A person wishes to show beyond reasonable doubt that he has

A person wishes to show beyond reasonable doubt that he has psychic powers. He takes the
test described in Trosset exercise 4.5.14, except he tries to identify 100 symbols, with a 20%chance of being right for each symbol if he does not have psychic powers. The result is thathe correctly identies 25 out of 100 symbols.

(a) Write down appropriate null and alternative hypotheses for this test.
(b) For someone without psychic powers, what is the probability of correctly identifying 25or more symbols out of 100? (Use the binomial.)
(c) Has the psychic demonstrated his powers beyond reasonable doubt?

Solution

a)

Ho: p <= 0.20
Ha: p > 0.20 [ANSWER]

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b)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    100      
p = the probability of a success =    0.2      
x = our critical value of successes =    25      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   24   ) =    0.868646783
          
Thus, the probability of at least   25   successes is  
          
P(at least   25   ) =    0.131353217 [ANSWER]

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c)

No. The probability in b) is still high, so it may just have happened by chance. It is not unusual.