1. A multiple-choice test has 30 questions each with five responses, one of whic
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Question
1. A multiple-choice test has 30 questions each with five responses, one of which is correct.
The lowest passing grade is, Find the probability of obtaining this grade by random guessing. Write your answer to seven decimal places.
2. A fair die is rolled 10 times. Compute the probability that a “one” appears exactly once.
3. If two dice are tossed six times, find the probability of obtaining a sum of 7 two or three times.
4. Determine the critical region and critical values for z that would be used to test the null hypothesis H o: = 25 vs. Ha : 25, at the level of significance = 0.10. Sketch a normal curve to display the results.
Explanation / Answer
1. (In the question, the lowest passing grade is missing. The probability cannot be calculated)
2. Let us first calculate the probability that one occurs only on the first roll.
This is equal to 1/6 * 5/6 * 5/6 ..... 5/6 = 59 / 610.
Similarly the probability that one occurs only on the second roll is 59 / 610.
This is true for all 10 rolls.
Thus the probability that one appears exactly * 59 / 610 = 0.323
3. A sum of 7 can be sum of 1 and 6, 2 and 5 or 3 and 4 in either order. They are 6 in total and the total number of combinations for a roll of two dice is 36.
Thus the probability of getting a sum of 7 in one roll = 6/36 = 1/6.
First considering 7 appearing two times, the two times can come in 6C2 ways.
The probability on each role is is (1/6)2 (5/6)4
=> Probaility of a sum of 7 appearing twice = 6C2 * (1/6)2 * (5/6)4 = 0.2
Next considering 7 appearing three times, the three times can come in 6C3 ways.
The probability on each role is is (1/6)3 (5/6)3
=> Probaility of a sum of 7 appearing twice = 6C3 * (1/6)3 * (5/6)3 = 0.054
Total probability = 0.2 + 0.054 = 0.254
4. Critical Values: z = +/-1.645
Critical Region: z < -1.645 or z > +1.645
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