An experimenter suspects that a certain die is "loaded;" that is, the chances that the die lands on different faces are not all equal. Recall that dice are made with the sum of the numbers of spots on opposite sides equal to 7: 1 and 6 are opposite each other, 2 and 5 are opposite each other, and 3 and 4 are opposite each other.
The experimenter decides to test the null hypothesis that the die is fair against the alternative hypothesis that it is not fair, using the following test. The die will be rolled 40 times, independently. If the die lands with one spot showing 11 times or more, or 1 time or fewer, the null hypothesis will be rejected.
1. the standard error of the number of times the die lands showing one spot is _____.
2. The significance level of this test is ____.
3. The power of this test against the alternative hypothesis that the chance the die lands with one spot showing is 13.62%, the chance the die lands with six spots showing is 19.71%, and the chances the die lands with two, three, four, or five spots showing each equal 1/6, is _____.
4. The power of this test against the alternative hypothesis that the chance the die lands with two spots showing is 17.96%, the chance the die lands with five spots showing is 15.37%, and the chances the die lands with one, three, four, or six spots showing each equal 1/6, is ____.