Question

According to a survey conducted by Deloitte in 2017, 0.4775 of U.S. smartphone owners have made an effort to limit their phone use in the past. In a sample of 72 randomly selected U.S. smartphone owners, what is the probability that no more than 30 will have attempted to limit their cell phone use in the past?

1) 0.1801
2) 0.0278
3) 0.05554) 0.1246
5) 0.8199

Answer 1

`P(X \leq 30)= P(X=0 + P(X=1) +....+ P(X=30)`

And it's easier to compute this in excel with the following formula:

"=BINOM.DIST(30,72,0.4775,TRUE)"

And we got:

`P(X \leq 30)= 0.1801`

So the correct option would be:

1) 0.1801

Explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Let X the random variable of interest, on this case we now that:

`X \sim Binom(n=72, p=0.4775)`

The probability mass function for the Binomial distribution is given as:

`P(X)=(nCx)(p)^x (1-p)^(n-x)`

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`

Solution to the problem

For this case we want to find the probability that no more than 30 will have attempted to limit their cell phone, so we want to find this:

`P(X \leq 30)`

Becuase no more than 30 means that the maximum value can be 30.

And we can find this probability like this:

`P(X \leq 30)= P(X=0 + P(X=1) +....+ P(X=30)`

And it's easier to compute this in excel with the following formula:

"=BINOM.DIST(30,72,0.4775,TRUE)"

And we got:

`P(X \leq 30)= 0.1801`

So the correct option would be:

1) 0.1801