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A random-number generator is supposed to produce a
Chapter 10, Problem 10.1.3(choose chapter or problem)
A random-number generator is supposed to produce a sequence of 0s and 1s with each value being equally likely to be a 0 or a 1 and with all values being independent. In an examination of the random-number generator, a sequence of 50,000 values is obtained of which 25,264 are 0s.
(a) Formulate a set of hypotheses to test whether there is any evidence that the random-number generator is producing 0s and 1s with unequal probabilities, and calculate the corresponding p-value.
(b) Compute a two-sided 99% confidence interval for the probability p that a value produced by the random-number generator is a 0.
(c) If a two-sided 99% confidence interval for this probability is required with a total length no larger than 0.005, how many additional values need to be investigated?
Questions & Answers
QUESTION:
A random-number generator is supposed to produce a sequence of 0s and 1s with each value being equally likely to be a 0 or a 1 and with all values being independent. In an examination of the random-number generator, a sequence of 50,000 values is obtained of which 25,264 are 0s.
(a) Formulate a set of hypotheses to test whether there is any evidence that the random-number generator is producing 0s and 1s with unequal probabilities, and calculate the corresponding p-value.
(b) Compute a two-sided 99% confidence interval for the probability p that a value produced by the random-number generator is a 0.
(c) If a two-sided 99% confidence interval for this probability is required with a total length no larger than 0.005, how many additional values need to be investigated?
ANSWER:Step 1 of 6
(a) Formulate a set of hypotheses to test whether there is any evidence that the random-number generator is producing 0s and 1s with unequal probabilities, and calculate the corresponding p-value.
Let p be the probability that the random number generator produces a zero,
Thus the null hypothesis stating that the random number generator produces equal probabilities of 1’s and 0’s is:
\(H_{0}=p=0.5\)
The hypothesis which states that the random number generator produces unequal probabilities of 1’s and 0’s is:
\(H_A:\ p\ne0.5\)
We know random variable x = 25264, and sample size n = 50,000