4. A bag contains a large number of marbles, each of which is blue or red.
A random sample of 3 marbles is taken from the bag.
The random variable \(D\) represents the number of blue marbles taken minus the number of red marbles taken.
Given that 20\% of the marbles in the bag are blue,
- show that \(\mathrm { P } ( D = - 1 ) = 0.384\)
- find the sampling distribution of \(D\)
- write down the mode of \(D\)
Takashi claims that the true proportion of blue marbles is greater than 20\% and tests his claim by selecting a random sample of 12 marbles from the bag.
- Find the critical region for this test at the 10\% level of significance.
- State the actual significance level of this test.
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