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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,

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

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