1.  We're counting the number of reds, so we have a box with "1"s for reds and "0"s for others.  The question is what proportion of 1's and 0's are in the box.
a) The null hypothesis is that the wheel is balanced, so the box contains 18 "1"s and 20 "0"s.  Or you might state this in terms of the porporttion of 1's  (18/38) and 0's (20/38).  The alternative hypothesis if that the wheel favors reds, so the proportion of 1's is MORE than 18/38.

Note that the alternative hypothesis makes no reference to the OBSERVED proportion of reds.  In practice, the null and alternative are announced before we've done our experiment to collect the data.

b) Under the null hyp., we expect  6000*(18/38) = 2842.105 Reds.  The SE is sqrt(6000)*SD(Box) = Sqrt(6000)*(1-0)*sqrt((18/38)*(20/38)) =  38.67615.  We observed 2806, which is fewer than we expected.  The question is, is it so much fewer that we suspect foul play, or is this the usual sort of variation we expect due to chance?

The test statistic is z = ( obs - expected)/SE = (2806 - 2842.105)/38.67615 =  -0.933521.

c) The p-value is the probability of getting a test statistic as extreme or more extreme than this, which in this case means seeing a test statistic LESS THAN or equal to -.933521.  The closest value in the table in the book is .95, which gives a probability of (100-65.79)/2 = 17.105%

d) If we reject H0 and claim the roullette wheel is unfair, then there's a 17.105% chance we're wrong.  This is a large probability of making so serious a mistake, and so it would be reasonable to NOT Reject and conclude that there's insufficient evidence to suspect the wheel is unfair.  (Traditaionlly, we would reject H0 if the p-value were less than 5%.  17% is NOT less than 5%, so don't reject.)

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2. Because there are 5 different symbols that can be chosen, and each symbol is equally likely, the "psychic" has a 1/5 or 20% chance of getting it right just by guessing.  The null hypothesis, which is skeptical, assumes the psychic is just guessing, and therefore will be successful 20% of the time.  If s/he is psychic, the success rate will be greater than 20%.

a) Null Hyp:  The box has 1 "1" and 4 "0"s.  (Or has 20% "1"s and 80% "0"'s)
    Alt. Hyp:  The box has MORE Than 20% 1's and so Less than 80% 0's.

b) If the Null hyp is true, we would expect 150*(1/5) = 30 correct answers.  The SE on this count is sqrt(150)*(1-0)*sqrt((1/5)*(4/5)) = 4.898979.  We observed 32 correct, so the test statistic is:
(32 - 30)/4.898979 =  0.4082483

c) Teh pvalue is the probability of getting an even BIGGER number, the probability that we see something greater than or equal to .4082.  The closest the table comes to .4082 is .40, and so the probability is (100-31.08)/2 = 34.46 %.

d) This is a big probability, suggesting that this event (observing 32 successes when a person is just guessing) is not at all unusual.    Don't reject the null hypothesis.
 
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