Question

Assume the reaction time for participant taking a placebo follows a normal distribution with a mean of 1.2 seconds and variance of 0.25 seconds.

What is the probability that a randomly selected participant’s reaction time is longer than 2.2 second if he/she took the placebo?

What is the probability that the same participant’s reaction time is between 0.8 seconds and 1.7 seconds?

What is the probability of that same participant’s reaction time being exactly 1.2 second?

What is the probability of that same participant’s reaction time be less than 1 second?

We would like to further study those participants who had a reaction time less than 0.5 seconds. Therefore, across a sample of 25 participants, we categorized (yes/no) those participants who had a reaction time less than 0.5. It is thought that, in general, 25% have a reaction time less than 0.5 seconds.

What is the probability that at least 7 participants have reaction times less than 0.5 seconds?

What is the probability at most 4 participants have reaction times less than 0.5 seconds?

What is the probability that exactly 5 or exactly 6 people have reaction times less than 0.5 seconds?

If we have a sample of 25 people and 25% have the disease, how many people do we expect to have the disease, round to the nearest whole ‘person’?

Answer 1

mean = 1.2 , s = sqrt(0.25) = 0.5

a)

P(x > 2.2)

z = (x -mean)/s
= ( 2.2 -1.2)/0.5
= 2

P(x > 2.2) = P(z >2) = 0.0228

b)
P(0.8 < x < 1.7)
= P((0.8 - 1.2)/0.5 < z < ( 1.7 - 1.2)/0.5)
= P(-0.8 < z < 1)
= 0.6295

c)
P(x=1.2)

d)
P(x<1)

z = (x -mean)/s
= ( 1 -1.2)/0.5
= -0.4
P(x<1)= P(z <-0.4) = 0.3446

As per binomial distribution,
P(X=r= nCr * p^r * (1-p)^(n-r)

n = 25 , p = 0.25

a)
P(x > =7) = 1 - P(x < =6)
= 1 - 0.5611
= 0.4389

b)

P(x < =4) = P(x =0) + P(x =1) + .... + P(x =4)
= 25C0 * 0.25^0 * 0.75^25 + 25C1 * 0.25^1* 0.75^24 + ...... + 25C4* 0.25^4 * 0.75^21

= 0.2137

c)
P(x = 5) or P(x =6) = P(x =6 ) - P(x =5)
= 25C6 * 0.25^6 * 0.75^19 - 25C5 * 0.25^5 * 0.75^20

= 0.1828 - 0.1645
= 0.0183

d)

n = 25 , p = 0.25

np = 0.25 * 25 = 6.25 = 6