Registration is now open for this year's LibreFest! Join us virtually the week of July 13.

5.8: Practice (Chapter 5)

Last updated
Save as PDF

Page ID: 59137

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\dsum}{\displaystyle\sum\limits} $

$ \newcommand{\dint}{\displaystyle\int\limits} $

$ \newcommand{\dlim}{\displaystyle\lim\limits} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$\newcommand{\longvect}{\overrightarrow}$

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

5.1: Random Variables – Discrete and Continuous

Identify whether each of the following is a random variable or not:
1. The amount of rain, in inches, that will fall next Friday in Denver, CO.
2. The major of a randomly drawn student from Red Rocks Community College.
3. The number of books purchased next year by your local library.
4. Whether people prefer cheese pizza or pepperoni pizza.
5. The number of credits a student takes in a semester.
6. The average weight of a cat.
In your own words, what is a random variable?
Identify whether each variable below is discrete or continuous:
1. Number of emails you receive in a day
2. Temperature outside (in °F)
3. Cost of a cup of coffee at a local shop
4. Number of correct answers on a 10-question quiz
5. Height of students in a class
Consider the following experiment. You are one of 100 people enlisted to take part in a study to determine the percent of nurses in America with an R.N. (registered nurse) degree. You ask nurses if they have an R.N. degree. The nurses answer “yes” or “no.” You then calculate the percentage of nurses with an R.N. degree. You give that percentage to your supervisor.
1. What part of the experiment will yield discrete data?
2. What part of the experiment will yield continuous data?
When age is rounded to the nearest year, do the data stay continuous, or do they become discrete? Why?
What is the difference between a discrete and a continuous random variable? Include one real-world example of each.
Why can’t we list individual probabilities for continuous random variables the way we do for discrete ones?
Determine whether or not the table is a valid probability distribution of a discrete random variable. Explain fully.
1. \[\begin{array}{c|c c c c} x &-2 &0 &2 &4 \\ \hline P(x) &0.3 &0.5 &0.2 &0.1\\ \end{array}\]
2. \[\begin{array}{c|c c c} x &0.5 &0.25 &0.25\\ \hline P(x) &-0.4 &0.6 &0.8\\ \end{array}\]
3. \[\begin{array}{c|c c c c c} x &1.1 &2.5 &4.1 &4.6 &5.3\\ \hline P(x) &0.16 &0.14 &0.11 &0.27 &0.22\\ \end{array}\]
Determine whether or not the table is a valid probability distribution of a discrete random variable. Explain fully.
1. \[\begin{array}{c|c c c c c} x &0 &1 &2 &3 &4\\ \hline P(x) &-0.25 &0.50 &0.35 &0.10 &0.30\\ \end{array}\]
2. \[\begin{array}{c|c c c } x &1 &2 &3 \\ \hline P(x) &0.325 &0.406 &0.164 \\ \end{array}\]
3. \[\begin{array}{c|c c c c c} x &25 &26 &27 &28 &29 \\ \hline P(x) &0.13 &0.27 &0.28 &0.18 &0.14 \\ \end{array}\]
What is a “probability distribution”? Describe it in your own words.
Create a discrete probability distribution for each of the following. (Make reasonable estimates where necessary.)
1. The number of heads in two tosses of a coin.
2. The number of newborn babies born in a particular county in a month.
3. The number of 12-ounce cans of soft drink someone drinks.
4. The number of games in the next World Series.
5. The number of cards that match when three cards are drawn from a deck.
Suppose that the probability distribution for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given in the following table.
1. In words, define the random variable X.
2. What does it mean that the values zero, one, and two are not included for X in the distribution?

X	P(X)
3	0.05
4	0.40
5	0.30
6	0.15
7	0.10

A physics professor wants to know what percent of physics majors will spend the next several years doing post-graduate research. She has the following probability distribution.

X	P(X)
1	0.35
2	0.20
3	0.15
4
5	0.10
6	0.05

Define the random variable X
What should the column P(X) add up to and why?
Find the probability that a physics major will do post-graduate research for four years
Find the probability that a physics major will do post-graduate research for at least three years.
Find the probability that a physics major will no do research for one year.

A ballet instructor is interested in knowing what percent of each year's class will continue on to the next, so that they can plan what classes to offer. Over the years, they have established the following probability distribution.

X	P(X)
1	0.10
2	0.05
3	0.10
4
5	0.30
6	0.20
7	0.10

In words, describe the random variable X.
Determine P(X=4)
Calculate and interpret P(x<4)
Calculate and interpret P(x\=7)

5.2: Expected Value of Discrete Distributions

In your own words, what is expected value? What does it represent?
Complete the following table and calculate the expected value.

X	P(X)	X P(X)
2	0.1
4	0.3	4(0.3) = 1.2
6	0.4	6(0.4) = 2.4
8	0.2	8(0.2) = 1.6

Complete the following table and calculate the expected value.

Expected values
$x$	$P(x)$	$x*P(x)$
0	0.2
1	0.2
2	0.4
3	0.2

Identify the mistake in the probability distribution table.

Find the mistake
$x$	$P(x)$	$x*P(x)$
1	0.15	0.15
2	0.25	0.50
3	0.30	0.90
4	0.20	0.80
5	0.15	0.75

Suppose that the probability distribution for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given as in the table.

B.S. completion times
$x$	$P(x)$
3	0.05
4	0.40
5	0.30
6	0.15
7	0.10

On average, how many years do you expect it to take for an individual to earn a B.S.?

You play a game where you win $10 with probability 0.2, win $5 with probability 0.3, and lose $2 with probability 0.5.
1. Create a probability distribution table for this game.
2. What is the expected value of your winnings?
3. Would you recommend playing this game? Why or why not?
A lottery ticket costs $3. The possible prizes (net winnings) and associated probabilities are as follows: What is the expected value of this ticket (accounting for the cost to play)?
1. $0 (prob = 0.85)
2. $5 (prob = 0.10)
3. $50 (prob = 0.04)
4. $500 (prob = 0.01)
A theater group holds a fund-raiser. It sells 100 raffle tickets for $5 each. Suppose you purchase four tickets. The prize is two passes to a Broadway show, worth a total of $150.
1. What are you interested in here?
2. Define the random variable $X$.
3. List the values that $X$ may take on.
4. Construct a probability distribution table for this situation.
5. If this fund-raiser is repeated often and you always purchase four tickets, what would be your expected average winnings per raffle?
A game involves selecting a card from a regular 52-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails.
1. If the card is a face card, and the coin lands on Heads, you win $6
2. If the card is a face card, and the coin lands on Tails, you win $2
3. If the card is not a face card, you lose $2, no matter what the coin shows.
4. Find the expected value for this game (expected net gain or loss).
5. Explain what your calculations indicate about your long-term average profits and losses on this game.
6. Should you play this game to win money?
A venture capitalist, willing to invest $1,000,000, has three investments to choose from. The first investment, a social media company, has a 20% chance of returning $7,000,000 profit, a 30% chance of returning no profit, and a 50% chance of losing the million dollars. The second company, an advertising firm has a 10% chance of returning $3,000,000 profit, a 60% chance of returning a $2,000,000 profit, and a 30% chance of losing the million dollars. The third company, a chemical company has a 40% chance of returning $3,000,000 profit, a 50% chance of no profit, and a 10% chance of losing the million dollars.
1. Construct a probability distribution table for each investment. In your table the X column is the net amount of profit/loss for the venture capitalist and the P(X) column is the likelihood given above.
2. Find the expected value for each investment.
3. Which investment has the highest expected return?
4. Which is the safest investment and why?
5. Which is the riskiest investment and why?
A company estimates that 3% of their products will fail after the original warranty period but within 2 years of the purchase, with a replacement cost of $250. If they want to offer a 2 year extended warranty, what price should they charge so that they'll break even (in other words, so the expected value will be 0).
Write an expression (but do not calculate) for the variance of the following random variable:
1. x = {–1, 0, 5}, with P(x) = {0.2, 0.5, 0.3}

Find the standard deviation for the following probability distribution

X	P(X)	X P(X)	(X-µ)²P(X)
2	0.1	2(0.1) = 0.2	(2–5.4)²(0.1) = 1.156
4	0.3	4(0.3) = 1.2	(4–5.4)²(0.3) = 0.588
6	0.4	6(0.4) = 2.4	(6–5.4)²(0.4) = 0.144
8	0.2	8(0.2) = 1.6	(8–5.4)²(0.2) = 1.352

Red Rocks Community College has 14 statistics classes scheduled for its Fall 2026 term. One class has space available for 30 students, eight classes have space for 60 students, one class has space for 70 students, and four classes have space for 100 students.
1. What is the average class size assuming each class is filled to capacity?
2. Space is available for 980 students. Suppose that each class is filled to capacity and select a statistics student at random. Let the random variable X equal the size of the student’s class. Create a probability distribution for X.
3. Find the mean of X.
4. Find the standard deviation of X.
Borachio works in an automotive tire factory. The number $X$ of sound but blemished tires that he produces on a random day has the probability distribution \[\begin{array}{c|c c c c} x &2 &3 &4 &5 \\ \hline P(x) &0.48 &0.36 &0.12 &0.04\\ \end{array}\]
1. Find the probability that Borachio will produce more than three blemished tires tomorrow.
2. Find the probability that Borachio will produce at most two blemished tires tomorrow.
3. Compute the mean and standard deviation of $X$. Interpret them in the context of the problem.
The number $X$ of days in the summer months that a construction crew cannot work because of the weather has the probability distribution \[\begin{array}{c|c c c c c} x &6 &7 &8 &9 &10\\ \hline P(x) &0.03 &0.08 &0.15 &0.20 &0.19 \\ \end{array}\] \[\begin{array}{c|c c c c } x &11 &12 &13 &14 \\ \hline P(x) &0.16 &0.10 &0.07 &0.02 \\ \end{array}\]
1. Find the probability that no more than ten days will be lost next summer.
2. Find the probability that from $8$ to $12$ days will be lost next summer.
3. Find the probability that no days at all will be lost next summer.
4. Compute the mean and standard deviation of $X$. Interpret them in the context of the problem.

Ice cream usually comes in 1.5 quart boxes (48 fluid ounces), and ice cream scoops hold about 2 ounces. However, there is some variability in the amount of ice cream in a box as well as the amount of ice cream scooped out. We represent the amount of ice cream in the box as X and the amount scooped out as Y. Suppose these random variables have the following means, standard deviations, and variances

	mean	SD	variance
X	48	1	1
Y	2	0.25	0.0625

An entire box of ice cream, plus 3 scoops from a second box is served at a party. How much ice cream do you expect to have been served at this party? What is the standard deviation of the amount of ice cream served?
How much ice cream would you expect to be left in the box after scooping out one scoop of ice cream? That is, find the expected value of X - Y . What is the standard deviation of the amount left in the box?
Using the context of this exercise, explain why we add variances when we subtract one random variable from another.

5.3: Discrete Distributions – Bernoulli and Binomial

Suppose you ask 40 individuals whether they ride their bicycle to work and 2 individuals say "yes". What proportion of the individuals in the sample responded "yes"? What proportion responded "no"?
Suppose you are a caterer and planning an event for 250 attendees. The event is held for a population that you know to be 15% vegan. How many attendees would you expect to be vegan?
Determine if each trial can be considered an independent Bernouilli trial for the following situations.
1. Cards dealt in a hand of poker.
2. Outcome of each roll of a die.
3. The number of coins that match when three coins are flipped.
4. The response from an individual to a yes/no question on a survey.
5. The number of times you roll a die until you get a six.
Determine whether or not the random variable $X$ is a binomial random variable. If so, give the values of $n$ and$p$. If not, explain why not.
1. $X$ is the number of dots on the top face of fair die that is rolled.
2. $X$ is the number of hearts in a five-card hand drawn (without replacement) from a well-shuffled ordinary deck.
3. $X$ is the number of defective parts in a sample of ten randomly selected parts coming from a manufacturing process in which $0.02\%$ of all parts are defective.
4. $X$ is the number of times the number of dots on the top face of a fair die is even in six rolls of the die.
5. $X$ is the number of dice that show an even number of dots on the top face when six dice are rolled at once.
Determine whether or not the random variable $X$ is a binomial random variable. If so, give the values of $n$ and $p$. If not, explain why not.
1. $X$ is the number of black marbles in a sample of $5$ marbles drawn randomly and without replacement from a box that contains $25$ white marbles and $15$ black marbles.
2. $X$ is the number of black marbles in a sample of $5$ marbles drawn randomly and with replacement from a box that contains $25$ white marbles and $15$ black marbles.
3. $X$ is the number of voters in favor of proposed law in a sample $1,200$ randomly selected voters drawn from the entire electorate of a country in which $35\%$ of the voters favor the law.
4. $X$ is the number of coins that match at least one other coin when four coins are tossed at once.
What are the requirements for a binomial experiment?
When dealing with the binomial distribution, why are the possible values for the random variable always 0,1,2,3,…,n where n is the number of trials or sample size? Why can't we use negative values, or fractions, or numbers greater than n?
Under what conditions is a binomial distribution symmetric? Skewed left? Skewed right? Why?
How do you know when to use the binomial distribution to model a situation?
How are the areas in the bars of a binomial histogram related to the probability of choosing those X values? (Hint: figure it out for a single bar)
Compare and contrast the Bernoulli and Binomial distributions.
A fair coin is flipped once. Define a random variable $ X $ to represent success = heads. Construct a table showing the probability distribution for this Bernoulli variable.
A basketball player makes 80% of their free throws. If they shoot 3 times, what is the probability they make exactly 2?
The Substance Abuse and Mental Health Services Administration estimated that 70% of 18-20 year olds consumed alcoholic beverages in 2008.
1. Suppose a random sample of ten 18-20 year olds is taken. Is the use of the binomial distribution appropriate for calculating the probability that exactly six consumed alcoholic beverages? Explain.
2. Calculate the probability that exactly 6 out of 10 randomly sampled 18-20 year olds consumed an alcoholic drink.
3. What is the probability that exactly four out of the ten 18-20 year olds have not consumed an alcoholic beverage?
4. What is the probability that at most 2 out of 5 randomly sampled 18-20 year olds have consumed alcoholic beverages?
5. What is the probability that at least 1 out of 5 randomly sampled 18-20 year olds have consumed alcoholic beverages?
The National Vaccine Information Center estimates that 90% of Americans have had chickenpox by the time they reach adulthood.
1. Suppose we take a random sample of 100 American adults. Is the use of the binomial distribution appropriate for calculating the probability that exactly 97 had chickenpox before they reached adulthood? Explain.
2. Calculate the probability that exactly 97 out of 100 randomly sampled American adults had chickenpox during childhood.
3. What is the probability that exactly 3 out of a new sample of 100 American adults have not had chickenpox in their childhood?
4. What is the probability that at least 1 out of 10 randomly sampled American adults have had chickenpox?
5. What is the probability that at most 3 out of 10 randomly sampled American adults have not had chickenpox?
A 2005 Gallup Poll found that that 7% of teenagers (ages 13 to 17) suffer from arachnophobia and are extremely afraid of spiders. At a summer camp there are 10 teenagers sleeping in each tent. Assume that these 10 teenagers are independent of each other.
1. Calculate the probability that at least one of them suffers from arachnophobia.
2. Calculate the probability that exactly 2 of them suffer from arachnophobia?
3. Calculate the probability that at most 1 of them suffers from arachnophobia?
4. If the camp counselor wants to make sure no more than 1 teenager in each tent is afraid of spiders, does it seem reasonable for him to randomly assign teenagers to tents?
About 24% of flights departing from New York's John F. Kennedy International Airport were delayed in 2009. Assuming that the chance of a flight being delayed has stayed constant at 24%, we are interested in finding the probability of 10 out of the next 100 departing flights being delayed. Noting that if one flight is delayed, the next flight is more likely to be delayed, which of the following statements is correct?
1. We can use the binomial distribution with n = 100, k = 10, and p = 0.24 to calculate this probability.
2. We can use the geometric distribution with n = 100, k = 10, and p = 0.24 to calculate this probability.
3. We cannot calculate this probability using the binomial distribution since whether or not one flight is delayed is not independent of another.
4. We can use the binomial distribution with n = 10, k = 100, and p = 0.24 to calculate this probability.
Write out the full binomial distribution table for $ n = 4 $, $ p = 0.5 $. Label columns clearly.
In a binomial experiment with $ n = 6 $, $ p = 0.3 $, what is the probability of 0 successes? One success? (Use binomial probabilities)
Suppose you are a lawyer representing a Hispanic individual in a criminal trial. Twelve jurors are randomly selected by the state from a population that you know to be 50% Hispanic.
1. How many jurors would you expect to be Hispanic?
2. The selection process produces a jury with 3 Hispanic individuals. What proportion of the jury is Hispanic?
3. Use the binomial distribution and find the probability that 3 or fewer jurors would be Hispanic.
4. Create a graph (using technology) of the distribution with the area corresponding to the previous answer highlighted
5. Use the above results to write an argumentative essay to the court on behalf of your client.
A small regional carrier accepted 21 reservations for a particular flight with 19 seats. 17 reservations went to regular customers who will arrive for the flight. Each of the remaining passengers will arrive for the flight with a 41% chance, independently of each other.
1. Find the probability that overbooking occurs.
2. Find the probability that the flight has empty seats.
After being rejected for employment, Kim Kelly learns that the Bellevue Credit Company has hired only four women among the last 17 new employees. She also learns that the pool of applicants is very large, with an approximately equal number of qualified men as qualified women.
1. Help her address the charge of gender discrimination by finding the probability of getting four or fewer women when 17 people are hired, assuming that there is no discrimination based on gender.
2. Because this is a serious claim, we will use a stricter cutoff value for unusual events. We will use 0.5% as the cutoff value (1 in 200 chance of happening by chance). With this in mind, does the resulting probability really support such a charge?
Among patients claiming to have the flu, the chance that someone has the flu (and not just a nasty cold) is only about 4%. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu.
1. Create a probability distribution table for this situation
2. Find the probability that at least four of the 25 patients actually have the flu.
3. On average, for every 25 patients calling in, how many do you expect to have the flu?
A student takes a ten-question true-false quiz, but did not study and randomly guesses each answer. Find the probability that the student passes the quiz with a grade of at least 70% of the questions correct.
According to The World Bank, only 9% of the population of Uganda had access to electricity as of 2009. Suppose we randomly sample 150 people in Uganda. Let X represent the number of people who have access to electricity.
1. Create a probability distribution for X?
2. Calculate the mean and standard deviation of X.
3. Use technology to find the probability that 15 people in the sample have access to electricity.
4. Find the probability that at most ten people in the sample have access to electricity.
5. Find the probability that more than 25 people in the sample have access to electricity.
If you flip a fair coin 10 times, what is the probability of
1. getting all tails?
2. getting all heads?
3. getting at least one tails?

5.4: The Normal Distribution

Which type of distribution does the graph illustrate?
Which type of distribution does the graph illustrate?
What does the shaded area represent?
What does the shaded area represent?
How is probability determined from a continuous distribution? Why is this easy for the uniform distribution and not so easy for the normal distribution?
Why is the probability that a continuous random variable is equal to a single number zero? (i.e. Why is P(X=a)=0 for any number a)
For a continuous probability distribution, 0<X<15.
1. What is P(X>15)?
2. What is P(X=7)?
Find the probability that falls in the shaded area.
Births are approximately uniformly distributed between the 52 weeks of the year. They can be said to follow a uniform distribution from one to 53 (spread of 52 weeks).Graph the probability distribution.
1. What is the mean?
2. What is the standard deviation?
3. Find the probability that a person is born at the exact moment week 19 starts.
4. Find the probability that a person is born after week 40.
5. Find the 70^th percentile.
6. Find the minimum for the upper quarter.
What two quantities do we need to fully describe a normal distribution?
Sketch and label a normal distribution. Indicate the mean and one, two, and three standard deviations on either side.
What does it mean for a distribution to be “normal”? List 3 key properties.
What does the symmetric bell shape of the normal curve imply about the distribution of individuals in a normal population?
The length of time to complete a survey is normally distributed with a mean of 10 minutes and standard deviation of 2 minutes.
1. What proportion of people take less than 10 minutes?
2. What proportion take between 8 and 12 minutes?
3. What is the median completion time?
The heights of the 430 National Basketball Association players were listed on team rosters at the start of the 2005–2006 season. The heights of basketball players have an approximate normal distribution with mean, µ = 79 inches and a standard deviation, σ = 3.89 inches. For each of the following heights, calculate the z-score and interpret it using complete sentences.
1. 77 inches
2. 85 inches
3. If an NBA player reported his height had a z-score of 3.5, would you believe him? Explain your answer.
Compare a normal distribution to a skewed distribution. How might their histogram shapes differ?
Suppose you randomly select an individual from a population that is normally distributed and they are above average. When you find out the probability of randomly selecting that individual is very very small, what are some possible explanations? In other words what does this very very small probability suggest?
A continuous random variable $X$ has a normal distribution with mean $100$ and standard deviation $10$. Sketch a qualitatively accurate graph of its density function.
A continuous random variable $X$ has a normal distribution with mean $73$ and standard deviation $2.5$. Sketch a qualitatively accurate graph of its density function.
A continuous random variable $X$ has a normal distribution with mean $73$. The probability that $X$ takes a value greater than $80$ is $0.212$. Use this information and the symmetry of the density function to find the probability that $X$ takes a value less than $66$. Sketch the density curve with relevant regions shaded to illustrate the computation.
A continuous random variable $X$ has a normal distribution with mean $169$. The probability that $X$ takes a value greater than $180$ is $0.17$. Use this information and the symmetry of the density function to find the probability that $X$ takes a value less than $158$. Sketch the density curve with relevant regions shaded to illustrate the computation.
A continuous random variable $X$ has a normal distribution with mean $50.5$. The probability that $X$ takes a value less than $54$ is $0.76$. Use this information and the symmetry of the density function to find the probability that $X$ takes a value greater than $47$. Sketch the density curve with relevant regions shaded to illustrate the computation.
A continuous random variable $X$ has a normal distribution with mean $12.25$. The probability that $X$ takes a value less than $13$ is $0.82$. Use this information and the symmetry of the density function to find the probability that $X$ takes a value greater than $11.50$. Sketch the density curve with relevant regions shaded to illustrate the computation.
The figure provided shows the density curves of three normally distributed random variables $X_A,\; X_B\; \text{and}\; X_C$. Their standard deviations (in no particular order) are $15$, $7$, and $20$. Use the figure to identify the values of the means $\mu _A,\: \mu _B,\; \text{and}\; \mu _C$ and standard deviations $\sigma _A,\: \sigma _B,\; \text{and}\; \sigma _C$ of the three random variables.

Graph depicting three distinct peaks labeled XA, XB, XC.

The figure provided shows the density curves of three normally distributed random variables $X_A,\; X_B\; \text{and}\; X_C$. Their standard deviations (in no particular order) are $20$, $5$, and $10$. Use the figure to identify the values of the means $\mu _A,\: \mu _B,\; \text{and}\; \mu _C$ and standard deviations $\sigma _A,\: \sigma _B,\; \text{and}\; \sigma _C$ of the three random variables.

Three probabilities distributions labeled XA, XB and XC.

The amount $X$ of orange juice in a randomly selected half-gallon container varies according to a normal distribution with mean $64$ ounces and standard deviation $0.25$ ounce.
1. Sketch the graph of the density function for $X$.
2. What proportion of all containers contain less than a half gallon ($64$ ounces)? Explain.
3. What is the median amount of orange juice in such containers? Explain.
The weight $X$ of grass seed in bags marked $50$ lb varies according to a normal distribution with mean $50$ lb and standard deviation $1$ ounce ($0.0625$ lb).
1. Sketch the graph of the density function for $X$.
2. What proportion of all bags weigh less than $50$ pounds? Explain.
3. What is the median weight of such bags? Explain

5.5: The Empirical Rule and Standard Normal (Z) Distribution

Find the z-score for the following:
1. x = 88, μ = 75, σ = 5
2. x = 64, μ = 70, σ = 3
The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days. What is the z-score for a patient who takes ten days to recover?
The widths of platinum samples manufactured at a factory are normally distributed, with a mean of 1.2 cm and a standard deviation of 0.5 cm. Find the z-scores that correspond to each of the following widths.
1. 1.9 cm
2. 0.5 cm
The heights of the 430 National Basketball Association players were listed on team rosters at the start of the 2005–2006 season. The heights of basketball players have an approximate normal distribution with mean, µ = 79 inches and a standard deviation, σ = 3.89 inches. For each of the following heights, calculate the z-score and interpret it using complete sentences.
1. 77 inches
2. 85 inches
3. If an NBA player reported his height had a z-score of 3.5, would you believe him? Explain your answer.
Two students take different exams:
1. Adrian: Score = 87, μ = 75, σ = 6
2. Blair: Score = 83, μ = 70, σ = 8
3. Who performed better, relative to their class?
Kyle’s doctor told him that the z-score for his systolic blood pressure is 1.75. Which of the following is the best interpretation of this standardized score?
Explain what a z-score of –2.1 tells us about a data value.
In general, what does a z-score tell you about a number in a data set?
Which of the following is false? Hint: It might be useful to sketch the distributions.
1. Calculating percentiles based on the Z table is only appropriate for observations that come from a nearly normal distribution.
2. The Z score for the median of a left skewed distribution is negative.
3. The Z score for the median of a symmetric distribution is approximately 0.
4. Z scores are defined for observations from distributions of any shape and skew.
According to the Empirical Rule, about what % of data fall within 1 standard deviation of the mean in a normal distribution?
If a dataset is normally distributed with mean = 60 and standard deviation = 4:
1. What interval contains approximately 68% of the data?
2. What interval contains approximately 95% of the data?
How can the empirical rule be restated in terms of z-scores and percentiles? Restate it for each of the seven z-scores.
Suppose the systolic blood pressure (in mm) of adults has an approximately normal distribution with mean 125 and standard deviation 14.
1. Create an empirical rule graph for this situation. The use your graph to answer the following questions.
2. About 99.7% of adults will have blood pressure between what amounts?
3. What percentage of adults will have a systolic blood pressure outside the range 111 mm to 153 mm?
4. Suppose you are a health practitioner and an adult patient has systolic blood pressure of 171 mm. Use the above results to explain the gravity of their situation.
We flip a coin 100 times ($n = 100$) and note that it only comes up heads 20% ($p = 0.20$) of the time. The mean and standard deviation for the number of times the coin lands on heads is $\mu = 20$ and $\sigma = 4$ (verify the mean and standard deviation). Solve the following:
1. There is about a 68% chance that the number of heads will be somewhere between ___ and ___.
2. There is about a ____chance that the number of heads will be somewhere between 12 and 28.
3. There is about a ____ chance that the number of heads will be somewhere between eight and 32.
The maintenance department at the Lakewood campus of Red Rocks Community College receives daily requests to replace fluorecent lightbulbs. The distribution of the number of daily requests is bell-shaped and has a mean of 37 and a standard deviation of 8. Using the 68-95-99.7 rule, what is the approximate percentage of lightbulb replacement requests numbering between 37 and 53?
A NUMMI assembly line, which has been operating since 1984, has built an average of 6,000 cars and trucks a week. Generally, 10% of the cars were defective coming off the assembly line. Suppose we draw a random sample of n = 100 cars. Let X represent the number of defective cars in the sample. What can we say about X in regard to the 68-95-99.7 empirical rule (one standard deviation, two standard deviations and three standard deviations from the mean are being referred to)? Assume a normal distribution for the defective cars in the sample.

Thuy Dau, Ngoc Bui, Sam Su, and Lan Voung conducted a survey as to how long customers at Lucky claimed to wait in the checkout line until their turn. Let $X =$ time in line. The following table displays the ordered real data (in minutes):

Waiting data
0.50	4.25	5	6	7.25
1.75	4.25	5.25	6	7.25
2	4.25	5.25	6.25	7.25
2.25	4.25	5.5	6.25	7.75
2.25	4.5	5.5	6.5	8
2.5	4.75	5.5	6.5	8.25
2.75	4.75	5.75	6.5	9.5
3.25	4.75	5.75	6.75	9.5
3.75	5	6	6.75	9.75
3.75	5	6	6.75	10.75

Calculate the sample mean and the sample standard deviation.
Construct a histogram.
Draw a smooth curve through the midpoints of the tops of the bars.
In words, describe the shape of your histogram and smooth curve.
Let the sample mean approximate $\mu$ and the sample standard deviation approximate $\sigma$. The distribution of $X$ can then be approximated by $X \sim$
Use the distribution in part e to calculate the probability that a person will wait fewer than 6.1 minutes.
Determine the cumulative relative frequency for waiting less than 6.1 minutes.
Why aren’t the answers to part f and part g exactly the same?
Why are the answers to part f and part g as close as they are?
If only ten customers has been surveyed rather than 50, do you think the answers to part f and part g would have been closer together or farther apart? Explain your conclusion.

For normally distributed data, what proportion of observations have a z-score less than -0.14.
For normally distributed data, what proportion of observations have a z-score greater than 2.51.
What proportion of data fall between z = –1.5 and z = 2?
A z-score is given as 1.75. Use a z-table or calculator to approximate $ P(Z > 1.75) $.
What percent of a standard normal distribution is found in each region?
1. Z < -1:35
2. Z > 1:48
3. -0:4 < Z < 1:5
4. |Z| > 2

5.6: Applications of the Normal Distribution

A weightlifting competition has normally distributed scores with μ = 100 and σ = 15. What proportion of athletes score above 120?
The average gas mileage for a hybrid car is 48 mpg (σ = 4). What is the probability that a randomly selected car gets less than 43 mpg?
What SAT Math score corresponds to the 95th percentile if μ = 520 and σ = 100?
The empirical rule says that 95% of the population is within 2 standard deviations of the mean, but when I find the z-scores that mark off the middle 95% of the standard normal distribution I calculate -1.96 and 1.96. Is this a contradiction? Why or why not? In other words why are the normal distribution calculators not agreeing with the empirical rule?
Suppose that you are an elementary school teacher and you are evaluating the reading levels of your students. You find an individual that reads 42.3 word per minute. You do some research and determine that the reading rates for their grade level are normally distributed with a mean of 85 words per minute and a standard deviation of 23 words per minute.
1. At what percentile is the child's reading level.
2. Create a graph with a normal curve that illustrates the problem.
3. Write an essay to the parents of the child justifying the need for remediation.
Suppose that you are working for a chain restaurant and wish to design a promotion to disabuse the public of notions that the service is slow. You decide to institute a policy that any customer that waits too long will receive their meal for free. You know that the wait times for customers are normally distributed with a mean of 19 minutes and a standard deviation of 3.4 minutes. Use statistics to decide the maximum wait time you would advertise to customers so that you only give away free meals to at most 1% of the customers.
1. Determine an estimate of an advertised maximum wait time so that 1% of the customers would receive a free meal.
2. Create a graph illustrating the solution.
3. Write a response to the vice president explaining your prescribed maximum wait time.
The amount $X$ of beverage in a can labeled $12$ ounces is normally distributed with mean $12.1$ ounces and standard deviation $0.05$ ounce. A can is selected at random.
1. Find the probability that the can contains at least $12$ ounces.
2. Find the probability that the can contains between $11.9$ and $12.1$ ounces.
The systolic blood pressure $X$ of adults in a region is normally distributed with mean $112$ mm Hg and standard deviation $15$ mm Hg. A person is considered “pre-hypertensive” if his systolic blood pressure is between $120$ and $130$ mm Hg. Find the probability that the blood pressure of a randomly selected person is pre-hypertensive.
Heights $X$ of adult women are normally distributed with mean $63.7$ inches and standard deviation $2.71$ inches. Juliet, who is $69.25$ inches tall, wishes to date only women who are shorter than she is, but within $4$ inches of her height. Find the probability that the next woman she meets will have such a height.
Heights $X$ of adult men are normally distributed with mean $69.1$ inches and standard deviation $2.92$ inches. Romeo, who is $63.25$ inches tall, wishes to date only men who are taller than he is but within 6 inches of his height. Find the probability that the next man he meets will have such a height.
A regulation hockey puck must weigh between $5.5$ and $6$ ounces. The weights $X$ of pucks made by a particular process are normally distributed with mean $5.75$ ounces and standard deviation $0.11$ ounce.
1. Find the probability that a puck made by this process will meet the weight standard.
2. Find the maximum allowable standard deviation so that at most $0.005$ of all pucks will fail to meet the weight standard. (Hint: The distribution is symmetric and is centered at the middle of the interval of acceptable weights.)
A regulation golf ball may not weigh more than $1.620$ ounces. The weights $X$ of golf balls made by a particular process are normally distributed with mean $1.361$ ounces and standard deviation $0.09$ ounce. Find the probability that a golf ball made by this process will meet the weight standard.
The length of time that the battery in Hippolyta's cell phone will hold enough charge to operate acceptably is normally distributed with mean $25.6$ hours and standard deviation $0.32$ hour. Hippolyta forgot to charge their phone yesterday, so that at the moment they first want to use it today it has been $26$ hours $18$ minutes since the phone was last fully charged. Find the probability that the phone will operate properly.
The amount of non-mortgage debt per household for households in a particular income bracket in one part of the country is normally distributed with mean $\$28,350$ and standard deviation $\$3,425$. Find the probability that a randomly selected such household has between $\$20,000$ and $\$30,000$ in non-mortgage debt.
Birth weights of full-term babies in a certain region are normally distributed with mean $7.125$ lb and standard deviation $1.290$ lb. Find the probability that a randomly selected newborn will weigh less than $5.5$ lb, the historic definition of prematurity.
The distance from the seat back to the front of the knees of seated adults is normally distributed with mean $23.8$ inches and standard deviation $1.22$ inches. The distance from the seat back to the back of the next seat forward in all seats on aircraft flown by a budget airline is $26$ inches. Find the proportion of adults flying with this airline whose knees will touch the back of the seat in front of them.
The distance from the seat to the top of the head of seated adults is normally distributed with mean $36.5$ inches and standard deviation $1.39$ inches. The distance from the seat to the roof of a particular make and model car is $40.5$ inches. Find the proportion of adults who when sitting in this car will have at least one inch of headroom (distance from the top of the head to the roof).
The useful life of a particular make and type of automotive tire is normally distributed with mean $57,500$ miles and standard deviation $950$ miles.
1. Find the probability that such a tire will have a useful life of between $57,000$ and $58,000$ miles.
2. Hamnet buys four such tires. Assuming that their lifetimes are independent, find the probability that all four will last between $57,000$ and $58,000$ miles. (If so, the best tire will have no more than $1,000$ miles left on it when the first tire fails.) Hint: There is a binomial random variable here, whose value of $p$ comes from part a.
A machine produces large fasteners whose length must be within $0.5$ inch of $22$ inches. The lengths are normally distributed with mean $22.0$ inches and standard deviation $0.17$ inch.
1. Find the probability that a randomly selected fastener produced by the machine will have an acceptable length.
2. Between what two sizes are the middle 50% of all fasteners?
The lengths of time taken by students on an algebra proficiency exam (if not forced to stop before completing it) are normally distributed with mean $28$ minutes and standard deviation $1.5$ minutes.
1. Find the proportion of students who will finish the exam if a $30$-minute time limit is set.
2. Between what two scores are the middle 70% of all students?
The amount of gasoline $X$ delivered by a metered pump when it registers $5$ gallons is a normally distributed random variable. The standard deviation $\sigma$ of $X$measures the precision of the pump; the smaller $\sigma$ is the smaller the variation from delivery to delivery. A typical standard for pumps is that when they show that $5$ gallons of fuel has been delivered the actual amount must be between $4.97$ and $5.03$ gallons (which corresponds to being off by at most about half a cup). Supposing that the mean of $X$ is $5$, find the largest that $\sigma$ can be so that $P(4.97 < X < 5.03)$ is $1.0000$ to four decimal places. (Hint: The $z$-score of $5.03$ will be the smallest value of $Z$ so that (P(Z<z)=1.0000\)).
Scores on a national exam are normally distributed with mean $382$ and standard deviation $26$.
1. Find the score that is the $50^{th}$ percentile.
2. Find the score that is the $90^{th}$ percentile.
Heights of adults are normally distributed with mean $63.7$ inches and standard deviation $2.47$ inches.
1. Find the height that is the $10^{th}$ percentile.
2. Find the height that is the $80^{th}$ percentile.
The monthly amount of water used per household in a small community is normally distributed with mean $7,069$ gallons and standard deviation $58$ gallons. Find the three quartiles for the amount of water used.
The quantity of gasoline purchased in a single sale at a chain of filling stations in a certain region is normally distributed with mean $11.6$ gallons and standard deviation $2.78$ gallons. Find the three quartiles for the quantity of gasoline purchased in a single sale.
Scores on the common final exam given in a large enrollment multiple section course were normally distributed with mean $69.35$ and standard deviation $12.93$. The department has the rule that in order to receive an $A$ in the course his score must be in the top $10\%$ of all exam scores. Find the minimum exam score that meets this requirement.
The average finishing time among all high school boys in a particular track event in a certain state is $5$ minutes $17$ seconds. Times are normally distributed with standard deviation $12$ seconds.
1. The qualifying time in this event for participation in the state meet is to be set so that only the fastest $5\%$ of all runners qualify. Find the qualifying time. (Hint: Convert seconds to minutes.)
2. In the western region of the state the times of all boys running in this event are normally distributed with standard deviation $12$ seconds, but with mean $5$ minutes $22$ seconds. Find the proportion of boys from this region who qualify to run in this event in the state meet.
Tests of a new tire developed by a tire manufacturer led to an estimated mean tread life of $67,350$ miles and standard deviation of $1,120$ miles. The manufacturer will advertise the lifetime of the tire (for example, a “$50,000$ mile tire”) using the largest value for which it is expected that $98\%$ of the tires will last at least that long. Assuming tire life is normally distributed, find that advertised value.
Tests of a new light led to an estimated mean life of $1,321$ hours and standard deviation of $106$ hours. The manufacturer will advertise the lifetime of the bulb using the largest value for which it is expected that $90\%$ of the bulbs will last at least that long. Assuming bulb life is normally distributed, find that advertised value.
The lengths $X$ of hardwood flooring strips are normally distributed with mean $28.9$ inches and standard deviation $6.12$ inches. Strips whose lengths lie in the middle 80% of the distribution of lengths of all strips are classified as “average-length strips.” Find the maximum and minimum lengths of such strips. (These lengths are endpoints of an interval that is symmetric about the mean and in which the lengths of $80\%$ of the hardwood strips lie.)
All students in a large enrollment multiple section course take common in-class exams and a common final, and submit common homework assignments. Course grades are assigned based on students' final overall scores, which are approximately normally distributed. The department assigns a $C$ to students whose scores constitute the middle $2/3$ of all scores. If scores this semester had mean $72.5$ and standard deviation $6.14$, find the interval of scores that will be assigned a $C$.
Researchers wish to investigate the overall health of individuals with abnormally high or low levels of glucose in the blood stream. Suppose glucose levels are normally distributed with mean $96$ and standard deviation $8.5\; mg/dl$, and that “normal” is defined as the middle $90\%$ of the population. Find the interval of normal glucose levels, that is, the interval centered at $96$ that contains $90\%$ of all glucose levels in the population.
A machine for filling $2$-liter bottles of soft drink delivers an amount to each bottle that varies from bottle to bottle according to a normal distribution with standard deviation $0.002$ liter and mean whatever amount the machine is set to deliver.
1. If the machine is set to deliver $2$ liters (so the mean amount delivered is $2$ liters) what proportion of the bottles will contain at least $2$ liters of soft drink?
2. Find the minimum setting of the mean amount delivered by the machine so that at least $99\%$ of all bottles will contain at least $2$ liters.
A nursery has observed that the mean number of days it must darken the environment of a species poinsettia plant daily in order to have it ready for market is $71$ days. Suppose the lengths of such periods of darkening are normally distributed with standard deviation $2$ days. Find the number of days in advance of the projected delivery dates of the plants to market that the nursery must begin the daily darkening process in order that at least $95\%$ of the plants will be ready on time. (Poinsettias are so long-lived that once ready for market the plant remains salable indefinitely.)

Practice problems include parts of:

Introductory Statistics 2e by OpenStax is licensed under CC BY 4.0

Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction

and

OpenIntro Statistics by David Diez, Christopher Barr, & Mine Çetinkaya-Rundel is licensed under CC BY 3.0

Search

Text Color

Text Size

Margin Size

Font Type