Thirty students are asked to choose a random number between 0 and 9, inclusive, to create a data set of n = 30 digit.If the numbers are truly random, we would expect about 3 zeros, 3 ones, 3 twos, and so on. If the data set includes 8 sevens, how unusual is that? If we look exclusively at the number of sevens, we expect the proportion of sevens to be 0.1 (since there are 10 possible numbers) and the number of sevens to be 3 in a sample size of 30. We are testing H0: p = 0.1 vs Ha: 0.1 where p is the proportion of sevens. We can generate the randomization distribution by generating 1000 sets of 30 random digits and recording X = the number of sevens in each simulated sample. See the figure below.

(a)Notice that this randomition distribution is not symmetric. This is a two - tailed test, so we need to consider both "tails". How far is X =9 from the expected value of 3? What number would be equally far out on the other side?

(b)What is the p- for the observed statistic of X =8 sevens when doing the two-tail test?

(c)The randomization distribution in the above figure would apply to any digits (not just sevens) if the null hypothesis is H0: p =0.1. Suppose we want to test if students tend to avoid choosing zero when picking a random digit. If we now let p be the proportion of zeros all students choose, the alternative would be Ha: p < 0.1. What is the smallest p - value we could get using the randomization distribution given above?