Probability Question

Question

Assume one of your finals has 50 questions on it, and lucky for you, it's all T/F. Find the approximate probability of guessing right on at least 30 of the 50 questions. 

a. verify that you can use the normal approximation.

b. find the mean

c. find the SD

d. get the z-score

e. state the probability of guessing correctly on at least 30 of the 50 questions

Answer 1

A) This is the easiest step! You perform two calculations every time. It's like having a door with two keys:

n*p = 50 * (1/2) = 25

n*q = 50 * (1/2) = 25

Both are greater than 5, so you have permission to move along.

B) The mean? You already found it because:

μ = n*p = 25

C) Slightly more work, because:

σ = √(n*p*q) = √(50 * 1/2 * 1/2) = √12.5 ≈3.54

D) Okay, tread carefully here! You can't use the question you were actually asked! When approximating binomial with a normal, you must always use the Continuity Correction Factor which basically says, "Be wrong by .5!" Your original question was:

P( at least 30)

...which means...

P(X ≥ 30)

You can NOT use an equal in the new version and you must change your number by 0.5 . Since we wanted to touch 30 while increasing, we change our 30 to 29.5 so they meet in passing:

P(X > 29.5)

Now we can finally find the Z-score using the usual formula:

        X - μ         29.5 - 25    4.5

Z = --------  =  ----------- = ----- = 1.27

          σ                3.54       3.54

This means we were really asking for:

E)   P (Z > 1.27) = 1 - .8980 = 0.1020  or 10.2% chance of guessing correctly on at least 30 out of 50.