Math 141 Chapter 6 Homework
When a certain type of thumbtack is flipped, the probability of its landing tip up (U) is 0.71 and the probability of its landing tip down (D) is 0.29. Suppose you flip two such thumbtacks, one at a time. The probability distribution for the possible outcomes of these flips is shown below. a. Find the probability of getting 0 ups, 1 up, or 2 ups when flipping two thumbtacks. b. Make a probability distribution graph of this.
A. 0 Ups= 0.0841 1 Up= 0.4118 2 Ups= 0.5041 B. Graph B
A coin will be flipped three times, and the number of heads recorded. Explain why this is a binomial experiment. Check all four required conditions.
A. Determine whether there is a fixed number of trials, n. (There is a fixed number of trials, n. The coin will be flipped three times, which is a fixed number.) B. Determine whether the only two outcomes are success and failure. (The only two outcomes are success and failure. In this experiment, a heads is considered a success, and a tails is considered a failure.) C. Determine whether the probability of success, p, is the same at each trial. (The probability of success, p, is the same at each trial. The probability of flipping heads is always 0.5.) D. Determine whether the trials are independent. (The trials are independent. The result of any flip is independent of the result of the other flips.)
A married couple plans to have four children, and they are wondering how many boys they should expect to have. Assume none of the children will be twins or other multiple births. Also assume the probability that a child will be a boy is 0.50. Explain why this is a binomial experiment. Check all four required conditions.
A. Do the possible trial outcomes meet the binomial conditions? (Yes, there are two complementary outcomes, either a boy or a girl) B. Does the number of trials meet the binomial conditions? (Yes, there are 4 fixed trials, since there are 4 children.) C. Does the probability of success meet the binomial conditions? (Yes, the probability of having a boy is 0.50 for each child.) D. Do the trials meet the binomial conditions? (Yes, because it is assumed there are no twins, the gender of one child does not affect the gender of another.)
The three-year recidivism rate of parolees in a certain state is about 20%; that is, 20% of parolees end up back in prison within three years. Assume that whether one parolee returns to prison is independent of whether any of the others returns. Complete parts a and b below.
A. Find the probability that exactly 4 out of 20 parolees will end up back in prison within three years. (The probability that exactly 4 out of 20 parolees will end up back in prison is (0.218) B. Find the probability that 4 or fewer out of 20 parolees will end up back in prison within three years. (The probability that 4 or fewer out of 20 parolees will end up back in prison is (0.630)
Eric wants to go skiing tomorrow, but only if there are 7 inches or more of new snow. According to the weather report, any amount of new snow between 5 inches and 9 inches is equally likely. The probability density curve for tomorrow's new snow depth is shown. Find the probability that the new snow depth will be seven inches or more tomorrow. Copy the graph, shade the appropriate area, and calculate its numerical value to find the probability. The total area is 1.
A. Graph B B. 0.5
Brian wants to go skiing tomorrow, but not unless there is between 3 and 5 inches of new snow. According to the weather report, any amount of new snow between 2 inches and 7 inches is equally likely. The probability density curve for tomorrow's new snow depth is shown. Find the probability that the new snow depth will be within Brian's ideal range. Copy the graph, shade the appropriate area, and calculate its numerical value to find the probability. The total area is 1.
A. Graph C B. 0.4
Determine whether each of the following variables would best be modeled as continuous or discrete. a. The length of time to run a marathon b. The number of socks in a drawer c. The time spent in line for a rollercoaster d. The number of mint condition baseball cards in a collection
A. The random variable is continuous. B. The random variable is discrete. C. The random variable is continuous. D. The random variable is discrete.
Determine whether each of the following variables would best be modeled as continuous or discrete. a. The number of bald eagles in the country. b. The floor area of a house. c. The number of people with blood type Upper A in a random sample of 34 people. d. The number of hits to a website in a day. e. The number of points scored during a basketball game.
A. The random variable is discrete. B. The random variable is continuous C. The random variable is discrete. D. The random variable is discrete. E. The random variable is discrete.
Assume a standard Normal distribution. Draw a well-labeled Normal curve for each part. a. Find the z-score that gives a left area of 0.6579. b. Find the z-score that gives a left area of 0.1682.
A. z= 0.41 Graph A B. z= -0.96 Graph B
A city's mean minimum daily temperature in February is 21 degrees Upper F. Suppose the standard deviation of the minimum temperature in February is 6 degrees Upper F and that the distribution of minimum temperatures in February is approximately Normal. What percentage of days in February has minimum temperatures below freezing left parenthesis 32 degrees Upper F right parenthesis?
Approximately (97)% of the days in February have minimum temperatures below freezing.
Which of the following cannot be used to represent a discrete probability distribution? Tables Equations Graphs Areas under curves
Areas Under Curves
What does a probability distribution indicate?
Both A and B A. All the possible outcomes of a random experiment B. The probability of each outcome of a random experiment
What are numerical variables with outcomes that cannot be listed or counted because they occur over a range called?
Continuous
The Normal model N(270,9) describes the length of pregnancies, in days, for women who go into spontaneous labor. Which of these statements is asking for a measurement (is an inverse Normal question) and which is asking for a probability? A. If 60% of women have a pregnancy length below this number of days, what is the number of days? B. If we select, at random, a woman who goes into spontaneous labor, what is the probability that her pregnancy will be 279 days or fewer?
(Statement A) asks for a measurement(inverse) and (Statement B) asks for a probability
Toss a fair eight-sided die. The probability density function (pdf) in table form is given below. Make a graph of the pdf for the die.
Graph D
A study of a population showed that males' body temperatures are approximately Normally distributed with a mean of 98.2 degrees Upper F and a population standard deviation of 0.40degreesF. What body temperature does a male have if he is at the 95 th percentile? Draw a well-labeled sketch to support your answer.
He has a body temperature of ( 98.9) Graph C
According to an insurance company, 30% of drivers aged 60-65 fail the written driver's test. This is the lowest failure rate of any age group. If 500 people aged 60-65 take the exam, how many would you expect to pass? Give or take how many?
If 500 people aged 60-65 take the exam, you would expect (350) to pass, give or take (10)
What is the most widely used probability model for continuous numerical variables?
Normal Distribution
A magician has shaved an edge off one side of a six-sided die, and as a result, the die is no longer "fair." The figure shows a graph of the probability density function (pdf). Show the pdf in table format by listing all six possible outcomes and their probabilities.
Number of spots 1 2 3 4 5 6 ------------------ Probability 0.05, 0.225, 0.225, 0.225, 0.225, 0.05
Suppose that the probability that a randomly selected person who has recently married for the first time will be divorced within 3 years is 0.2, and that the probability that a randomly selected person who has recently married for the second time will be divorced within 3 years is 0.3. Take a random sample of 25 people married for the first time and 25 people married for the second time. The sample is chosen such that no one in the sample is married to anyone else in the sample. Why is the binomial model inappropriate for finding the probability that exactly 10 of the 50 people in the sample will be divorced within 3 years? List all of the binomial conditions that are not met. A. The trials are independent.Th B. The only two outcomes are success and failure. C. The probability of success, p, is the same at each trial. D. There are a fixed number of trials: n.
Select all that apply. The probability of success, p, is the same at each trial.
Suppose the heights of women at a college are approximately Normally distributed with a mean of 65 inches and a population standard deviation of 2.5 inches. What height is at the 20 th percentile? Include an appropriately labeled sketch of the Normal curve to support your answer.
The 20th percentile is (62.9) inches Graph A
What is another name for the expected value of a probability distribution?
The mean
Assume that adults have IQ scores that are normally distributed with a mean of mu equals 105 and a standard deviation sigma equals 20. Find the probability that a randomly selected adult has an IQ less than 137.
The probability that a randomly selected adult has an IQ less than 137 is (0.9452)
Assume that adults have IQ scores that are Normally distributed with a mean of mu equals 95 and a standard deviation sigma equals 15. Find the probability that a randomly selected adult has an IQ of 110 or above.
The probability that a randomly selected adult has an IQ of 110 or more is (0.159)
Scores on a particular standardized test are approximately Normally distributed with a mean of 21 and a standard deviation of 5, as shown in the figure to the right. What is the probability that a randomly selected person scores 29 or more?
The probability that a randomly selected person will score 29 or more is (0.0548)
According to a newspaper, 40 % of bicycles stolen in a certain country are recovered. Find the probability that, in a sample of 5 randomly selected cases of bicycles stolen in the country, exactly 4 out of 5 bikes are recovered.
The probability that exactly 4 out of 5 stolen bicycles are recovered is (0.77) used statcrunch
What is true of the shape of a binomial distribution?
The shape depends on both the number of trials, n, and the probability of success, p.
What are the values of the mean and the standard deviation for the standard Normal model?
The standard normal model has a mean of 0 and a standard deviation of 1.
What determines the exact shape of a Normal distribution?
The values of the mean and the standard deviation
The normal distribution is symmetric and _____.
Unimodal
IQ scores have a population mean of 100, a population standard deviation of 15, and are approximately Normally distributed. Use one of the StatCrunch outputs below to find the probability that a randomly selected person will have an IQ of 105 or above. State whether Figure (A) or Figure (B) is the correct representation of a person with an IQ of 105 or above.
Use Figure (A) to find the probability. The probability that a randomly selected person will have an IQ of 105 or above is (0.3694)
Assume critical reading scores for a standardized test are distributed as N(550, 50). Complete parts (a) through (d) below.
a. Find the test score at the 75th percentile. The 75th percentile is (584) b. Find the test score at the 25th percentile. The 25th percentile is (517) c. Find the interquartile range for test scores. The interquartile range is (67) d. Is the interquartile range larger or smaller than the standard deviation? The interquartile range is (larger) than the standard deviation. SD=50
Wechsler IQs are approximately Normally distributed with a mean of 100 and a standard deviation of 15. Do not use the Normal table or technology. You may want to label the figure with Empirical Rule probabilities to help you think about this question. Complete parts (a) through (f).
a. Graph B b. Roughly what percentage of people have IQs more than 100? (50%) c. Roughly what percentage of people have IQs between 100 and 115? (34%) d. Roughly what percentage of people have IQs below 55? (About 0%) e. Roughly what percentage of people have IQs between 70 and 130? (95%) f. Roughly what percentage of people have IQs above 130? (2.5%) g. Roughly what percentage of people have IQs above 145? (About 0%)
In a certain state, about 80 % of drivers who are arrested for driving while intoxicated (DWI) are convicted. Complete parts a through c below.
a. If 12 independently selected drivers were arrested for DWI, how many of them would you expect to be convicted? (10) b. What is the probability that exactly 10 out of 12 independently selected drivers are convicted? (0.283) c. What is the probability that 10vor fewer are convicted? (0.725)
For each situation, identify the sample size n, the probability of success p, and the number of successes x. Give the answer in the form b(n,p,x). Do not go on to find the probability. Assume the four conditions for a binomial experiment are satisfied. Complete parts (a) and (b) below.
a. In the 2008 presidential election, 56% of the voters voted for a certain candidate. What is the probability that 67 out of 100 independently chosen voters voted for this candidate? (The probability is b(n,p,x)=b(50,0.52,17) b. The manufacturer of a certain vehicle recovery system claims that the probability that a stolen vehicle using its product will be recovered is 91%. What is the probability that exactly 19 out of 25 independently stolen vehicles with this product will be recovered? (The probability is b(n,p,x)=b(25,0.91,19)
The Empirical Rule applies rough approximations to probabilties for any unimodal, symmetric distribution. But for the Normal distribution we can be more precise, as the figure shows. Use the figure and the fact that the Normal curve is symmetric to answer the questions. Do not use a Normal table or technology. Complete parts a through e.
a. Roughly what percentage of z-scores are between minus2 and 2? (95%) b. Roughly what percentage of z-scores are between minus3 and 3? (Almost All) c. Roughly what percentage of z-scores are between minus1 and 1? (68%) d. Roughly what percentage of z-scores are more than 0? (50%) e. Roughly what percentage of z-scores are between 1 and 2? (13.5)
A fair coin is flipped 50 times. Complete parts a through c below.
a. What is the expected number of heads? (25) b. Find the standard deviation for the number of heads. (4) c. How many heads should you expect, give or take how many? Give the range of the number of heads based on these numbers. (You should expect between (21) and (29))
In a large city, 65% of people pass the drivers' road test. Suppose that every day, 100 people independently take the test. Complete parts (a) through (d) below.
a. What is the number of people who are expected to pass? (Expected number is (65)) b. What is the standard deviation for the number expected to pass? (The standard deviation is (5) c. After a great many days, according to the Empirical Rule, on about 95% of these days, the number of people passing will be as low as _____ and as high as _____. (Hint: Find two standard deviations below and two standard deviations above the mean.) (After a great many days, according to the Empirical Rule, on about 95% of these days, the number of people passing will be as low as (55) and as high as (75) d. If you found that one day, 37 out of 100 passed the test, would you consider this to be a very low number? (Yes), because 37 is (more than 3 standard deviations) below the mean.
A certain state has the highest high school graduation rate of all states at 85%. a. In a random sample of 20 high school students from the state, what is the probability that 18 will graduate? b. In a random sample of 20 high school students from the state, what is the probability than 17 or fewer will graduate? c. What is the probability that at least 18 high school students in our sample of 20 will graduate?
a. What is the probability that 18 out of 20 high school students will graduate? (Probability that 18 students will graduate is (0.229) b. What is the probability that 17 or fewer out of 20 high school students will graduate? (Probability that 17 or fewer will graduate is (0.595) What is the probability that at least 18 out of 20 high school students will graduate? (Probability that 18 out of 20 will graduate is (0.405)
In a certain state, the recent average critical reading standardized test score was 537. Assume that the standard deviation is 100 and that standardized test scores are Normally distributed. Complete parts (a) and (b) below. Include a Normal curve for each part.
a. What percentage of standardized test takers scored 500 or less? Choose the Normal curve that represents the percentage of standardized test takers that scored 500 or less. (Graph D) b. The percentage of standardized test takers who scored 500 or less is (35.6%) c. What percentage of standardized test takers scored more than 537? Choose the Normal curve that represents the percentage of standardized test takers that scored more than 537. (Graph B) The percentage of standardized test takers who scored more than 537 is (50%)
Use technology to find the indicated area under the standard Normal curve. Include an appropriately labeled sketch of the Normal curve and shade the appropriate region. a. nbsp Find the area in a standard Normal curve to the left of 1.58. b. nbsp Find the area in a standard Normal curve to the right of 1.58. Remember that the total area under the curve is 1.
a. Which graph below shows the area in a standard Normal curve to the left of 1.58? (Graph B) b. The area in a standard Normal curve to the left of 1.58 is (0.9429) Used Statcrunch c. Which graph below shows the area in a standard Normal curve to the right of 1.58? (Graph D) d. The area in a standard Normal curve to the right of 1.58 is (0.0571) Used Statcrunch
The total area under a probability density curve _____.
equals 1, because it represents the probability that the outcome will be somewhere on the x-axis.
The Normal model is a good first-choice to model data if the data are suspected to be _____.
symmetric and unimodal.
In a standard Normal distribution, if the area to the left of a z-score is about 0.2000, what is the approximate z-score? Draw a sketch of the Normal curve, showing the area and z-score.
z=(-0.84) Graph A
Variables have outcomes that you can list or count.
Discrete
The binomial probability model is useful in many situations with variables of what kind?
Discrete-valued numerical variables
