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When we test a simple null hypothesis against a composite alternative, a critical region is said to be unbiased if the corresponding power function takes on its minimum value at the value of the parameter assumed under the null hypothesis. In other words, a critical region is unbiased if the probability of rejecting the null hypothesis is least when the null hypothesis is true. Given a single observation of the random variable X having the density
where -1 θ 1, show that the critical region x α provides an unbiased critical region of size α for testing the null hypothesis θ = 0 against the alternative hypothesis θ 0.
the scores on a standardized test had mean of 2.711 and a standard deviation of 1.335. a teacher had 61 students take
A random variable is normally distributed, with (population mean)= 300 pounds and (standard deviation) = 50 pounds. How small can a simple random sample be if the standard error of the mean can be no more than 20 pounds?
Explain how cards can be used to generate random numbers. Explain how a pair of dice can be used to generate random numbers.
Repeat the analysis for the new alternative being considered by the manager.
question the manufacturer of a steel-belted radial tire claims that the mean mileage the tire can be driven before the
The probability density of a one-parameter Weibull distribution is given by r2αxe-αx2 , x > 0 f(x) = 0, otherwise. (a) Using a random sample of size n, obtain a moment estimator for α.
Kyle averages 7 finished accounts a day. His standard deviation is 2 account per day and we wish the probability that Kyle will finish day with more than 8 finished accounts. Probability will be?
suppose the conditional distribution of y given x x is gamma with ? 4 and ? 1x and the marginal distribution of x is
A fair die is rolled until win or loss occurs: For k=1 to 5: if a 6 occurs on the kth roll you win N dollars and game ends; if a number
suppose that the distance from one end of a pole and the other be modeled by a normal distribution with mean 47.5 and
an experiment involves selecting a random sample of 256 middle managers at a random for study. one item of interest is
roger and harry play the following game each of them rolls a fair die. their rolls are independent. roger wins if the
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