Concept explainers
An educational consulting firm is trying to decide whether high school students who have never before used a hand-held calculator can solve a certain type of problem more easily with a calculator that uses reverse Polish logic or one that does not use this logic. A sample of 25 students is selected and allowed to practice on both calculators. Then each student is asked to work one problem on the reverse Polish calculator and a similar problem on the other. Let p = P(S), where S indicates that a student worked the problem more quickly using reverse Polish logic than without, and let X = number of S's.
- a. If p = .5, what is P(7 ≤ X ≤ 18)?
- b. If p = .8. what is P(7 ≤ X ≤ 18)?
- c. If the claim that p = .5 is to be rejected when either x ≤ 7 or x ≥ 18, what is the
probability of rejecting the claim when it is actually correct? - d. If the decision to reject the claim p = .5 is made as in part (c), what is the probability that the claim is not rejected when p = .6? When p = .8?
- e. What decision rule would you choose for rejecting the claim p = .5 if you wanted the probability in part (c) to be at most .01?
a.
Find the value of
Answer to Problem 102SE
The value of
Explanation of Solution
Given info:
An educational consulting firm wants to check whether the high school students who have never used a hand held calculator can solve a certain type of problem more easily with a calculator that uses reverse polish logic or one that does not use the logic. 25 students were selected as a sample. The both type of calculators was used by 25 students.
Calculation:
The value of
Let X be the number of S’s. p = 0.5.
Hence,
It is known that
Using this formula
Where,
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n = 25, choose x = 18, 6
- Then obtain the table value corresponding to p = 0.5
The value of
Hence,
Thus, the value of
b.
Find the value of
Answer to Problem 102SE
The value of
Explanation of Solution
Calculation:
The value of
Let X be the number of S’s. p = 0.8.
Hence,
Where,
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n = 25, choose x = 18, 6
- Then, obtain the table value corresponding to p = 0.8
The value of
Hence,
Thus, the value of
c.
Find the probability of rejecting the claim when it is actually correct.
Answer to Problem 102SE
The probability of rejecting the claim when it is actually correct is 0.044.
Explanation of Solution
Given info:
The claim is that p = 0.5 is to be rejected when either
Calculation:
The probability of rejecting the claim when it is actually correct:
Where,
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n= 25, choose x = 7, 17
- Then, obtain the table value corresponding to p = 0.5
The value of
Hence,
Thus, the probability of rejecting the claim when it is actually correct is 0.044.
d.
Find the probability that the claim in part c is not rejected for p = 0.6 and p = 0.8
Answer to Problem 102SE
The probability that the claim in part c is not rejected for p = 0.6 is 0.845.
The probability that the claim in part c is not rejected for p = 0.8 is 0.109.
Explanation of Solution
Calculation:
Here, the values of p = 0.6 and p = 0.8
The claim is that p = 0.6 is to be rejected when either
The rule for non-rejection is
The probability of not rejecting the claim when p = 0.6:
Where,
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n= 25, choose x = 7, 17
- Then, obtain the table value corresponding to p = 0.6
The value of
Hence,
Thus, the probability that the claim in part c is not rejected for p = 0.6 is 0.845.
The probability of not rejecting the claim when p = 0.8:
Where,
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n= 25, choose x = 7, 17
- Then, obtain the table value corresponding to p = 0.8
The value of
Hence,
Thus, the probability that the claim in part c is not rejected for p = 0.8 is 0.109.
e.
Find the decision rule for rejecting the claim if the probability in part c is at most 0.01.
Answer to Problem 102SE
The decision rule for rejecting the claim is
Explanation of Solution
Calculation:
Here, the probability in part c, is at most 0.01.
The condition is
For the range,
For the range,
Where,
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n= 25, choose x = 6, 18
- Then, obtain the table value corresponding to p = 0.5
The value of
Hence,
The value 0.014 is too large.
For the range,
Where,
Procedure for binomial distribution table value:
From the table A.1 of Cumulative Binomial probabilities,
- Locate n = 25
- Along with n= 25, choose x = 5, 19
- Then, obtain the table value corresponding to p = 0.5
The value of
Hence,
The value 0.004 is less than 0.01.
Hence, the decision rule for rejecting the claim if the probability in part c is at most 0.01 is
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