Question

I am convinced that my friend is cheating in our board game. I think that their die is not a normal six sided die because it rolls six far too often. Let ? be the proportion of rolls that are sixes. I plan on recording their result on their next 80 turns. (I will keep track of the number of sixes.) Draw the sampling distribution.

Answer 1

We just need here some basic concepts from discrete probability distribution. The values on the die are 1, 2,3,4,5,6. If the die is unbiased then let x be the random variable that takes those 6 values will follow uniform distribution. Ie. Each number has probability of 1/6 of occurrence ie. Out of 80 trials, a number will occur near about 13. 33 times. (approx 13

But here it is unbiased ie. Die gives the 6 number too often, ie. It has probability of occurrence greater than 1/6 ie. Out of 80 trials it may occur more than 13 (or 14 you can take) times.

Ie. Here the proportion p is

p =(number of times 6 occurred) /80

We can writ this p as probability in sampling distribution also since 0 <p<1

And other number will occur equally likely with probability

(1-p) /5

Hence p will be greater than (1-p)/5

And p + 5*(1-p)/5 =1 (5 is multi0lied because 5 values have same probability or proportion)

Hence we can draw sampling distribution as follows :