Chapter 4, Problem 4.156LM

For each of the following examples, decide whether x is a binomial random variable and explain your decision:

1. a. A manufacturer of computer chips randomly selects 100 chips from each hour’s production in order to estimate the proportion defective. Let x represent the number of defectives in the 100 sampled chips.
2. b. Of five applicants for a job, two will be selected. Although all applicants appear to be equally qualified, only three have the ability to fulfill the expectations of the company. Suppose that the two selections are made at random from the five applicants and let x be the number of qualified applicants selected.
3. c. A software developer establishes a support hotline for customers to call in with questions regarding use of the software. Let x represent the number of calls received on the support hotline during a specified workday.
4. d. Florida is one of a minority of states with no state income tax. A poll of 1,000 registered voters is conducted to determine how many would favor a state income tax in light of the state’s current fiscal condition. Let x be the number in the sample who would favor the tax.

To determine

Whether X is a binomial random variable or not.

Answer to Problem 4.156LM

The random variable X is not a Binomial random variable.

Explanation of Solution

Given info:

A sample of computer chips of size 100 are randomly selected from each hour’s production to estimate the proportion defective. The X represent the number of defectives in the 100 sampled chips.

Justification:

Binomial Distribution:

A random variable X is said to follow Binomial distribution if X has only two outcomes “success” and “failure” and the mass function of X is,

$p\left(x\right)=\left(\begin{array}{l}n\\ x\end{array}\right)p^x{q}^{n-x},x=0,1,2,\mathrm{...},n$,

Where n is the number of trials and p is the probability of success, $q=1-p$.

The mean is $\mu =np$ and the standard deviation is $\sigma =\sqrt{(npq)}$.

Here, X is said to be binomial random variable.

Requirements of a binomial random variable:

• The experiment consists of n independent trials.
• There are only 2 possible outcomes for each trial, success and failure.
• The probability of success (S) remains the same for all trials.
• The trials are independent of each other.
• The random variable in the experiment has binomial distribution.

In the study, the following characteristic are observed:

• There are 100 trials, identical in nature. Thus, taking a sample of reasonable size $n=100$ from a very large population will result in trials being essentially identically.
• There are two possible outcomes, one is success, cheap is defective and another is failure, cheap is not defective.
• The number of chips produced in one hour is larger than 100, then the probability of defective chip is same on each trial and the trials are independent.
• If the number of chips produced in one hour is larger than 100, then the probability of defective chip is not same on each trial and the trials are not independent.

Thus, the distribution of X is not a binomial probability distribution.

To determine

Whether X is a binomial random variable or not.

Answer to Problem 4.156LM

The random variable X is not a Binomial random variable.

Explanation of Solution

Given info:

Among five applicants only two will be selected whereas all applicants appear to be equally qualified. Only three have the ability to fulfill the expectations. The two selections are made at random from the five applicants and the random variable X is the number of qualified applicants selected.

Justification:

Hypergeometric distribution:

A random variable x is said to follow a Hypergeometric distribution, if the probability distribution is

$p\left(x\right)=\frac{\left(\begin{array}{l}r\\ x\end{array}\right)\left(\begin{array}{l}N-r\\ n-x\end{array}\right)}{\left(\begin{array}{l}N\\ n\end{array}\right)}$

Where,

$\begin{array}{c}x=\text{Maximum }\left[0,n-\left(N-r\right)\right],\mathrm{....},\text{Minimum}\left(r,n\right)\\ =\text{ Number of "success", }S,\text{drawn in the }n\text{elements}\end{array}$

$N\text{ = Total number of elements}$.

$r\text{ = Number of }S\text{'}s\text{in the }N\text{elements}$.

$n=\text{ Number of elements drawn}$.

The mean of the distribution is $\mu =\frac{nr}{N}$ and the standard deviation is $\sigma =\sqrt{\frac{r\left(N-r\right)n\left(N-n\right)}{N^2\left(N-1\right)}}$.

Requirements for a hypergeometric random variable:

• The experiment consists of drawing n individuals without replacement from a set of N individuals, r of which are successes and $\left(N-r\right)$ of which are failures.
• The random variable X in the experiment is the number of successes in the experiment and has hypergeometric distribution.

In the study, the following characteristics are observed:

• The experiment consists of two trials and for each trial there are two outcomes; applicant qualified or not.
• The probability of selecting a qualified applicant on the first trial is 3 out of 5 and the probability of selecting a qualified applicant on the second trial is depends on the outcome in first trail. Thus, X is not a Binomial random variable.

Thus, the distribution of X is a hypergeometric probability distribution but not a Binomial random variable.

To determine

Whether X is a binomial random variable or not.

Answer to Problem 4.156LM

The random variable X is not a Binomial random variable.

Explanation of Solution

Given info:

The random variable X defines the number of calls received on the support hotline during a specified workday.

Justification:

Binomial Distribution:

A random variable X is said to follow Binomial distribution if X has only two outcomes “success” and “failure” and the mass function of X is,

$p\left(x\right)=\left(\begin{array}{l}n\\ x\end{array}\right)p^x{q}^{n-x},x=0,1,2,\mathrm{...},n$,

Where n is the number of trials and p is the probability of success, $q=1-p$.

The mean is $\mu =np$ and the standard deviation is $\sigma =\sqrt{(npq)}$.

Here, X is said to be binomial random variable.

Requirements of a binomial random variable:

• The experiment consists of n independent trials.
• There are only 2 possible outcomes for each trial, success and failure.
• The probability of success (S) remains the same for all trials.
• The trials are independent of each other.
• The random variable in the experiment has binomial distribution.

In the study, the following characteristic are observed:

• The random variable X is counting the number of calls received. The number of trials is not a specified number in this experiment, thus X is not a Binomial random variable.

Thus, the distribution of X is not a binomial probability distribution.

To determine

Whether X is a binomial random variable or not.

Answer to Problem 4.156LM

The random variable X is a Binomial random variable.

Explanation of Solution

Given info:

In Florida, a poll of 1,000 registered voters is conducted to determine the number of voters would favor a state income tax in light of the state’s current. The random variable X be the number of sample who would favor the tax.

Justification:

Binomial Distribution:

A random variable X is said to follow Binomial distribution if X has only two outcomes “success” and “failure” and the mass function of X is,

$p\left(x\right)=\left(\begin{array}{l}n\\ x\end{array}\right)p^x{q}^{n-x},x=0,1,2,\mathrm{...},n$,

Where n is the number of trials and p is the probability of success, $q=1-p$.

The mean is $\mu =np$ and the standard deviation is $\sigma =\sqrt{(npq)}$.

Here, X is said to be binomial random variable.

Requirements of a binomial random variable:

• The experiment consists of n independent trials.
• There are only 2 possible outcomes for each trial, success and failure.
• The probability of success (S) remains the same for all trials.
• The trials are independent of each other.
• The random variable in the experiment has binomial distribution.

In the study, the following characteristic are observed:

• There are 100 trials, identical in nature. Thus, taking a sample of reasonable size $n=1000$ from a very large population will result in trials being essentially identically.
• There are two possible outcomes, one is success, favor sate income tax and another is failure, not favor state income tax.
• As 1,000 is small compared to the number of registered voters in Florida, the probability of selecting voter in favor of the state income tax is the same from trail to trial.
• The trials are independent to each other,

Thus, the distribution of X is a binomial probability distribution.