Question

Dorothy, Blanche, and Rose were observed and it was found that they have a 35% chance of staying up late to eat cheesecake. Suppose you consider 15 randomly selected nights.

1. a) State the random variable.

2. b) Check if this is a binomial experiment, by running through the following questions:· What is a trial and how many are there?

3. Are the trials independent? Why? What is Success? What is a Failure?
4. What is the probability of success? What is the probability of failure?

5. Find the probability that exactly 5 nights, the gals stayed up late to eat cheesecake. Round to four decimal places.

6. d) Find the probability that at most 3 nights out of the 15, the gals stayed up to eat cheesecake. Round to three decimal places.

7. Find the probability that at least 4 nights out of the 15, the gals stayed up to eat cheesecake. Round to four decimal places.

8. f) Find the expected value, μ, and the standard deviation, σ. Round to four decimal places.

9. g) Find the usual range. Round to three decimal places.

10. h) Is it usual for the gals to stay up 6 out 15 nights to eat cheesecake

Answer 1

(a)

The random variable : The number of nights the gals staying up to eat cheesecake

(b)

(i)

What is a trial and how many are there?

The trial is selected nights

There are n = 12 trials

(ii)

Are the trials independent? Why?

The trials sre independent because eachnight has no effect on any other night.

(iii)

What is Success

the gals stay up the night to eat cheesecake

(iv)

What is a Failure?

the gals do not stay up the night to eat cheesecake

(v)

Find the probability that exactly 5 nights, the gals stayed up late to eat cheesecake. Round to four decimal places.

Binomial Distribution

n = 12

p = 0.35

q = 1 - p =0.65

So,

Answer is:

0.2039

(d)

P(X3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)            (1)

Subsrtituting (1) becomes:

P(X3) = P(X=0) + P(X=1) + P(X=2) + P(X=3) = 0.3467

So,

Answer is:

0.3467

(e)
P(X4) = 1 - P(X3) = 1 - 0.3467 = 0.6533

So,

Answer is:

0.6533

(f)

(i) = np = 12 X 0.35 = 4.2000

(ii)

(g)

So,

Answer is:

(0.8955, 7.5045)

(h)

Since 6 is included in the usual range (0.8955, 7.5045), we conclude that it is usual for the gals to stay up 6 out 15 nights to eat cheesecake