A manufacturer of 40-amp fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customerswill complain because the fuses require replacement too often. If higher, the manufacturer might be liable for damage. To verify the amperage of the fuses, a sample offuses is to be selected and inspected. If a hypothesis test were to be performed on the resulting data, what null and alternative hypothesis would be of interest tothe manufacturer? Describe type I and type II errors in the context of this problem situation.

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An article reported that, in a study of a particular wafer inspection process, 356 dies were examined by an inspection probe and 210 of these passed the probe. Assuming a stable process, calculate a 95% (two-sided) confidence interval for the proportion of all dies that pass the probe. (Round your answers to three decimal places.

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Is the law of large numbers a phenomenon? If something is random, then how can we define an average outcome?

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What is the "law of large numbers" in insurance theory?

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Suppose the diagram of an electrical system is as given in Figure. What is the probability that the system works? Assume the components fail independently.

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Let X be a random variable with mean 11 and variance 9. Using Tchebysheff's theorem,find

(a) a lower bound for \( P(6 < X < 16) \)

(b) the value of C such that \( P(|X-11|\geq C)\leq 0.09 \)

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The number of customers per day at a sales counter, Y , has been observed for a long period of time and found to have mean 20 and standard deviation 2. The probability distribution of Y is not known. What can be said about the probability that, tomorrow, Y will be greater than 16 but less than 24?

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[Chebyshev inequality] For any random variable X and any \( a > 0 \), we have

\( P(|X-E(X)|\geq a)\leq\frac{V(X)}{a^2} \)

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[Markov inequality] If \( X \geq 0, i.e. X \) takes only nonnegative values, then for an \( a>0 \) we have \\( \hspace{3mm} P(X\geq a)\leq \frac{E(X)}{a} \)

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Could we use the special addition rule for determining the probability that for one draw from a deck of cards, that the card is either a Queen or a Heart? Why or why not?

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What is the difference between combinations and permutations? Give details

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In how many ways can you arrange 5 books in a bookshelf? Give details

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In how many ways can you rearrange the letters A, B, C, D, E?

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In an NBA (National Basketball Association) championship series, the team that wins four games out of seven is the winner. Suppose that teams A and B face each other in the championship games and that team A has probability 0.55 of winning a game over team B.

(a) What is the probability that team A will win the series in 5 games?

(b) What is the probability that team A will win the series?

(c) What is the probability that team B will win the series in 6 games?

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Suppose that p = P(male birth) = 0.3, A couple wishes to have exactly two female children in their family. They will have children until this condition is fulfilled.

(a) What is the probability that the family has x male children?

(b) What is the probability that the family has four children?

(c) What is the probability that the family has at most four children?

(d) How many male children would you expect this family to have? How many children would you expect this family to have?

In: Statistics and Probability