I need help with the last three questions to this problem. 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 and 5. Call the random number generated Y. Then the density curve of the random variable Y has constant height between 0 and 5, and height 0 elsewhere. (a) What is the height of the density curve between 0 and 5? (Enter your answer to two decimal places.) 0.20 Draw a graph of the density curve. (b) Use your graph from (a) and the fact that probability is area under the curve to find P(Y 1.2). (Enter your answer to three decimal places.) (c) Find P(0.2 < Y < 1.6). (Enter your answer to three decimal places.) (d) Find P(Y 0.5). (Enter your answer to three decimal places.) I do not understand these questions: Role-playing games like Dungeons & Dragons use many different types of dice. Suppose that a eight-sided die has faces marked 1, 2, 3, 4, 5, 6, 7, 8. 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? How spread out are their intelligences, as measured by the standard deviation of the distribution? (Round your answer to four decimal places.) 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 scores Xwith mean 459 and standard deviation 101. Scores Y of children of parents with graduate degrees have mean 557 and standard deviation 105. Perhaps we should standardize to a common scale for equity. Find numbers a, b, c, and d such that a + bX and c + dY both have mean 500 and standard deviation 100. (Round your answers to two decimal places.) = a b c d = = = 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 has 20 symbols, all equally likely to show when the wheel stops spinning; the three wheels are independent of each other. Suppose that the middle wheel has 7 bells among its 20 symbols, and the left and right wheels have 1 bell each. What is the probability that the wheels stop with exactly 2 bells showing? (Round your answer to four decimal places.) It is difficult to conduct sample surveys on sensitive issues because many people will not answer questions if the answers might embarrass them. Randomized response is 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 is 0.3 that 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 were 0.24, what would be the probability of a "No" response on the poll? Now suppose that you get 32% "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? % 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 = 129 mg/dl and = 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 on 2 separate 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.) Here is a simple probability model for multiple-choice tests. Suppose that each student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher p than a weak student.) The correctness of answers to different questions are independent. Jodi is a good student for whom p = 0.78. (a) Use the Normal approximation to find the probability that Jodi scores 73% 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 score 73% 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? Does delaying oral practice hinder learning a foreign language? Researchers randomly assigned 22 beginning students of Russian to begin speaking practice immediately and another 22 to 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 the N(31, 8) distribution and scores for delayed speaking have the N(25, 8) distribution. (a) What is the sampling distribution of the mean score x in the early speaking group in many repetitions of the experiment? (Round your answers for s to two decimal places.) Mean s = = What is the sampling distribution of the mean score y in the delayed speaking group? Mean s = = (b) If the experiment were repeated many times, what would be the sampling distribution of the difference y - x between the mean scores in the two groups? (Round your answer for s to 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.) "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 is p = 0.49. What is the probability that a sample proportion pp falls between 0.46 and 0.52 (that is, within 3 percentage points of the true p) if the sample is an SRS of size n = 350? (Round your answer to four decimal places.) What is the probability that a sample proportion pp falls between 0.46 and 0.52 if the sample is an SRS of size n = 5000? (Round your answer to four decimal places.) he standard deviation of a sample proportion pp gets smaller as the sample size n increases. If the population proportion is p = 0.58, how large a sample is needed to reduce the standard deviation of pp to = 0.006? (The 689599.7 rule then says that about 95% of all samples will have pp within 0.01 of the true p. Round your answer to up to the next whole number.) The number of flaws per square yard in a type of carpet material varies with mean 1.6 flaws per square yard and standard deviation 1.2 flaws per square yard. This population distribution cannot be normal, because a count takes only whole-number values. An inspector studies 170 square yards of the material, records the number of flaws found in each square yard, and calculates x, 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 exceeds 1.7 per square yard. (Round your answer to four decimal places.) 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 mean 52 pounds and standard deviation 2.9 pounds. The same type of fabric after treatment has normally distributed breaking strength with mean 20.5pounds and standard deviation 1.7 pounds. A clothing manufacturer tests 4 specimens of each fabric. All 8 strength measurements are independent. (Round your answers to four decimal places.) (a) What is the probability that the mean breaking strength of the 4 untreated specimens exceeds 50 pounds? (b) What is the probability that the mean breaking strength of the 4 untreated specimens is at least 25 pounds greater than the mean strength of the 4 treated specimens? 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 are 7 kettles, all of which receive dye liquor from a common source. Past data show that pH varies according to a Normal distribution with = 4.28 and = 0.157. 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 size 7. The mean pH x is compared with "control limits" given by the 99.7 part of the 689599.7 rule for normal distributions, namely x 3x. What are the numerical values of these control limits for x? (Round your answers to three decimal places.) (smaller value) (larger value) First question: Here are the projected numbers (in thousands) of earned degrees in a certain country during one academic year, classified by level and by the sex of the degree recipient: Bachelor's Master's Professional Doctorate Female 935 404 52 28 Male 661 260 43 28 Use these data to answer the following questions. (a) What is the probability that a randomly chosen degree recipient is a man? (Round your answer to four decimal places.) (b) What is the conditional probability that the person chosen received a bachelor's degree, given that he is a man? (Round your answer to four decimal places.) (c) Use the multiplication rule to find the probability of choosing a male bachelor's degree recipient. Check your result by finding this probability directly from the table of counts. (Round your answer to four decimal places.) Second Question: Joe buys a ticket in the Tri-State Pick 3 lottery every day, always betting on 956. He will win something if the winning number contains 9, 5, and 6 in any order. Each day, Joe has probability 0.006 of winning, and he wins (or not) independently of other days because a new drawing is held each day. What is the probability that Joe's first winning ticket comes on the 27th day? (Round your answer to four decimal places.) Third Question: 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 = 126 mg/dl and = 10 mg/dl. What is the level L such that there is probability only 0.05 that the mean glucose level of 3 test results falls above L for Sheila's glucose level distribution? (Round your answer to one decimal place.) Fourth Question: It is a striking fact that the first digits of numbers in legitimate records often follow a distribution known as Benford's Law, shown below. First digit 1 2 3 4 5 6 7 8 9 Proportion 0.319 0.155 0.1 0.085 0.067 0.061 0.033 0.033 0.147 Fake records usually have fewer first digits 1, 2, and 3. What is the approximate probability, if Benford's Law holds, that among 1243 randomly chosen invoices there are no more than 682 in amounts with first digit 1, 2, or 3? (Round your answer to four decimal places.) Fifth Question: The weight of the eggs produced by a certain breed of hen is normally distributed with mean 66.2 grams (g) and standard deviation 4.4 g. If cartons of such eggs can be considered to be SRSs of size 12 from the population of all eggs, what is the probability that the weight of a carton falls between 775 g and 825 g? (Round your answer to four decimal places.) Sixth Question: The mailing list of an agency that markets scuba-diving trips to the Florida Keys contains 60% males and 40% females. The agency calls 25 people chosen at random from its list. (a) What is the probability that 19 of the 25 people are men? (Use the binomial probability formula. Round your answer to four decimal places.) (b) What is the probability that the first woman is reached on the 5th call? (That is, the first 5 calls give MMMMF. Round your answer to four decimal places.) Seventh Question: Although cities encourage carpooling to reduce traffic congestion, most vehicles carry only one person. For example, 72% of vehicles on the roads are occupied by just the driver. (Round your answers to four decimal places.) (a) If you choose 10 vehicles at random, what is the probability that more than half (that is, 6 or more) carry just one person? (b) If you choose 106 vehicles at random, what is the probability that more than half (that is, 54 or more) carry just one person? (Use the normal approximation.)