight of the golf ball ball shall not be greater than 1.620 ounces​ (45.93 grams). The diameter of the ball shall not be less than 1.680 inches. The velocity of the ball shall not be greater than 250 feet per second. The golf association periodically checks the specifications of golf balls using random sampling.

Three dozen of each kind are​ sampled, and if more than three do not meet size or velocity​requirements, that kind of ball is removed from the golf​ association's approved list. Complete parts a and b.

a. What assumptions must be made in order to use the binomial probability distribution to calculate the probability that a particular kind of golf ball will be​ removed?

A.The experiment consists of n identical trials. The number of outcomes can vary. The probability of success can change. The trials are independent.

B.The experiment consists of n identical trials. There are only two possible outcomes on each trial. The probability of success remains the same from trial to trial. The trials are independent.

C.The experiment consists of n identical trials. There are only two possible outcomes on each trial. The probability of success can change from trial to trial. The trials are dependent.

What information must be known in order to use the binomial probability distribution to calculate the probability that a particular kind of golf ball will be​ removed?

A.The percentage of that kind of golf ball that meets all the requirements

B.The percentage of that kind of golf ball that meets the size requirements

C.The percentage of that kind of golf ball that fails to meet both size and velocity requirements

D.The percentage of that kind of golf ball that meets the velocity requirements

b. Suppose15​% of all balls produced by a particular manufacturer are less than 1.680 inches in​ diameter, and assume that the number of such​ balls, x, in a sample of three dozen balls can be adequately characterized by a binomial probability distribution. Find the mean of the binomial distribution.

μ=__________​(Round to the nearest tenth as​ needed.)           

Find the standard deviation of the binomial distribution.

σ=_________________​(Round to the nearest thousandth as​ needed.)                       

Answer 1

Solution

Back-up Theory

If an experiment is dichotomous, i.e., has only two possible outcomes, say success and failure, which are mutually exclusive and collectively exhaustive, n repetitions of the experiment are identical, outcomes are independent and probability of a success, say p, is constant, then the random variable X, that represents the number of successes of the experiment, is a binomial variable and it represented as: X ~ B(n, p) ………………………………………………………………….……..…….. (1)

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and p = probability of one success, then probability mass function (pmf) of X is given by p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2,. , n ..........(2)

Mean (average) of X = E(X) = µ = np……………………………………………........................................................…………..(3)

Variance of X = V(X) = σ² = np(1 – p)………………………………………......................................................….……………..(4)

Standatd Deviation of X = SD(X) = σ = √{ np(1 – p)} ……………….....................................................………………………...(5)

Now to work out the solution,

Part (a.1) Assumptions of Binomial distribution

Vide (1), answer option B Answer 1

Part (a.2) Information Required to apply Binomial distribution

Vide (2), answer option C Answer 2

Part (b)

Here n = 3 dozens = 12 x 3 = 36 and p = 0.15 (15%)

Vide (3), µ = 36 x 0.15

= 5.4 Answer 3

Vide (5), σ = √(36 x 0.15 x 0.85)

= 2.142 Answer 4

DONE