Question

week 7 worksheet name:

1. determine in which of the following situations a binomial distribution can be applied. if so, find the mean and standard deviation of x. if not, state which of the conditions to satisfy the binomial distribution requirements has been violated. round to 2 decimals.

(a) linda is interested in toilet paper pulling preferences. she takes a simple random sample of 5 people and asks each whether they always pull from the top or not. the probability that a person pulls from the top is 0.53 and x = the number of people who pull from the top.
(b) i roll a fair, 6 - sided die until i get a two. x is the number of rolls it takes before i obtain a roll of two.
(c) you have a bag containing 4 red chips and 6 white chips and you draw 4 chips. let random variable y be the number of red chips drawn from the bag out of 4 draws without replacement.

Explanation:

Step1: Check binomial conditions for (a)

There are \(n = 5\) independent trials (asking 5 people), each trial has 2 outcomes (pull from top or not), and the probability \(p=0.53\) is constant for each trial. So, it is a binomial distribution.

Step2: Calculate the mean for (a)

The mean of a binomial distribution is \(\mu=np\). Substituting \(n = 5\) and \(p = 0.53\), we get \(\mu=5\times0.53 = 2.65\).

Step3: Calculate the standard - deviation for (a)

The standard deviation of a binomial distribution is \(\sigma=\sqrt{np(1 - p)}\). Substituting \(n = 5\) and \(p = 0.53\), we have \(1-p=1 - 0.53=0.47\), then \(\sigma=\sqrt{5\times0.53\times0.47}=\sqrt{1.2455}\approx1.11\).

Step4: Check binomial conditions for (b)

This is a geometric distribution, not a binomial distribution. In a binomial distribution, the number of trials \(n\) is fixed. Here, the number of trials is not fixed as we roll the die until we get a two. The condition of a fixed - number of trials is violated.

Step5: Check binomial conditions for (c)

Since the draws are made without replacement, the probability of drawing a red chip changes for each draw. In a binomial distribution, the probability of success \(p\) must be constant for all trials. So, the constant - probability condition is violated.

Answer:

(a) Binomial distribution. Mean \(\mu = 2.65\), Standard deviation \(\sigma\approx1.11\)
(b) Not a binomial distribution. Violated condition: Number of trials is not fixed.
(c) Not a binomial distribution. Violated condition: Probability of success is not constant.