Question

An American roulette wheel has 38 slots and 18 of these are red. If the wheel is fair, then the probability of it landing on red is 18/38. At a certain roulette table, after 190 spins it landed red on 78 of the spins. This is less than the expected value. Could chance explain the difference, or is this sufficient evidence otherwise? Use a hypothesis test as follows.

(a) Compute the expected value of the number of "reds" after 190 spins, assuming the null hypothesis (that the wheel is fair).  
(b) Compute the SD of the number of "reds" after 190 spins, again assuming the null hypothesis. (You may round to two decimal places.)  
(c) Compute the z-statistic. (Round to two decimal places.)  
(d) What is the observed significance level? (Round to the nearest percentage. Use a Normal approximation; do not bother with a continuity correction.)  %
(e) Is the result statistically significant, highly significant, or neither?

A. Statistically significant but not highly significant.

B. Highly significant.

C. Neither

(f) Do we reject the null hypothesis at a 1% significance level?

A. No.

B. Yes.

Answer 1

(a) The expected value of the number of "reds" after 190 spins, assuming the null hypothesis (that the wheel is fair) is:

(b) The SD of the number of "reds" after 190 spins, again assuming the null hypothesis is:

(c) The z statistic is:

(d) The observed significance level is:

(e) Since observed significance level is slightly less than 0.05 or 5%, the result is Statistically significant but not highly significant.

(f) Since observed significance level is not less than 0.01 or 1%, we do not reject the null hypothesis at a 1% significance level.