Question

We are testing a new prescription drug that cures the common cold. One of the side effects of the drug is insomnia (can’t sleep). There is a 6% chance the patient will develop insomnia. We have selected 50 people and administered the drug to them and asked if they developed insomnia. Define the random variable x to be the number of people who develop insomnia.

Showing your work in Excel as demonstrated in class, what is the probability that x is:

Exactly 3?

Exactly 6?

Greater than or equal to 5?

Greater than 5?

Less than or equal to 7?

Less than 7?

Answer 1

Let ,

Random variable x be the number of people who develop insomnia.

Probabilit that the patient will develop insomnia = p =6%= 0.06

sample size = n = 50

1) Probability that x is exactly 3.

Using Excel function,

=BINOMDIST( x, n , p, 0 )  

This function gives exact probability. That is P( X = x )

Here we have to find P( x = 3 )

P( x = 3 ) = BINOMDIST( 3 , 50, 0.06, 0 ) = 0.2311

2 ) Probability that x is exactly 6

we have to find P( x = 6 )

Using Excel function,

P( x = 6 ) = BINOMDIST( 6 , 50, 0.06, 0 ) = 0.0487

3) Probability that x is greater than or equal to 5

We have to find P( x >= 5 )

P( x >= 5 ) = 1 - P( x < 5 ) = 1 - P( x <= 4 )

Using Excel function,  

=BINOMDIST( x, n, p, 1 )

This function gives probability less than or equal to x. That is P( X <= x )

So, P( x <= 4 ) = BINOMDIST( 4 , 50, 0.06, 1 ) = 0.820596

P( x >= 5 ) = 1 - 0.820596 = 0.1794

4) Probability that x is greater than 5

we have to find P( x > 5 )

P( x > 5 ) = 1 - P( x <= 5 )

P( x <= 5 ) = BINOMDIST( 5 , 50, 0.06, 1 ) = 0.922359

So, P( x > 5 ) = 1 - 0.922359 = 0.0776

5 ) Probability that x is less than or equal to 7

we have to find P( x <= 7 )

P( x <= 7 ) = BINOMDIST( 7 , 50, 0.06, 1 ) = 0.9906

6 ) Probability that x is less than 7

we have to find P( x < 7 )

P( x < 7 ) = P( x <= 6 )

P( x <= 6 ) = BINOMDIST( 6 , 50, 0.06, 1 ) = 0.9711