Math 200 E3 Study Guide

Imagine that Alina's shower leaks and she calls the plumber. The plumber tells her that the problem is caused by either a leaky drain or a cracked shower pan, but never both. He also tells her that, in his experience, a leaky drain is the cause of the problem in about 90%90% of cases. If he discovers that the drain is the problem, he can fix it for $250$250. If he discovers that the shower pan is the problem, he must charge $2500$2500 instead. What is the expected value of the repair price? Please round your answer to the nearest whole dollar.

$475

A random variable 𝑥x has a Normal distribution with an unknown mean and a standard deviation of 12. Suppose that we take a random sample of size 𝑛=36n=36 and find a sample mean of 𝑥¯=98x¯=98 . What is a 95% confidence interval for the mean of 𝑥x ?

(94.08,101.92)

A government sample survey plans to measure the total cholesterol level of an SRS of men aged 20-3420-34 . Suppose that, in fact, the total cholesterol level of all men aged 20-3420-34 follows the Normal distribution with mean 𝜇=182μ=182 milligrams per deciliter (mg/dL) and standard deviation 𝜎=37σ=37 mg/dL. Use Table A to answer the questions, where necessary.

(a) Choose an SRS of 100100 men from this population. What is the sampling distribution of 𝑥¯x¯ ? (Use the units of mg/dL.) the 𝑁(182,3.7)N⁡(182,3.7) distribution What is the probability that 𝑥¯x¯ takes a value between 180180 and 184184 mg/dL? This is the probability that 𝑥¯x¯ estimates 𝜇μ within ±2±2 mg/dL. 0.4108 (b) Choose an SRS of 10001000 men from this population. Now what is the probability that 𝑥¯x¯ falls within ±2±2 mg/dL of 𝜇μ ? (Enter your answer rounded to three decimal places.) 0.912

A government sample survey plans to measure the total cholesterol level of an SRS of men aged 20-3420-34 . Suppose that, in fact, the total cholesterol level of all men aged 20-3420-34 follows the Normal distribution with mean 𝜇=182μ=182 milligrams per deciliter (mg/dL) and standard deviation 𝜎=37σ=37 mg/dL. Use Table A to answer the questions, where necessary.

(a) Choose an SRS of 100100 men from this population. What is the sampling distribution of 𝑥¯x¯ ? (Use the units of mg/dL.) the 𝑁(182,3.7)N⁡(182,3.7) distribution What is the probability that 𝑥¯x¯ takes a value between 180180 and 184184 mg/dL? This is the probability that 𝑥¯x¯ estimates 𝜇μ within ±2±2 mg/dL. 0.4108 (b) Choose an SRS of 10001000 men from this population. Now what is the probability that 𝑥¯x¯ falls within ±2±2 mg/dL of 𝜇μ ? (Enter your answer rounded to three decimal places.) 0.912

The 2015 American Time Use survey contains data on how many minutes of sleep per night each of 10,90010,900 survey participants estimated they get. The times follow the Normal distribution with mean 529.9529.9 minutes and standard deviation 135.6135.6 minutes. An SRS of 100100 of the participants has a mean time of 𝑥¯=514.4x¯=514.4 minutes. A second SRS of size 100100 has mean 𝑥¯=539.3x¯=539.3 minutes. After many SRSs, the values of the sample mean 𝑥¯x¯ follow the Normal distribution with mean 529.9529.9 minutes and standard deviation 13.5613.56 minutes.

(a) What is the population? The population is the 10,900 respondents to the American Time Use Survey. What values does the population distribution describe? What is this distribution? The population distribution describes the minutes of sleep per night for the individuals in this population. This distribution is Normal with mean 529.9 minutes and standard deviation 135.6 minutes. (b) What values does the sampling distribution of 𝑥¯x¯ describe? What is the sampling distribution? The sampling distribution describes the distribution of the average sleep time for 100100 randomly selected individuals from this population. This distribution is Normal with mean 529.9 minutes and standard deviation 13.56 minutes.

Typing 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%70% of word errors. You ask a fellow student to proofread an essay in which you have deliberately made 1010 word errors.

(a.) If the student matches the usual 70%70% rate, what is the distribution of the number of errors caught? binomial, with 𝑛=10n=10 and 𝑝=0.7 (b.) If the student matches the usual 70% rate, what is the distribution of the number of errors missed? binomial, with 𝑛=10n=10 and 𝑝=0.3

Typing 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%70% of word errors. You ask a fellow student to proofread an essay in which you have deliberately made 1010 word errors.

(b) Missing 33 or more out of 10 errors10 errors seems a poor performance. What is the probability that a proofreader who catches 70%70% of word errors misses exactly 33 out of 10? 0.2668 (c) What is the probability that a proofreader who catches 70%70% of word errors misses 33 or more out of 10?10? Use software. 0.6172

In each of the following situations, is it reasonable to use a binomial distribution for the random variable 𝑋?X? Give reasons for your answer in each case.

(b) The pool of potential jurors for a murder case contains 100100 persons chosen at random from the adult population of a large city. Each person in the pool is asked whether he or she opposes the death penalty; 𝑋X is the number who say "Yes." Yes, a binomial distribution is reasonable

In each of the following situations, is it reasonable to use a binomial distribution for the random variable 𝑋?X? Give reasons for your answer in each case.

(c) Joe buys a ticket in his state's Pick 33 lottery game every week; 𝑋X is the number of times in a year that he wins a prize. Yes, a binomial distribution is reasonable.

A sales representative makes visits to customers. Based on his history, the probability that he makes a sale on any visit is 0.15. It is reasonable to assume that customers' decisions are independent of one another. If the sales representative makes 10 visits in a day, what is the chance he makes at least five sales?

0.0099

There are five multiple choice questions on an exam, each having responses a, b, c, and d. Each question is worth 5 points, and only one option per question is correct. Suppose the student guesses the answer to each question, and these guesses, from question to question, are independent. If the student needs at least 20 points to pass the test, the probability that the student passes is closest to:

0.0156

The random variable 𝑋X denotes the time taken for a computer link to be made between the terminal in an executive's office and the computer at a remote factory site. 𝑋X is known to have a Normal distribution, with a mean of 15 seconds and a standard deviation of 3 seconds. 𝑃(𝑋>20)P(X>20) has a rounded value of:

0.048

Let 𝑋X be a binominal random variable with 𝑛=9n=9 and 𝑝=0.2p=0.2 . What is the probability of four successes; that is, 𝑃(𝑋=4)P(X=4) ?

0.066

The number of years of education of self‑employed individuals in the United States has a population mean of 13.6 years and a population standard deviation of 3 years. If we survey a random sample of 100 self‑employed people to determine the average number of years of education for the sample, what is the standard deviation of the sampling distribution of 𝑥¯x¯ , the sample mean?

0.3 years

Consider the following probability distribution for a random variable 𝑋:X: 𝑋X34567𝑃(𝑋)P(X)0.150.100.200.250.30

0.45

A fair coin is tossed 10 times. If 𝑋X is the number of times that heads is tossed, what is 𝑃(3<𝑋≤6)P(3<X≤6) ?

0.65625

As part of a promotion for a new type of cracker, free samples areoffered to shoppers in a local supermarket. The probability that a shopper will buy a packet of crackers after tasting the free sample is 0.2. Different shoppers can be regarded as independent trials. If 𝑋X is the number among the next 100 shoppers who buy a packet of crackers after tasting a free sample, then the probability that fewer than 30 buy a packet after tasting a free sample is approximately:

0.9938

Suppose that 𝑥x is a Normally distributed random variable with an unknown mean 𝜇μ and known standard deviation 6. If we take repeated samplesof size 100 and compute the sample means 𝑥¯x¯ , 95% of all of these values of 𝑥¯x¯ should lie within a distance of _ from 𝜇μ . (Use the 68‑95‑99.7 rule.)

1.2

There are five multiple choice questions on an exam, each having responses a, b, c,and d. Each question is worth 5 points, and only one option per question is correct. Suppose the student guesses the answer to each question, and these guesses, from question to question, are independent. The student's mean number of questions correct on the exam should be:

1.25

Suppose we have a loaded die that gives the outcomes 1 through 6 according to the following probability distribution. Die outcome123456Probability0.10.20.30.2?0.1 What is the probability of rolling a 5?

1/10

Using a standard Normal table or technology, find the critical value 𝑧∗z∗ for a 98% confidence level.

2.326

Imagine that a traffic intersection has a stop light that repeatedly cycles through the normal sequence of traffic signals (green light, yellow light, and red light). In each cycle the stop light is green for 30 s, yellow for 3 s, and red for 50 s. Assume that cars arrive at the intersection uniformly, which means that in any one interval of time, approximately the same number of cars arrive at the intersection at any other time interval of equal length. Determine the probability that a car arrives at the intersection while the stop light is yellow. Give your answer as a percentage precise to two decimal places.

3.61%

Students at University X must be in one of the following class ranks: freshman, sophomore, junior, or senior. At University X, 35% of the students are freshman and 30% are sophomores. If a student is selected at random, the probability that he or she is either a junior or a senior is:

35%

At a certain driver's license testing station,only 40% of all new drivers pass the behind‑the‑wheel test the first time they take it. A sample of 50 new drivers from a certain high school found that 36% of them had passed the test the first time. Which of these numbers is a statistic?

36%

A refrigerator contains 6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11 plums, and 2 mangos. Imagine you stick your hand into the refrigerator and pull out a piece of fruit at random. What is the chance you don't get an apple?

38/44

In a certain high school, 20% of the graduating seniors have chosen to attend The Ohio State University. If there are 265 seniors in the graduating class, the number who will go to The Ohio State University is a binominal random variable. What is the standard deviation of the number of students who will attend Ohio State?

6.51

A box at a miniature golf course contains contains 88 red golf balls, 77 green golf balls, and 66 yellow golf balls. What is the probability of taking out a golf ball and having it be a red or a yellow golf ball? Express your answer as a percentage and round it to two decimal places.

66.67

The level of nitrogen oxides (NOX) and nonmethane organic gas (NMOG) in the exhaust over the useful life (150,000150,000 miles of driving) of cars of a particular model varies Normally with mean 8585 mg/mi and standard deviation 66 mg/mi. A company has 1616 cars of this model in its fleet. Using Table A, find the level 𝐿L such that the probability that the average NOX + NMOG level 𝑥¯x¯ for the fleet greater than 𝐿L is only 0.010.01 ? (Enter your answer rounded to three decimal places. If you are using CrunchIt, adjust the default precision under Preferences as necessary. See the instructional video on how to adjust precision settings.)

88.489

Suppose that we compute a 90% 𝑧z confidence interval for an unknown population mean 𝜇μ . Which of the following is a correct interpretation?

90% of all possible 𝑧z confidence intervals computed from samples of the same size would contain 𝜇μ .

Which of the following would have a binomial distribution?

A fair coin is tossed 10 times. 𝑋X is the number of heads tossed in these 10 flips.

In each of the following situations, is it reasonable to use a binomial distribution for the random variable 𝑋?X? Give reasons for your answer in each case.

An auto manufacturer chooses one car from each hour's production for a detailed quality inspection. One variable recorded is the count 𝑋X finish defects (dimples, ripples, etc.) in the car's paint. Is it reasonable to use a binomial distribution for the random variable 𝑋? - A binomial distribution is not reasonable because trials are not independent and 𝑝p is likely not constant. - A binomial distribution is not reasonable because 𝑛n is not fixed. - A binomial distribution is not reasonable because there are more than two outcomes of interest.

A student is chosen at random from a statistics class. Which of the following events are disjoint?

Event 𝐴A is that the student is a junior. Event 𝐵B is that the student is a senior.

If we roll a single six‑sided die, the probability of rolling a 6 is 1/6.1/6. If we roll the die 60 times, how many times will we roll a 6?

It is impossible to determine from the information given.

The probability of event 𝐴A is 𝑃(𝐴)=0.3,P(A)=0.3, and the probability of event 𝐵B is 𝑃(𝐵)=0.25.P(B)=0.25. Are 𝐴A and 𝐵B disjoint?

It is impossible to determine from the information given.

Many young men in North America and Europe (but not in Asia) tend to think they need more muscle to be attractive. One study presented 200200 young American men with 100100 images of men with various levels of muscle. Researchers measure level of muscle in kilograms per square meter (𝑘𝑔/𝑚2)(k⁢g/m2) of fat‑free body mass. Typical young men have about 2020 𝑘𝑔/𝑚2k⁢g/m2 . Each subject chose two images, one that represented his own level of body muscle and one that he thought represented "what women prefer." The mean gap between self‑image and "what women prefer" was 2.35 𝑘𝑔/𝑚22.35 k⁢g/m2 . Suppose that the "muscle gap" in the population of all young men has a Normal distribution with standard deviation 2.5 𝑘𝑔/𝑚22.5 k⁢g/m2 . Give a 90%90% confidence interval for the mean amount of muscle young men think they should add to be attractive to women. (Enter your answers rounded to four decimal places.)

LL: 2.059 UL: 2.6408

According to the Center for Disease Control and Prevention (CDC), the mean life expectancy in 2015 for Hispanic females was 84.3 years.84.3 years. Assume that the standard deviation was 15 years,15 years, as suggested by the Bureau of Economic Research. The distribution of age at death, 𝑋,X, is not normal because it is skewed to the left. Nevertheless, the distribution of the mean, 𝑥⎯⎯⎯,x¯, in all possible samples of size 𝑛n is approximately normal if 𝑛n is large enough, by the central limit theorem. Let 𝑥⎯⎯⎯x¯ be the mean life expectancy in a sample of 100100 Hispanic females. Determine the interval centered at the population mean 𝜇μ such that 95%95% of sample means 𝑥⎯⎯⎯x¯ will fall in the interval. Give your answers precise to one decimal. You may need to use software or a table of 𝑧-z-critical values.

LL: 81.4 UL: 87.2

A set of four cards consists of two red cards and two black cards. The cards are shuffled thoroughly and I am dealt two cards. I found the number of red cards (𝑋)(X) in these two cards. The random variable 𝑋X has which of the given probability distributions?

Neither answer option is correct.

A student reads that a recent poll finds a 95%95% confidence interval for the mean ideal weight given by adult American women is 139±1.3139±1.3 pounds. Asked to explain the meaning of the confidence interval for mean ideal weight, the student answers: "We can be 95%95% confident that future samples of adult American women will say that their mean ideal weight is between 137.7137.7 and 140.3140.3 pounds." Is this explanation correct?

No. If we repeated the sample over and over, 95%95% of all future sample means would be within 1.961.96 standard deviations of 𝜇μ , the true, unknown value of the mean ideal weight for American women. Future samples will not depend on the results of a previous sample.

Asked what the central limit theorem says, a student replies, "As you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal." Is the student right? Explain your answer.

No. The central limit theorem says nothing about the histogram of the sample values. It deals only with the distribution of the sample means.

Boxes of 6-inch6-inch slate flooring tile contain 40 tiles40 tiles per box. The count 𝑋X is the number of cracked tiles in a box. You have noticed that most boxes contain no cracked tiles, but if there are cracked tiles in a box, then there are usually several. Does 𝑋X have a binomial distribution?

No. The trials are not independent. If one tile in a box is cracked, there are likely more tiles cracked.

The figure displays several possible finite probability models for rolling a die. We can learn which model is actually accurate for a particular die only by rolling the die many times. However, some of the models are not valid. That is, they do not obey the rules. Which are valid and which are not? Select the best answer, with the correct explanation of what is wrong in the case of the invalid models.

Only Model 2 is valid. Models 1,, 3,, and 4 have probabilities that do not sum to 1.1. Model 4 has some probabilities that are greater than 1.

Suppose Gabe, an elementary school student, has just finished dinner with his mother, Judy. Eyeing the nearby cookie jar, Gabe asks his mother if he can have a cookie for dessert. She tells Gabe that she needs to check his backpack to make sure that he finished his homework. Gabe cannot remember where he left his backpack, but he knows for sure that he did not complete his homework and will not be allowed to eat a cookie. Gabe believes his only option is to quickly steal a cookie while his mother is out of the room. Judy then leaves the room to look for Gabe's backpack. Assume that Judy could return at any time in the next 6060 seconds with equal probability. For the first 1010 seconds, Gabe sheepishly wonders if he will get caught trying to grab a nearby cookie. After waiting and not seeing his mother, Gabe decides that he needs a cookie and begins to take one from the jar. Assuming it takes Gabe 2020 seconds to grab a cookie from the jar and devour it without a trace, what is the probability that his mother returns in time to catch Gabe stealing a cookie? Please round your answer to the nearest 2.

P(gabe gets caught) 0.40

Statisticians prefer large samples. Select the correct explanation of the effect of increasing the size of a sample on the margin of error of a 95%95% confidence interval.

Regardless of the level of confidence (the 95%95% confidence level has nothing to do with it), larger samples reduce margins of error, which provides greater precision in estimating 𝜇μ .

Suppose that fish inhabiting the river in the town of Glenmeadow have a 30% rate of parasite incidence. Carl is a fisherman who wants to verify the parasite infection rate of Glenmeadow's fish. He catches 1000 fish at random and sequentially tests them for parasites. Assuming that the true parasite incidence rate is 30%, which of the following statements is the most accurate?

The more fish Carl tests, the more likely he is to find a 30% parasite incidence rate.

In 2015, over 30,00030,000 Americans died from opioid overdoses, and the number of inpatient stays and emergency visits to hospitals related to opiods increased dramatically compared to previous years. The National Institutes of Health (NIH) have partnered with the Healthcare Cost and Utilization Project (HCUP) to gather data on opioid-related deaths and hospitalizations. These data include information about inpatient stays, emergency department visits, age, sex, income, and opiod-related use. For a given patient admitted to a hopsital for an opioid-related event, identify which of these events are disjoint.

The patient's case was critical and they were admitted to the emergency room, The patient did not visit the emergency room.

Ramon is interested in whether the global rise in temperature is also showing up locally in his town, Centerdale. He plans to look up the average annual temperature for Centerdale for five recent randomly selected years. He wants to report the number of years whose temperature was higher than the previous year's temperature. What is the random variable in Ramon's study, and what are its possible values?

The random variable is the number of years in which the temperature increased from the previous year. Its possible values are {0,1,2,3,4,5}.

A researcher is planning to construct a one-sample 𝑧z‑confidence interval for a population mean 𝜇.μ. Select the statements that would lead to a smaller margin of error, assuming the other factors remain the same.

The researcher lowers the confidence level. The population standard deviation turns out to be lower than expected. The researcher increases the sample size.

Suppose that a manager is interested in estimating the average amount of money customers spend in her store. After sampling 36 transactions at random, she found that the average amount spent was $35.2535.25. She then computed a 9595% confidence interval to be between $31.8431.84 and $38.6638.66. Which statement gives a valid interpretation of the interval?

The store manager is 9595% confident that the average amount spent by all customers is between $31.84 and $38.66

In the following hypothetical scenarios, classify each of the specified numbers as a parameter or statistic.

There have been 44 Presidents of the United States, and 36% of them were Democrats. The 36% here is a PARAMETER 1936, Literary Digest polled 2.3 million adults in the United States, and 57% of them said they would vote for Alf Landon for the Presidency. The 57% here is a STATISTIC The survey results from 12 local parks show the mean height of 60 ft for mature oak trees. The mean height of 60 ft is a STATISTIC The 36 students in Ms. Dunham's second grade class have a mean height of 64 inches with a standard deviation of 1.3 inches. The 1.3 inches is a PARAMETER In a random sample of homeowners in the United States, it is found that 34% of the sampled homeowners renew their home warranty. The 34% here is a STATISTIC

In the following hypothetical scenarios, classify each number as a parameter or a statistic.

There have been 44 Presidents of the United States, and 36% of them were Democrats. The 36% here is a PARAMETER In a 2011 Gallup poll of 1008 adults living in the United States, 11% said they are satisfied with the condition of the national economy. The 11% here is a STATISTIC A survey of hospital records in 120 hospitals throughout the world shows the mean height of 180 cm for adult males. The mean height of 180 cm is a STATISTIC The 36 students in Ms. Dunham's second grade class have a mean height of 64 inches with a standard deviation of 1.3 inches. The 1.3 inches is a PARAMETER In a random sample of business owners in the United States, it is found that 63% of the sampled business owners attend at least one business conference per year. The 63% here is a STATISTIC

Suppose that 𝑋X is the count of successes in a binomial distribution with 𝑛n fixed observations and a probability 𝑝p of success on any given single observation. Let 𝑌Y be the number of failures in the same 𝑛n observations. Will the binomial distribution for 𝑋X and 𝑌Y necessarily have the same mean and/or standard deviation?

They will always have the same standard deviation, but they might not have the same mean.

When an opinion poll uses random digit dialing to select respondents for polls, the response rate (the percentage who actually provide a usable response to the poll) is approximately 10%10% for people contacted by cell phone. A pollster dials 2020 cell phone numbers. 𝑋X is the number that respond to the pollster. Does 𝑋X have a binomial distribution?

Yes, the calls are independent, each one has two possibilities, and the probability of getting a usable response to the poll is the same for each call.

The Department of Motor Vehicles reports that 32% of all vehicles registered in a state are made by a Japanese or a European automaker. The number 32% is best described as:

a parameter

A confidence interval is constructed to estimate the value of:

a parameter.

For which of the following situations would the central limit theorem not imply that the sample distribution for 𝑥¯x¯ is approximately Normal?

a population is not Normal, and we use samples of size 𝑛=6.

As part of a promotion for a new type of cracker, free samples are offered to shoppers in a local supermarket. The probability that a shopper will buy a packet of crackers after tasting the free sample is 0.2. Different shoppers can be regarded as independent trials. If 𝑋X is the number among the next 100 shoppers who buy a packet of crackers after tasting a free sample, then 𝑋X has approximately:

an 𝑁N(20,4)(20,4) distribution.

To obtain a smaller margin of error:

choose a larger sample size.

To obtain a smaller margin of error:

choose a smaller confidence level.

Classify each described variable as discrete or continuous. Not every variable will be one or the other, but no variable can be both.

discrete: grade level of students in elementary school, outcome of rolling a six-sided number cube Continuous: neck diameter of eight-year olds, volume of a cube

For a simple random sample of size 𝑛n , the count 𝑋X of successes in the sample has a binomial distribution.

false

Personal probabilities are not important since they are based on personal judgment.

false

The random digits generated by a computer program are randomly generated.

false

A margin of error tells us:

how accurate the statistic is when using it to estimate the parameter.

The Stanford-Binet Intelligence Scale is an intelligence test, which, like many other IQ tests, is standardized in order to have a normal distribution with a mean of 100 and a standard deviation of 15 points. As an early intervention effort, a school psychologist wants to estimate the average score on the Stanford-Binet Intelligence Scale for all students with a specific type of learning disorder using a simple random sample of 36 students with the disorder. Determine the margin of error, 𝑚m, of a 99% confidence interval for the mean IQ score of all students with the disorder. Assume that the standard deviation IQ score among the population of all students with the disorder is the same as the standard deviation of IQ score for the general population, 𝜎=15σ=15 points. Give your answer precise to at least two decimal places.

m = 6.44 points

The probability of event 𝐴A is 𝑃(𝐴)=0.5P(A)=0.5 , and the probability of event 𝐵B is 𝑃(𝐵)=0.7P(B)=0.7 . Are 𝐴A and 𝐵B disjoint?

no

Suppose Jason read an article stating that in a 2005-2006 survey, the average American adult woman at least 19 years old drank an average of 1.061.06 liters of plain water per day with a standard deviation of 0.060.06 liters. Jason wants to find out if the women at his college drink a similar amount per day. He asks 6060 of his female classmates in his Introductory Economics class to record the amount of water they drink in one day, and he is willing to assume that the standard deviation at his college is the same as in the 2005-2006 survey. Jason wants to construct a 9595% confidence interval for 𝜇μ, the average amount of water the women at his college drink per day. Have the requirements for constructing a 𝑧z-confidence interval for a mean been met? Mark all of the following requirements that have been met with yes, and all the requirements that have not been met with no

no the sample is a simple random sample yes the population std dev is known yes the pop from which the data are obtained is normally distributed, or the sample size is large enough no the requirements for constructing a z-confidence interval has been met

Suppose that the height of Asian males aged 35-44 in the United States is normally distributed with a standard deviation of 3 in. Jeremy takes a simple random sample of 19 such men and notices that none of them are unusually tall or unusually short. He then calculates the average of the sample to be 67.5 in. and is planning on using this to find a 99% 𝑧z‑confidence interval for the true mean height. Can Jeremy use this information to find the 𝑧z‑confidence interval? Complete the following sentences. The population distribution is the sample and the population standard deviation is to find a 99% 𝑧z‑confidence interval. Therefore, normal, size is irrelevant, known.

normal, size is irrelevant, known, yes all the requirements are met

In 2017, the entire fleet of light‑duty vehicles sold in the United States by each manufacturer must emit an average of no more than 8484 milligrams per mile (mg/mi) of nitrogen oxides (NOX) and nonmethane organic gas (NMOG) over the useful life (150,000150,000 miles of driving) of the vehicle. NOX ++ NMOG emissions over the useful life for one car model vary Normally with mean 7878 mg/mi and standard deviation 66 mg/mi. (a) What is the probability that a single car of this model emits more than 8484 mg/mi of NOX ++ NMOG? (Enter your answer rounded to four decimal places.)

probability: 0.1587 (b) A company has 3636 cars of this model in its fleet. What is the probability that the average NOX ++ NMOG level 𝑥¯x¯ of these cars is above 8484 mg/mi? (Enter your answer rounded to four decimal places.) probabilty: 0

Almost all medical schools in the United States require students to take the Medical College Admission Test (MCAT). To estimate the mean score 𝜇μ of those who took the MCAT on your campus, you will obtain the scores of an SRS of students. The scores follow a Normal distribution, and from published information you know that the standard deviation is 10.410.4 . Suppose that, unknown to you, the mean score of those taking the MCAT on your campus is 500500 . In answering the questions, use 𝑧z‑scores rounded to two decimal places. (a) If you choose one student at random, what is the probability that the student's score is between 495495 and 505505 ? Use Table A, or software to calculate your answer. (Enter your answer rounded to four decimal places.)

probability: 0.3693 (b) You sample 2525 students. What is the standard deviation of the sampling distribution of their average score 𝑥¯x¯ ? (Enter your answer rounded to two decimal places.) standard deviation: 2.08 (c) What is the probability that the mean score of your sample is between 495495 and 505505 ? (Enter your answer rounded to four decimal places.) probability: 0.9836

The National Assessment of Educational Progress (NAEP) includes a mathematics test for eighth‑grade students. Scores on the test range from 00 to 500500 . Demonstrating the ability to use the mean to solve a problem is an example of the skills and knowledge associated with performance at the Basic level. An example of the knowledge and skills associated with the Proficient level is being able to read and interpret a stem‑and‑leaf plot. In 2015, 136,900136,900 eighth‑graders were in the NAEP sample for the mathematics test. The mean mathematics score was 𝑥¯=282x¯=282 . We want to estimate the mean score 𝜇μ in the population of all eighth‑graders. Consider the NAEP sample as an SRS from a Normal population with standard deviation 𝜎=110σ=110 . (a) If we take many samples, the sample mean 𝑥¯x¯ varies from sample to sample according to a Normal distribution with mean equal to the unknown mean score 𝜇μ in the population. What is the standard deviation of this sampling distribution? (Enter your answer rounded to four decimal places.)

standard deviation = 0.2973 (b) According to the 9595 part of the 68‑95‑99.768‑95‑99.7 rule, 95%95% of all values of 𝑥¯x¯ fall within how many points on either side of the unknown mean 𝜇μ ? (Enter your answer rounded to four decimal places.) points = 0.5946 (c) What is the 95%95% confidence interval for the population mean score 𝜇μ based on this one sample? (Enter your answer rounded to one decimal place.) UL: 281.4 LL: 282.6

To estimate the mean score 𝜇μ of those who took the Medical College Admission Test on your campus, you will obtain the scores of an SRS of students. From published information you know that the scores are approximately Normal with standard deviation about 6.36.3 . You want your sample mean 𝑥¯x¯ to estimate 𝜇μ with an error of no more than 1.31.3 point in either direction. (a) What standard deviation must 𝑥¯x¯ have so that 99.7%99.7% of all samples give an 𝑥¯x¯ within 1.31.3 point of 𝜇μ ? Use the 68-95-99.768-95-99.7 rule. (Enter your answer rounded to four decimal places.)

standard deviation of 𝑥¯=x¯= 0.433 (b) How large an SRS do you need in order to reduce the standard deviation of 𝑥¯x¯ to the value you found? (Enter your answer rounded to the nearest whole number.) SRS size: 212

Choose the correct definition of a sampling distribution. The sampling distribution of a statistic of size 𝑛n is

the distribution of all values of the statistic resulting from all samples of size 𝑛n taken from the same population.

The binomial coefficient, written (𝑛𝑘)(nk) =𝑛!𝑘!(𝑛−𝑘!)=n!k!(n−k!) , gives what information?

the number of ways in which 𝑘k successes in 𝑛n trials can be obtained

The probability distribution of a random variable is:

the possible values of the random variable and the frequency with which the variable takes each value.

Suppose there are three cards in a deck: one marked with a "1,"one marked with a "2,"and one marked with a "5." You draw two cards at random, without replacement from the deck of three cards. The sample space 𝑆={(1,2), (2,1),(1,5),(5,1),(2,5),(5,2)}S={(1,2), (2,1),(1,5),(5,1),(2,5),(5,2)} consists of these six equally likely outcomes. Let 𝑋X be the total of the two cards drawn. Which of the following is the correct set of probabilities for 𝑋X ?

x: 3, 6, 7 p: 1/3, 1/3, 1/3

STATE: How heavy a load (in pounds) is needed to pull apart pieces of Douglas fir 44 inches long and 1.51.5 inches square? Given are data from students doing a laboratory exercise. 33,19033,19031,86031,86032,59032,59026,52026,52033,28033,28032,32032,32033,02033,02032,03032,03030,46030,46032,70032,70023,04023,04030,93030,93032,72032,72033,65033,65032,34032,34024,05024,05030,17030,17031,30031,30028,73028,73031,92031,920 To access the complete data set, click the link for your preferred software format: We are willing to regard the wood pieces prepared for the lab session as an SRS of all similar pieces of Douglas fir. Engineers also commonly assume that characteristics of materials vary Normally. Suppose that the strength of pieces of wood like these follows a Normal distribution with standard deviation 30003000 pounds. PLAN: We will estimate 𝜇μ by giving a 98%98% confidence interval. SOLVE: Find the sample mean 𝑥¯x¯ . (Enter your answer rounded to the nearest whole number.)

xprime= 30841 Give a 98%98% confidence interval, [𝑙𝑜𝑤,ℎ𝑖𝑔ℎ][l⁢o⁢w,h⁢i⁢g⁢h] , for the mean load required to pull the wood apart. (Enter your answers rounded to the nearest whole number.) low: 29280 high: 32402

A simple random sample is drawn from a large population with a Normal distribution. What is the sampling distribution of the sample mean?

𝑁(𝜇,𝜎/𝑛√)

I select two cards from a standard deck of 52 cards and observe the color of each (26 cards in the deck are red and 26 are black). Which of the following is an appropriate sample space 𝑆S for the possible outcomes?

𝑆={(red, red), (red, black), (black, red), (black, black)}

A refrigerator contains 6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11 plums, and 2 mangos. Imagine you stick your hand into the refrigerator and pull out a piece of fruit at random. What is the sample space for your action?

𝑆={apple, orange, banana, pear, peach, plum, mango}

A random variable 𝑋X can take on the value 0, 1, 2, or 3. Which of the following is a possible probability model for 𝑋X ?

𝑋X0123𝑃(𝑋)P(X)0.50.30.10.1

Which of the following values of 𝑛n and 𝑝p would give a binomial distribution for which we should avoid using the Normal approximation?

𝑛=30 , 𝑝=0.8

The critical value 𝑧∗z∗ for confidence level 75%75% is not in Table C. Use Table A of standard Normal probabilities to find 𝑧∗.z∗. Select the correct critical value.

𝑧∗=1.15 Select the plot that correctly shows the critical value and the area left in each tail when the central area matches the confidence level of 75%. graph description: normal, mean 0, std dev 1 x axis: -1.15 , 0, 1.15 filled in both sides red 0.125