The text cites an example in which researcher carried out 77

The text cites an example in which researcher carried out 77 separate significance tests, of which two were significant at 5% level. Suppose that these tests are independent of each other. If all of the null hypothesis are true, each test has probability 0.05 being significant at the 5% level.

1. What is the distribution of the number X of tests that are significant?

2. Find the probability that two or more of the test are significant.

Solution

1. It is a binomial distribution, with n = 77, p = 0.05.

2.

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 =    77      
p = the probability of a success =    0.05      
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   1   ) =    0.097327426
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.902672574 [ANSWER]