Applied Statistics for Engineers and Scientists - Practice Problems 2 | STAT 541, Assignments of Statistics

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Practice Problems # 02

1. In a random pattern of eight bits used to test a microcircuit, each bit is equally likely to be 0 or 1. Assume the values of the bits are independent.

a. What is the probability that all eight bits are 1? b. What is the probability that exactly three of the bits are 1? c. What is the probability that at least six of the bits are 1? d. What is the probability that at least two of the bits are 1?

2. Of the bolts manufactured for a certain application, 90% meet the length requirements and can be used immediately, 6% are too long and can be used after being cut, and 4% are too short and must be scrapped.

a. Find the probability that a randomly selected bolt can be used (either immediately or after being cut). b. Find the probability that fewer than 9 out of a random sample of 10 bolts can be used (either immediately or after being cut).

3. The number of messages received by a computer bulletin board is a Poisson random variable with a mean rate of 8 messages per hour.

a. What is the probability that five messages are received in a given hour? b. What is the probability that more than 15 messages are received in a given hour? c. What is the probability that fewer that three messages are received in one-half hour?

4. Suppose the error involved in making a certain measurement is a continuous random variable X with pdf

( )

( )

0

x x f x otherwise

⎧⎪ − − ≤ ≤

= ⎨

⎪⎩

a. Compute P[ X > 0 ]. b. Compute P [ -1 < X < 1 ]. c. Compute E[X].

5. The fill volume of cans filled by a certain machine is normally distributed with mean 12.05 oz and standard deviation 0.03 oz.

a. What proportion of cans contain less than 12 oz? b. The process can be adjusted through calibration. To what value should the mean be set so that 99% of the cans will contain 12 oz or more? c. If the process mean remains at 12.05 oz, what must the standard deviation be so that 99% of the cans will contain 12 oz or more?