1) Many random number generators allow users to specify the range of the random numbers to be produced. Suppose that you specify that the range is to be all numbers between 0 and4. Call the random number generatedY. Then the density curve of the random variableYhas constant height between 0 and4, and height 0 elsewhere.

(a) What is the height of the density curve between 0 and4? (Enter your answer to two decimal places.)

0.25 (I figured out this answer)

Draw a graph of the density curve.

(I already figured out the graph)

(b) Use your graph from (a) and the fact that probability is area under the curve to find

P (Y1.2). (Enter your answer to three decimal places.)

(c) FindP (0.3<Y<1.7). (Enter your answer to three decimal places.)

(d) FindP (Y0.5). (Enter your answer to three decimal places.)

2) Role-playing games like Dungeons & Dragons use many different types of dice. Suppose that asix-sided die has faces marked

1, 2, 3, 4, 5, 6.

A)The intelligence of a character is determined by rolling this die twice and adding 1 to the sum of the spots. The faces are equally likely, and the two rolls are independent. What is the average (mean) intelligence for such characters?

B)How spread out are their intelligences, as measured by the standard deviation of the distribution? (Round your answer to four decimal places.)

3) The business of selling insurance is based on probability and the law of large numbers. Consumers buy insurance because we all face risks that are unlikely but carry high cost. Think of a fire destroying your home. So we form a group to share the risk: we all pay a small amount, and the insurance policy pays a large amount to those few of us whose homes burn down. The insurance company sells many policies, so it can rely on the law of large numbers.

In fact, the insurance company sees that in the entire population of homeowners, the mean loss from fire is= $300 and the standard deviation of the loss is= $400.What are the mean and standard deviation of the average loss for7policies? (Losses on separate policies are independent. Round your standard deviation to two decimal places.)

_X=$

_X= $

What are the mean and standard deviation of the average loss for14policies? (Round your standard deviation to two decimal places.)

_X= $

_X= $

4) An ancient Korean drinking game involves a 14-sided die. The players roll the die in turn and must submit to whatever humiliation is written on the up-face: something like "Keep still when tickled on face." Six of the 14 faces are squares. Let's call them A, B, C, D, E, and F for short. The other eight faces are triangles, which we will call 1, 2, 3, 4, 5, 6, 7, and 8. Each of the squares is equally likely. Each of the triangles is also equally likely, but the triangle probability differs from the square probability. The probability of getting a square is0.65. Give the probability model for the 14 possible outcomes. (Round your answers to three decimal places.)

P(any one square)=

P(any one triangle)=

5) The College Board finds that the distribution of students' SAT scores depends on the level of education their parents have. Children of parents who did not finish high school have SAT math scoresXwith mean454and standard deviation109. ScoresYof children of parents with graduate degrees have mean553and standard deviation102. Perhaps we should standardize to a common scale for equity. Find numbersa,b,c, anddsuch thata+bXandc+dYboth have mean 500 and standard deviation 100. (Round your answers to two decimal places.)

a =

b=

c=

d=

6) Slot machines are now video games, with winning determined by electronic random number generators. In the old days, slot machines were like this: you pull the lever to spin three wheels; each wheel has25symbols, all equally likely to show when the wheel stops spinning; the three wheels are independent of each other. Suppose that the middle wheel has14bells among its25symbols, and the left and right wheels have 1 bell each.

(a) You win the jackpot if all three wheels show bells. What is the probability of winning the jackpot? (Round your answer to four decimal places.)

(b) What is the probability that the wheels stop with exactly 2 bells showing? (Round your answer to four decimal places.)

7) It is difficult to conduct sample surveys on sensitive issues because many people will not answer questions if the answers might embarrass them.Randomized responseis an effective way to guarantee anonymity while collecting information on topics such as student cheating or sexual behavior. Here is the idea. To ask a sample of students whether they have plagiarized a term paper while in college, have each student toss a coin in private. If the coin lands heads and they have not plagiarized, they are to answer "No." Otherwise they are to give "Yes" as their answer. Only the student knows whether the answer reflects the truth or just the coin toss, but the researchers can use a proper random sample with follow-up for nonresponse and other good sampling practices. Suppose that in fact the probability is0.4that a randomly chosen student has plagiarized a paper. Draw a tree diagram in which the first stage is tossing the coin and the second is the truth about plagiarism. The outcome at the end of each branch is the answer given to the randomized-response question.

_____ _____ ____ _____ 0.5

What is the probability of a "No" answer in the randomized-response poll?

If the probability of plagiarism were0.28, what would be the probability of a "No" response on the poll?

Now suppose that you get33%"No" answers in a randomized-response poll of a large sample of students at your college. What do you estimate to be the percent of the population who have plagiarized a paper?

% (Answer needs to be in a percent)

8) Sheila's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Sheila's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with=125mg/dland= 10 mg/dl.

(a) If a single glucose measurement is made, what is the probability that Sheila is diagnosed as having gestational diabetes? (Round your answer to four decimal places.)

(b) If measurements are made instead on2separate days and the mean result is compared with the criterion 140 mg/dl, what is the probability that Sheila is diagnosed as having gestational diabetes? (Round your answer to four decimal places.)

9) Here is a simple probability model for multiple-choice tests. Suppose that each student has probabilitypof correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higherpthan a weak student.) The correctness of answers to different questions are independent. Jodi is a good student for whomp=0.83.

(a) Use the Normal approximation to find the probability that Jodi scores76%or lower on a 100-question test. (Round your answer to four decimal places.)

(b) If the test contains 250 questions, what is the probability that Jodi will score76%or lower? (Use the normal approximation. Round your answer to four decimal places.)

(c) How many questions must the test contain in order to reduce the standard deviation of Jodi's proportion of correct answers to half its value for a 100-item test?

questions

(d) Laura is a weaker student for whomp=0.78. Does the answer you gave in (c) for standard deviation of Jodi's score apply to Laura's standard deviation also?

Yes, the smallerpfor Laura has no effect on the relationship between the number of questions and the standard deviation.No, the smallerpfor Laura alters the relationship between the number of questions and the standard deviation.

10) Does delaying oral practice hinder learning a foreign language? Researchers randomly assigned27beginning students of Russian to begin speaking practice immediately and another27to delay speaking for 4 weeks. At the end of the semester both groups took a standard test of comprehension of spoken Russian. Suppose that in the population of all beginning students, the test scores for early speaking vary according to theN(32,9) distribution and scores for delayed speaking have theN(30,5) distribution.

(a) What is the sampling distribution of the mean scorexin the early speaking group in many repetitions of the experiment? (Round your answers forsto two decimal places.)

Mean=

s=

What is the sampling distribution of the mean scoreyin the delayed speaking group?

Mean=

s=

(b) If the experiment were repeated many times, what would be the sampling distribution of the differencey-xbetween the mean scores in the two groups? (Round your answer forsto two decimal places.)

Mean=

s=

(c) What is the probability that the experiment will find (misleadingly) that the mean score for delayed speaking is at least as large as that for early speaking? (Round your answer to four decimal places.)

11) "What do you think is the ideal number of children for a family to have?" A Gallup Poll asked this question of 1016 randomly chosen adults. Almost half (49%) thought two children was ideal.We are supposing that the proportion of all adults who think that two children is ideal isp= 0.49.

What is the probability that a sample proportionpfalls between 0.46 and 0.52 (that is, within3 percentage points of the truep) if the sample is an SRS of sizen=250?(Round your answer to four decimal places.)

What is the probability that a sample proportionpfalls between 0.46 and 0.52 if the sample is an SRS of sizen= 5000?(Round your answer to four decimal places.)

Combine these results to make a general statement about the effect of larger samples in a sample survey. Highlight the appropriate answer.

A) Larger samples give a larger probability thatpwill be close to the true proportionp.

B) Larger samples have no effect on the probability thatpwill be close to the true proportionp.

C)Larger samples give a smaller probability thatpwill be close to the true proportionp.

12) The standard deviation of a sample proportionpgets smaller as the sample sizenincreases. If the population proportion isp=0.51, how large a sample is needed to reduce the standard deviation ofpto=0.004?(The689599.7rule then says that about 95% of all samples will havepwithin 0.01 of the truep. Round your answer to up to the next whole number.)

13) The number of flaws per square yard in a type of carpet material varies with mean1.8flaws per square yard and standard deviation1.1flaws per square yard. This population distribution cannot be normal, because a count takes only whole-number values. An inspector studies175square yards of the material, records the number of flaws found in each square yard, and calculatesx, the mean number of flaws per square yard inspected. Use the central limit theorem to find the approximate probability that the mean number of flaws exceeds1.9per square yard. (Round your answer to four decimal places.)

14) "Durable press" cotton fabrics are treated to improve their recovery from wrinkles after washing. Unfortunately, the treatment also reduces the strength of the fabric. The breaking strength of untreated fabric is normally distributed with mean52pounds and standard deviation2.3pounds. The same type of fabric after treatment has normally distributed breaking strength with mean22.8pounds and standard deviation2pounds. A clothing manufacturer tests3specimens of each fabric. All6strength measurements are independent. (Round your answers to four decimal places.)

(a) What is the probability that the mean breaking strength of the3untreated specimens exceeds 50 pounds?

(b) What is the probability that the mean breaking strength of the3untreated specimens is at least 25 pounds greater than the mean strength of the3treated specimens?

15) Typographic errors in a text are either nonword errors (as when "the" is typed as "teh") or word errors that result in a real but incorrect word. Spell-checking software will catch nonword errors but not word errors. Human proofreaders catch 70% of word errors. You ask a fellow student to proofread an essay in which you have deliberately made12-word errors. What is the smallest number of missesmwithP (Xm) no larger than 0.05? You might considermor more misses as evidence that a proofreader actually catches fewer than 70% of word errors.

A) (misses)

16) The unique colors of the cashmere sweaters your firm makes result from heating undyed yarn in a kettle with a dye liquor. The pH (acidity) of the liquor is critical for regulating dye uptake and hence the final color. There are5kettles, all of which receive dye liquor from a common source. Past data show that pH varies according to a Normal distribution with=4.23and=0.128.You use statistical process control to check the stability of the process. Twice each day, the pH of the liquor in each kettle is measured, giving a sample of size5. The mean pHxis compared with "control limits" given by the 99.7 part of the 689599.7 rule for normal distributions, namely_x3_x.

What are the numerical values of these control limits forx? (Round your answers to three decimal places.)

A) (smaller value)

B) (larger value)