he national norm for a junior high chemistry achievement test is a score of 90, with ?x = 18. The research director of a school district wants to know whether the students in her district are substantially higher or lower than this national norm. She selects a random sample of 81 students who took this exam and finds a mean of 93.

a) State the null and alternative hypotheses.

b) Using the z table, test the null hypothesis at the .05 level of significance and state your conclusions.

c) Using the z table, determine the probability of obtaining a random sample from the national norm sample (with a mean of 90) that would be as extreme or more extreme (either above or below the national mean) than the obtained sample mean of 93. (Hint: This is really a Chap 11 type of problem).

3. with 40 degrees of freedom, what is the probability in random sampling of obtaining a t ? -2.021?

b) with 20 degrees of freedom, what is the probability in random sampling of obtaining a t ? +1.325?

c) with 500 degrees of freedom, what is the probability in random sampling of obtaining a t ? +1.648 or t ? -1.648? 4. Imagine that we

31. Suppose 100 people all toss a hat into a box and then proceed to randomly pick out

a hat. What is the expected number of people to get their own hat back.

Hint: express the number of people who get their own hat as a sum of random variables

whose expected value is easy to compute.

32. Suppose I play a gambling game where I either win or lose k dollars. Suppose further

that the chance of winning is p = .5.

I employ the following strategy to try to guarantee that I win some money.

I bet $1; if I lose, I double my bet to $2, if I lose I double my bet again. I continue until

I win. Eventually I'm sure to win a bet and net $1 (run through the first few rounds and

you'll see why this is the net).

If this really worked casinos would be out of business. Our goal in this problem is to

understand the flaw in the strategy.

(a) Let X be the amount of money bet on the last game (the one I win). X takes values

1, 2, 4, 8, . . . . Determine the probability mass function for X. That is, find p(2k), where k

is in {0, 1, 2, . . . }.

(b) Compute E(X).

(c) Use your answer in part (b) to explain why the strategy is a bad one.

[3] A chip manufacturing company manufactures chips independently. Each manufactured chip is either good or defective. Based on the company's data, with probability 0.18, one out of every two manufactured chips is defective. It is also known that the probability of a defective chip is smaller than 0.5. Find the probability that (a) [9 pts] more than two chips are defective out of 100 manufactured chips (b) [8 pts] the 34" chip and one other chip is defective out of the first 100 manufactured chips (c) [8 pts] 2 chips among the first 50 manufactured chips and I chip among the second 50 manufactured chips are defective3. A company producing electric relays has three manufacturing plants producing 50, 30, and 20 percent, respectively, of its product. Suppose that the probabilities that a relay manufactured by these plants is defective are 0.02, 0.05, and 0.01, respectively. (a) If a relay is selected at random from the output of the company, what is the probability that it is defective? (b) If a relay selected at random is found to be defective, what is the probability that it was manufactured by plant 2?[a) In a Galton-Watson branching process starting with one individual in generation zero, the offspring random variable X has a geometric distribution with mean 3: X ~ Go[p]. [ Find the value of the parameter p of the geometric distribution. [in) Calculate to four decimal places the probability that the process becomes extinct: By the third generation; [2) At the fourth generation. (iii) Calculate to four decimal places the probability that the process eventually becomes extinct. (b] Consider the branching process for which an individual has zero offspring with probability a, or 3 offspring with probability 1- a. The offspring random variable Y has the generating function n(s) = a - [1 - 0] $43. [i) For what values of a is extinction not certain? [ii) When a = 36, find the probability of extinction. (iin If there were four individuals in generation zero, what would be the probability that the process eventually becomes extinct?5) One of the manufacturing stages for a certain engine part involves drilling 3 different holes, A, B, and C. in a steel plate. The probability that the hole A is within specifications (i.e., its diameter is not more than 0.050 mm off the target value) is 0.9997, and the same number is 0.9995 and 0.9998 for holes B and C, respectively. S. a) If we can assume independence between the quality of drilling of the three holes, what is the probability that the manufacturing stage is completed successfully for a randomly selected engine part (i.e., all three holes are within specifications)? b) Suppose that 250 engine parts are to be manufactured during an 8-hour shift. If we can assume independence between manufacturing of parts, what is the probability that the manufacturing stage 23 in question is successful for all 250 parts? Think aboot likea series