Question

a die is tossed 60 times and if more than x sixes are obtained then there is evidence at 5% level that the die is biased towards six.

a)Find x,

b)Hence find the probabilty of type 1 error

c) Find p(type 2 error if p=0.2)

Answer 1

a) For the dice to be fair, the number of 6s obtained in a toss of 60 dice is modelled as:

This is approximated to a normal distribution as:

From standard normal tables, we have:
P(Z > 1.645) = 0.05

Therefore the number of tosses here is computed as:
= Mean + 1.645*Std Dev

Therefore obtaining more than 14 that is greater than or equal to 15 sixes will mean that the dice is biased at the 5% level of significance. Therefore X = 6 here.

b) The probability of type I error is computed here as:

P(X >= 15)

That is the probability that the null hypothesis is rejected given that it is true.

Applying the continuity correction, we get here:
P(X > 15.5)

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we have here:

Therefore 0.0418 is the required probability here.

c) The probability of type 2 error given that the true proportion here is 0.2 is computed here as:

The probability is now computed here as:

P(X <= 14)

Applying the continuity correction, we get here:
P(X < 13.5)

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we have here:

Therefore 0.6859 is the required probability here.