12. The proportion of students who own a cell phone on college campuses across the country has increased tremendously over the past few years. It is estimated that approximately 94% of students now own a cell phone. Twenty five students are to be selected at random from a large university. Assume that the proportion of students who own a cell phone at this university is the same as nationwide. Let X = the number of students in the sample of 15 who own a cell phone.

Question

12. The proportion of students who own a cell phone on college campuses across the country has increased tremendously over the past few years. It is estimated that approximately 94% of students now own a cell phone. Twenty five students are to be selected at random from a large university. Assume that the proportion of students who own a cell phone at this university is the same as nationwide. Let X = the number of students in the sample of 15 who own a cell phone.

QUESTION: What is the probability that if you choose 40 people exactly 38 will have cell phones?

Answer 1

"Let X = the number of students in the sample of 15 who own a cell phone. "

does not seem to be relevant to the question.

Situation:
1. Each outcome is a Bernoulli trial, i.e. success (owns a cell phone) or failure (does not own one)
2. The probability of success is constant throughout the trials (p=0.94, q=1-p=0.06)
3. Every trial has equal probability as the others.
4. Every trial is independent of the others
5. There is a fixed number of trials (n=40)

Under these conditions, the probability can be modelled using the binomial distribution.
Let i=number of successes (n-i=#of failures0, then
P(i)=C(n, i)pⁱq^n-i
and C(n, i) is the combination function, where
C(n,i)=n!/(i!(n-i)!)
Example:
for n=40, i=38, p=0.99, q=0.03
I get P(38)=0.2206

Answer 2

Sorry, for the example, p=0.97, q=0.03.

Answer 3

To find the probability that exactly 38 out of 40 people will have cell phones, we can use the binomial probability formula. The binomial probability formula is:

P(X = x) = C(n, x) * p^x * (1 - p)^(n - x)

Where:
P(X = x) = probability that exactly x successes will occur
C(n, x) = number of combinations (n choose x)
p = probability of success for each trial
n = number of trials

In this case, the probability of a student owning a cell phone is given as 94%, which can be written as 0.94. The number of trials is 40 (choosing 40 people), and the number of successes we want is 38.

Plugging the values into the formula:

P(X = 38) = C(40, 38) * 0.94^38 * (1 - 0.94)^(40 - 38)

To find the value of C(40,38), we can use the formula:

C(n, x) = n! / (x!(n - x)!)

Plugging the values:

C(40, 38) = 40! / (38!(40 - 38)!)

Simplifying further:

P(X = 38) = (40! / (38!(40 - 38)!) * 0.94^38 * (1 - 0.94)^(40 - 38)

Calculating this expression will give you the probability that exactly 38 out of 40 people will have cell phones.