the normal distribution common core algebra 2 homework

• Why MathPrep
• Free Math Practice

Algebra 2 - Normal Distributions

• Introduction - Normality of Data
• Data can be distributed in many ways. The data is arranged in a distribution to make it more organized and easier to analyze.
• How can we say that the data or scores are "normally distributed" ? Again, answers may be dependent on the behavior or type of scores. 
• If all measures of average (mean, median, mode) are equal, it is roughly considered as normally distributed.
• For instance, there are many real-life applications or naturally occurring phenomena with a distribution that has normal features like height, weight, test scores, the life expectancy of bulbs, crop yield in farms, etc. There are a few tall and a few small in terms of height. If there are many members in a certain population, there are numerous scores between the two extreme heights.
• If the scores are normally distributed, one good characteristic is these scores are closer to the mean and very close to each other. This is because the number of extreme low scores equals the number of extreme high scores. 
• By graphing the distribution of continuous random variables, it will be noticed that the shape of the graph resembles a bell-shaped curve.
• There are two tests to inspect the normality of the distribution using the SPSS (Statistical Package for Social Sciences): 1. Shapiro-Wilk Test and 2. Kolmogorov-Smirnov Test. 
• Using the p-value approach when using the Shapiro-Wilk Test is the default. If the computed p-value is less than or equal to 0.05, then the distribution is approximately normally distributed. If the computed p-value is greater than 0.05, then the distribution is not normally distributed.
• Properties of the Normal Distribution
• Normal Random Variables are continuous random variables whose distribution resembles a bell-shaped curve. 
• A continuous random variable is a random variable that takes on measurable characteristics. Examples are speed, time, height, weight, age, test scores, income, and the like.
• A continuous random variable whose probabilities are described by the normal distribution with mean  μ and standard deviation  σ is called a normally distributed random variable or normal random variable.
• Normally distributed scores are described by a symmetrical graph of a curve called the "NORMAL CURVE." Below is the sketch of a Standard Normal Curve.
• The equation (probability density function) of the normal curve is given by  f x = 1 σ 2 π e - 1 2 x - μ σ 2 or  f x = 1 σ 2 π e - 1 2 z 2 where x is any score in the distribution,  σ is the population standard deviation,  μ is the population mean,  e is the Euler's number, and  π ≈ 3 . 1416 .
• The distribution curve is bell-shaped.
• The curve is symmetrical about its center.
• The mean, median, and mode are at the center of the normal curve.
• The width of the curve is determined by the standard deviation of the distribution.
• The curve is asymptotic to the baseline, the x-axis.
• The total area under the normal curve is 1. The areas under the normal curve are expressed in probability values or percentages.
• Converting Normal Scores to Standard Scores
• Suppose that we have quantitative data that approximates a normal distribution with any mean μ and standard deviation  σ , we compute for probabilities of normal random variables by converting them to the standard normal distribution.
• z = X - μ σ   f o r   p o p u l a t i o n   d a t a
• z = X - x s   f o r   s a m p l e   d a t a
• These formulas are used to standardize normal raw scores from a distribution. The term "standardized" implies that normal scores will be converted to z-scores (standard scores).

Example 1. Given a raw score of 82 from the distribution, a population mean of 85, and a standard deviation of 1.20, what is its corresponding z-score? Assume that scores are normally distributed. 

X = 82 ,   μ = 85 ,   a n d   σ = 1 . 20

z = X - μ σ

z = 82 - 85 1 . 20 = - 2 . 50

Example 2. Find the normal raw score  X given that its corresponding z-score is 2.5 with a population mean of 84 and a standard deviation of 1.35.

We can use the formula  X = z σ + μ to find the normal raw score that corresponds to  z = 2 . 5 .

X = z σ + μ

X = 2 . 5 1 . 35 + 84

X = 87 . 375 ≈ 87 . 38

• Computing Probabilities using the Normal Curve
• To find the area or region under the normal curve is also to determine the probability value. 
• The area under the normal curve has values between 0 and 1. The total area under the normal curve is 1.
• Important Notations:

1.  P a < z < b means the probability that the z-score corresponding to a normal raw score is between the z-scores a   a n d   b .

2.  P z > a means the probability that the z-score corresponding to a normal raw score is greater than a .

3.  P ( z < b ) means the probability that the z-score corresponding to a normal raw score is less than b .

4.  P ( a < z < b ) and  P a ≤ z ≤ b are the same in terms of computing probabilities.

Using the Standard Normal Table

The above table helps you find the probability of normal scores. 

This provides the area (or probability) between  z = 0 and any value of  z .

Observe that the row entries are the z-scores. The column entries indicate the hundredth place of a z-score.

The z-score of 1.46 in the table is the probability that the z-score lies between z = 0 and  z = 1 . 46 .

Using the notion of symmetry in the normal distribution, the area between  z = 0 and any value to the left is equal to the area between  z = 0 and the point equidistant to the right.

Example 1. Calculate the probability  P 0 < z < 1 . 36 .

Use a table of the area under the normal curve or the Geogebra. 

The notation  P ( 0 < z < 1 . 36 ) implies we need to compute the area under the normal curve from 0 to 1.36. 

Since z-scores are positive, the area is located on the right portion of the curve.

Computing the area gives:  P ( 0 < z < 1 . 36 ) = 0 . 4131

Example 2. Find the area between  z = - 1 . 3   a n d   z = 2 . 4 .

The graph is shown below.

Since the area includes z = 0, then we add the areas  P ( - 1 . 3 < z < 0 )   a n d   P ( 0 < z < 2 . 4 )

P - 1 . 3 < z < 2 . 4 = P ( - 1 . 3 < z < 0 ) + P ( 0 < z < 2 . 4 ) = 0 . 4032 + 0 . 4918 = 0 . 8950   o r   89 . 50 %

Example 3. Compute the probability  P 1 . 25 < z < 2 . 0 .

The graph of  P ( 1 . 25 < z < 2 . 0 ) is presented below.

To compute the probability, we need to subtract  P ( 0 < z < 2 . 0 )   a n d   P ( 0 < z < 1 . 25 ) .

P ( 0 < z < 2 . 0 ) - P ( 0 < z < 1 . 25 ) = 0 . 4772 - 0 . 3944 = 0 . 0828   o r   8 . 28 %

Example 4. What is the area under a normal curve greater than z = 1 . 25 ?

The graph of this area is shown below:

P ( z > 1 . 25 ) is computed by subtracting the area from z = 0 to z = 1.25  P ( 0 < z < 1 . 25 ) from the half area of the normal curve, which is 0.50.

P ( z > 1 . 25 ) = P ( z > 0 ) - P 0 < z < 1 . 25 = 0 . 50 - 0 . 3944 = 0 . 1056   o r   10 . 56 %

• Solved Examples

Sample Problem: In a job fair, 2500 applicants applied for contractual work. Their mean age was 32, with a standard deviation of 8 years. What is the probability that applicants have ages between 25 and 35?

To determine the probability that the applicants have ages between 25 and 35, we convert the age of 25 and age of 35 to standard scores.

z = 25 - 32 8 = - 0 . 875   o r   - 0 . 88

z = 35 - 32 8 = 0 . 375   o r   0 . 38

The probability is written as  P ( - 0 . 88 < z < 0 . 38 ) .

P ( - 0 . 88 < z < 0 . 38 ) = P - 0 . 88 < z < 0 + P 0 < z < 0 . 38 = 0 . 4586   o r   45 . 86 %

• Cheat Sheet
• In computing the probabilities, we conventionally use the table of areas under normal curve. Geogebra is also applicable for automated results and graphical presentation of the normal curve areas or probabilities.
• To compute the probability P 0 < z < a , we determine the area under the normal curve from z = 0   t o   z = a . This portion of the normal curve is on the right side.
• To compute the probability  P - b < z < 0 , we find the area under the normal curve from  z = - b   t o   z = 0 . This portion of the normal cuve is on the left side.
• To calculate the probability  P - a < z < b , we find the area under the normal curve from  z = - a   t o   z = 0 and from  z = 0   t o   z = b . Add the obtained areas to get the total probability. The notation is  P ( - a < z < 0 ) + P ( 0 < z < b ) = P ( - a < z < b ) .
• When computing areas under normal curve involving a negative z-score and a positive z-score, we add the areas on the left of z = 0 and right of z = 0.
• When computing areas or probability in cases like  P a < z < b wherein both z-scores a   a n d   b are positive, we first compute the area from  z = 0   t o   z = b . Compute the area from  z = 0   t o   z = a . To find the area between z-scores  a   a n d   b , we subtract the area  z = 0   t o   z = a from the area z = 0   t o   z = b . In symbols, we have  P a < z < b = P 0 < z < b - P 0 < z < a .
• In cases like  P z > a , we subtract the probability  P 0 < z < a from 0.50 (half the area of the normal curve).
• In cases like  P z < - a , we subtract the probability  P - a < z < 0 from 0.50 (half the area of the normal curve).
• Blunder Areas
• In computing areas or probabilities in normal distribution, the result is always positive. The standard scores (z-scores) have negative and positive values.
• Normal distribution cannot be applicable to discrete random variables. All normal random variables are under continuous variables.

• 10 Comments

Quick Links

• Free eTextbooks
• Answer Keys

Category Archives: Algebra II Unit 13

New cc algebra ii statistics videos – by kirk.

A few weeks back, I published three new lessons to go with our final unit in Common Core Algebra II – Statistics. These lessons extended the work that we began in inferential statistics by introducing more formal ways of finding variation within sample statistics, such as sample means and proportions. By continuing to use simulation, we introduce students to the Central Limit Theorem for both sample means and sample proportions and extend this to a final lesson on margin of error. All of the lessons and their answer keys can be found at this post:

Simulation and Inferential Statistics – by Kirk

It took awhile, but I finally found time this weekend to record the videos for these lessons. I hope the videos are in time to help both students and teachers with this challenging material. I believe that these lessons will help students understand inferential statistics and how the results of the simulations can be extended to standard methods within statistics. Here are the YouTube links to the three new videos:

CC Alg II.Unit #13.Lesson #8.The Distribution of Sample Means

CC Alg II.Unit #13.Lesson #9.The Distribution of Sample Proportions

CC Alg II.Unit #13.Lesson #10.Margin of Error

I know that the numbering system here is in conflict with our final two lessons on regression. We will be re-numbering those last two lessons on regression as Lessons #11 and #12 respectively (as well as figuring out where they really make sense in the overall text).

Google Sheets for CC Alg II Statistical Simulation – by Kirk

As many of you know eMathInstruction offers online apps for performing the statistical simulations required in Common Core Algebra II. You can find links to those simulators at this post of mine:

Statistical Simulations and Inferential Statistics – by Kirk

The simulators work like a charm and I explain how to use them as well as how to use our TI based calculator programs in this YouTube video:

YouTube – Statistical Simulation Programs from eMathInstruction

One of the things I mention in the video is using Google Sheets to analyze the results of the simulations. Google Sheets is super convenient to use given that everyone has access to it. I also find it is much, much easier to use than Excel in terms of creating distribution histograms of the simulation results. Here’s a distribution of 1000 sample means that I created just now:

I show how to create these in the video. But, now I’ve created three Google Sheets for the simulators that people can just copy and paste their data into. I’ll give a links in a second. One of the nice things is that it will report the mean of the results, the standard deviation of the results, the 2.5% percentile, the 5% percentile, the 95% percentile, and the 97.5% percentile (hopefully you know why these are good to have). Here’s what a screen shot looks like:

As always, click on the images to see them much more clearly. Then hit the back arrow to come back to the article.

Anyhow, I’m going to supply links to all of these sheets. Now, and this is key, we don’t want to make it so that everyone is sharing and editing the same Sheet. So, I’m going to make the links “View Only.” All you have to do to have them all for yourself is make a copy:

Then, give it a cool new name and, as all things Google, it will magically now be in your Google Drive (assuming you have a Google account).

O.k. Here are the links to all of the Google Sheets:

Normal Sample (NORMSAMP) Google Sheet

Proportion Simulator (PSIMUL) Google Sheet

Difference in Sample Means (MEANCOMP) Google Sheet

Have fun with them. If you are going to share the links with students, consider making copies on your own drive and then sharing them from there.

So, I’ve made quite a few posts lately because we are near the time when teachers will be working through the very confusing topic of inferential statistics in Common Core Algebra II. With so little guidance from NYSED on the matter, we are left to sift through sample problems, standards, and (ugh!) the Modules. Ultimately, inferential statistics is a topic that is much, much too large to simply squish into Algebra II.

If anything, the state should almost consider renaming the course Algebra 2 with Statistics if they are honest about the content.

What’s ultimately very difficult is that the CC Standards and the GAISE report for inferential statistics emphasize (in fact insist) on using statistical simulation instead of formulas to develop things such as confidence intervals and margins of error. The theory, at least, is that the formulas are much more understandable (in future courses) if students develop a more intuitive grasp of inferential statistics by using probabilistic thinking generated through simulation.

I’m not sure I even understood what I just typed.

Still, I do get the idea of simulation. If we find that a random sample of 50 people have an average television view time per week of 18 hours per week with a standard deviation of 3.5 hours per week, it is unlikely that the population as a whole has an average view time outside of the interval 17 to 19 hours. Here are the results of running our online simulator where I assume a population mean of 18 with a standard deviation of 3.5 and a sample size of 50. Click on the image to see it more clearly:

Notice, there are no sample means under this simulation that fall outside of the range 17.0 to 19.2. This, in fact, would be a rough approximation for our confidence interval and half the width between these, i.e. (19.2-17.0)/2=1.1, would be a rough width for our margin of error. By the way, the actual margin of error is a theoretical 0.99 (2*stddev/sqrt(n)). Here’s a link to that simulator and our other two as well:

Sample Normal Distribution Web Based App (NORMSAMP)

Sample Proportion Simulator Web Based App (PSIMUL)

Difference of Sample Means Web Based App (MEANCOMP)

Now, there are more formulaic ways to grind out confidence intervals and margins of error. And, let’s face the fact, the state isn’t going to make them do statistical simulation on the Regents exam (June 1st); what it will do is make them interpret the results of those simulations.  I’ve created three new lessons that weren’t in Version 1 of our Common Core Algebra II text. I’ve posted them before, but I’m going to do it again, with the answer keys. Should you spend time on these more formulaic approaches to confidence interval and margin of error? That, I will leave to your professional judgement. I do tie the statistical simulation into these lessons, so that will get reinforced. Here are the lessons and their keys.

CCAlgII.Unit #13.Lesson #8.The Distribution of Sample Means

CCAlgII.Unit #13.Lesson #9.The Distribution of Sample Proportions

CCAlgII.Unit #13.Lesson #10.Margin of Error

CCAlgII.Unit #13.Lesson #8.The Distribution of Samples Means.Answer Key

CCAlgII.Unit #13.Lesson #9.The Distribution of Sample Proportions.Answer Key

CCAlgII.Unit #13.Lesson #10.Margin of Error.Answer Key

Sorry, but no videos on these yet. I am looking to do them before the end of April (so well before June 1st). Maybe consider flipping these or just giving kids the option to watch them and learn the content.

Statistical Simulators and Additional Lessons – Common Core Algebra II

Well, it’s getting to be crunch time in Common Core Algebra II now that April looms before us. As at Arlington, I’m sure that many districts are beginning Probability soon and will then turn to statistics in Common Core Algebra II.

I still am a little in the dark about the expectations that the state has in terms of what it wants students to know and be able to do when it comes to inferential statistics. Some of what we know comes from the CC Standards themselves, some comes from the Modules, and some comes from the sample questions. But, these three things are often contradictory.

At the end of the day, I feel like the original curriculum that I created does a good job at using Statistical Simulation to give students a good feel for how probability ties into inferential statistics. I’ve written statistical simulators for the TI-83/84 calculators and much, much faster simulators that are web-based. I describe all of these simulators in detail and supply links to the calculator code and web based apps in this earlier post of mine:

Statistical Simulation in Common Core Algebra II

I decided to add three additional lessons to the inferential statistics curriculum and am offering them up here in their pdf form. The answer keys will be available on the new answer key for CC Algebra II but aren’t available now. These three lessons take a more classic, formula based approach to inferential statistics, particularly margin of error for both means and proportions. The lessons are numbered a bit strangely because they will fit after all of the other simulation lessons but before our final two lesson on regression. Here they are:

TI Statistics Programs Updated for Common Core Algebra II

O.k. So, I’ve updated the statistics programs for the TI-83/84 that are needed to do the simulations for Unit #13 (Statistics) in the new Common Core Algebra II.

Here are the programs and a description of what they do. Click on the link to download one and then use TI-Connect (and a cable) to get them onto your calculator. If you are unsure how to do this, click on the link below:

Instructions for TI-Connect (Don’t download statistics programs from this link).

Here are the programs:

NORMSAMP is a very simple program. The user specifies a mean and standard deviation for an infinite population that is normally distributed. The program will then simulate a sample from this population of any size and perform multiple simulations. This output is a list of sample means and a list of sample standard deviations.

PSIMUL will pull a sample (of any size) from a population with a given proportion specified by the user. The population is assumed infinite (sampling with replacement essentially). The program will create as many samples as the user specifies and then exports a list of sample proportions.

This program allows the user to put in two lists with experimental results (Treatments 1 and 2). It then scrambles the data randomly and recalculates the differences in the treatment means. The user can do this as many times as they want in order to establish if the original difference was statistically significant.

Statistical Simulation – The Effect of the Treatment on Sample Means

More than any other topic, I was most worried about writing lessons for the statistics standards in Common Core Algebra II. I’ve been a high school math teacher in New York now for 16 years. The extent I’ve taught stat has been the normal distribution in half-standard deviation increments (all you Integrated Course III teachers know what I’m saying). I taught college statistics over a decade ago at Syracuse University, but it’s been awhile.

So the prospect of somehow trying to tackle a standard like:

S-IC.5 : Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

Whew! Right?

Well, these have been some of my very favorite lessons to write. That’s mostly because I’ve gotten to do math and understand math better than before. And that’s been thrilling.

Here’s the lesson and homework set that I just created to address this standard. In order to run do the lesson, you MUST download and install the TI83/84 program MEANCOMP. Place data in L1 and L2 and the program will do amazing simulations!

Unit #13.Lesson #7.The Difference in Sample Means

When the simulation is over it is good to SortA (sort ascending) your results that are in L3. In a newer version of the program I will be posting, this is done automatically.

Statistical Simulation Programs for Download

O.k. So, after a bit of legwork, it turns out that it is not hard to post TI-83/84 programs to this site and its not even hard to have you download them.

A few days ago, I posted a document detailing three Statistics programs I wrote for the TI-83/84 to do the statistical simulations that are required in the new Common Core Algebra II. I created the simulation programs based on the simulations recommended in the G.A.I.S.E. Report (Guidelines for the Assessment and Instruction in Statistics Education). The New York State Department of Education has been referencing that report highly whenever they speak of the statistics in Common Core Algebra II. Here’s another posting of the article I wrote explaining the three programs:

Unit #13.Simulation Code for TI Programs

But, now, you don’t need to type them in. I want to sincerely thank Jenn Sauer, a great educator from my neck of the woods over in the beautiful town of Saugerties. Jenn was kind enough to brainstorm the statistics a bit with me and give me super helpful suggestions on the programs themselves. Here they are for download.

Save them in a safe spot where you will know where they are.

O.k. Now, how do you get them onto your calculator especially if you are not used to doing this? First things first, you need to download TI Connect . It’s free and easy. Here’s the link:

TI Connect Link

Alright, once you have TI Connect you’ll see this main screen:

You’ll need to connect your TI-84 or 83 to the computer using a USB chord. Hopefully you didn’t throw it away when you got your calculator.

TI Connect will automatically find your calculator. But, if you want to make sure, once you have the calculator plugged into the computer, turned on, and you have TI Connect open, click on the icon:

This will allow you to find the calculator and browse its contents (programs, lists, apps). The first time it looks for your calculator, it may take a bit of time. Anyhow, go back to the home screen (the one up above). Now to get those programs onto your calculator. Click on the Send to Device icon:

You will then navigate to whatever folder you stored the programs in. Select them all at once and then send them to your calculator.

It will take a little bit (maybe a minute), but TI Connect will keep you up to date on the progress.

That’s it. I hope you can use these as you prepare for the coming of statistics in Common Core Algebra II. I will be putting out an entire statistics unit in late March, so wait until then to see the full power of these programs. Until then, play around with them.

But, be sure to print out the first document I posted in this entry. It serves as the defacto users guide to the programs. Please note that the code has been slightly changed from what is in the documents. Here it is again for your convenience:

P.S. I just clicked on the links, downloaded the programs and sent them to my absolutely older TI-83+ without a hitch.

So, the statistics standards in Common Core Algebra II specify a lot of use of simulation to establish confidence intervals and other statistical measures of variability. I like these standards, but was having a hard time thinking about how to really fulfill them in a meaningful way.

So, I attacked them head on. Based on work from the G.A.I.S.E. Report (google it), I created three programs for the TI-84+ operating system (they work fine on the 83+) that perform these simulations.

The first program allows you to pick a random sample from a population to calculate sample means and see their distributions. The second allows you to pull a sample from a population with a known proportion, and then see a distribution of sample proportions. Finally, the last program scrambles up treatment data to test the difference in treatment means.

I have not gotten them up on TI’s site, yet, but plan to later on in the spring. I’m offering them now, in case people want to type one or all of them into their calculators and start to really play with these simulation standards.

The document includes an explanation of the three programs and their code.

• Algebra I Projects and Labs
• Algebra I Unit 1
• Algebra I Unit 10
• Algebra I Unit 11
• Algebra I Unit 2
• Algebra I Unit 3
• Algebra I Unit 4
• Algebra I Unit 5
• Algebra I Unit 6
• Algebra I Unit 7
• Algebra I Unit 8
• Algebra I Unit 9
• Algebra II Unit 10
• Algebra II Unit 13
• Algebra II Unit 2
• Algebra II Unit 4
• Algebra II Unit 6
• Algebra II Unit 7
• Arlington Algebra Project
• Assessments
• Common Core Algebra I Unit #1
• Common Core Algebra I Unit #2
• Common Core Algebra I Unit #3
• Extended Problems
• Kirk's General Rambling
• Sharing Resources (Everything but assessments)
• The Business Side of Things
• The Common Core Standards
• Uncategorized

[email protected]

[email protected]

Fax orders to: (845)-758-6717

eMath Instruction Inc.

10 Fruit Bud Lane

Red Hook, NY, 12571

©2016 eMath Instruction Inc.

HIGH SCHOOL

• ACT Tutoring
• SAT Tutoring
• PSAT Tutoring
• ASPIRE Tutoring
• SHSAT Tutoring
• STAAR Tutoring

GRADUATE SCHOOL

• MCAT Tutoring
• GRE Tutoring
• LSAT Tutoring
• GMAT Tutoring
• AIMS Tutoring
• HSPT Tutoring
• ISAT Tutoring
• SSAT Tutoring

Search 50+ Tests

Loading Page

math tutoring

• Elementary Math
• Pre-Calculus
• Trigonometry

science tutoring

Foreign languages.

• Mandarin Chinese

elementary tutoring

• Computer Science

Search 350+ Subjects

• Video Overview
• Tutor Selection Process
• Online Tutoring
• Mobile Tutoring
• Instant Tutoring
• How We Operate
• Our Guarantee
• Impact of Tutoring
• Reviews & Testimonials
• About Varsity Tutors

Algebra II : Normal Distributions

Study concepts, example questions & explanations for algebra ii, all algebra ii resources, example questions, example question #1 : graphing data.

Not enough information to answer the question.

Example Question #491 : Algebra Ii

The scores for your recent english test follow a normal distribution pattern. The mean was a 75 and the standard deviation was 4 points. What percentage of the scores were below a 67?

Use the 68-95-99.7 rule which states that 68% of the data is within 1 standard deviation (in either direction) of the mean, 95% is within 2 standard deviations, and 99.7% is within 3 standard deviations of the mean.

In this case, 95% of the students' scores were between:

75-(2 x 4) and 75+(2 x 4)

or between a 67 and a 83, with equal amounts of the leftover 5% of scores above and below those scores. This would mean that 2.5% of the students scored below a 67% on the test.

Your class just took a math test. The mean test score was a 78 with a standard deviation of 2 points. With this being the case, 99.7% of the class scored between what two scores?

Use the 68-95-99.7 rule which states that 68% of the data is within 1 standard deviation (in either direction) of the mean, 95% is within 2 standard deviations, and 99.7% is within 3 standard deviations (in either direction) of the mean.

In this case, 99.7% of the students' scores were between 3 standard deviation above the mean and 3 standard deviations below the mean:

78-(2 x 3) and 78+(2 x 3)

or between a 72 and an 84. 

All of the following statements regarding a  Normal Distribution are true except:

All of these are true.

Between two graphs of normally-distributed data sets, the graph of the set with a higher standard deviation will be wider than the graph of the set with a lower standard deviation.

A graph of a normally-distributed data set is symmetrical.

A graph of a normally-distributed data set will have a single, central peak at the mean of the data set that it describes.

The shape of the graph of a normally-distributed data set is dependent upon the mean and the standard deviation of the data set that it describes.

The graph of a normally-distributed data set is symmetrical.

The graph of a normally-distributed data set has a single, central peak at the mean of the data set that it describes.

The graph of a normally-distributed data set will vary based only upon the mean and the standard deviation of the set that it describes.

The graph of a normally-distributed data set with a higher standard deviation will be wider than the graph of a normally-distributed data set with a lower standard deviation.

The question asks us to find the statement that is  not true ; however, all statements are true so the correct response is "All of these are true."

Report an issue with this question

If you've found an issue with this question, please let us know. With the help of the community we can continue to improve our educational resources.

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC 101 S. Hanley Rd, Suite 300 St. Louis, MO 63105

Or fill out the form below:

Contact Information

Complaint details.

• Share on Facebook
• Tweet This Resource
• Pin This Resource

Common Core Algebra 2, Unit 12

The four lessons in Common Core Algebra 2, Unit 12, introduce high schoolers to probability. Topics covered include compound probability and independence, as well as normal distributions. Each topic is introduced with a video, and then viewers practice concepts by completing problem worksheets. The end-of-unit review asks scholars to demonstrate what they have learned by identifying outcomes and sample spaces, determining compound and conditional probabilities, and investigating situations involving normal distributions.

Common Core

Introduction to Probability (.html)

Practice packet (.pdf), practice solutions (.pdf), corrective assignment (.pdf).

Introduction to Probability

Compound events and independence (.html), corrective assignment 1 (.pdf), corrective assignment 2 (.pdf).

Compound Probability and Independence

Normal distributions (.html).

Normal Distributions

Unit 12 review: probability and normal distributions (.html).

Unit 12 Review: Probability and Normal Distributions