According to a Gallup poll, 60% of American adults prefer saving over spending. Let X = the number of American adults out of a random sample of 50 who prefer saving to spending.

a. What is the probability distribution for X?

b. Use your calculator to find the following probabilities:

i. the probability that 25 adults in the sample prefer saving over spending

ii. the probability that at most 20 adults prefer saving

iii. the probability that more than 30 adults prefer saving

c. Using the formulas, calculate the

(i) mean and

(ii) standard deviation of X.

The proportion of times an event occurs out of a large number of trials is the likelihood of the event.

According to a Gallup poll $60\%$of American adults prefer saving over spending.

Let

$X=thenumberofAmericanadultsoutofarandomsampleof50whoprefersavingtospending$

In statistics, binomial distribution for a discrete random variables with two possible outcomes. A random variable is said to have a binomial distribution if the probability mass function is:

$P(X=x)=\frac{n!}{(n-x)!x!}{p}^x{(1-p)}^{n-x}$

The probability distribution of binomial distribution has two parameters $n=numberoftrials$and $p=probabilityofsuccess$.The binomial distribution is of the form:

$X~B(n,p)$

In the given binomial experiment $50$American adults and the probability that the American adults prefer saving over spending is $0.6$

The probability distribution for X is

$X~B(50,0.6)$

(i)

The steps to be followed in the calculator are:

Press $2^{nd}$and then press VARS

Scroll down to binompdf (n, p, x)

where,

n is the number of trials,

p is probability of success and

x is the number of successes required.

role="math" localid="1648217590381" $binompdf=(50,0.6,25)=0.0405$

(ii)

The steps to be followed in the calculator are:

Press$2^{nd}$and then press VARS

Scroll down to binompdf (n, p, x)

where,

n is the number of trials,

p is probability of success and

x is the number of successes required.

$binompdf=(50,0.6,20)=0.0034$

(iii)

The steps to be followed in the calculator are:

Press $2^{nd}$and then press VARS

Scroll down to binompdf (n, p, x)

where,

n is the number of trials,

p is probability of success and

x is the number of successes required.

$binompdf=(50,0.6,30)=0.4465$

In statistics, binomial distribution for a discrete random variables with two possible outcomes. A random variable is said to have a binomial distribution if the probability mass function is:

$P(X=x)=\frac{n!}{(n-x)!x!}{p}^x{(1-p)}^{n-x}$

The probability distribution of binomial distribution has two parameters $n=numberoftrials$and $p=probabilityofsuccess.$.The binomial distribution is of the form:

$X~B(n,p)$

The mean and standard deviation of binomial distribution is given below:

$$\begin{gathered} Mean=np \\ SD=\sqrt{np(1-p)} \end{gathered}$$

where,$n=50,p=0.6$

Therefore,

$$\begin{gathered} mean=50\times 0.6 \\ mean=30 \end{gathered}$$

$$\begin{gathered} SD=\sqrt{50\times 0.6(1-0.6)} \\ SD=3.4641 \end{gathered}$$