We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

AP Statistics : Measures of Independent Random Variables

Study concepts, example questions & explanations for AP Statistics

Create An Account

All AP Statistics Resources

4 Diagnostic Tests 140 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Find The Mean Of The Sum Of Independent Random Variables

If is a random variable with a mean of and standard deviation of , what is the mean and standard deviation of ?

Possible Answers:

$\mu = 20, \sigma = 10$

$\mu = 10, \sigma = 20$

$\mu = 20, \sigma = 40$

$\mu = 20, \sigma = \sqrt{20}$

$\mu = 10, \sigma = 10$

Correct answer:

$\mu = 20, \sigma = \sqrt{20}$

Explanation:

Remember how the mean and standard deviation of a random variable are affected when it is multiplied by a constant.

$E(cX)=c\cdot E(X)$

$\mu = 2\cdot 10 = 20$

$\sigma=\sqrt{c^2\cdot 5}$

$\sigma = \sqrt{2^2 \cdot 5} = \sqrt{20}$

Report an Error

Example Question #1 : How To Find The Mean Of The Sum Of Independent Random Variables

If you have ten independent random variables $\small X_1,X_2,...,X_{10}$ , normally distributed with mean $\small 0$ and variance $\small 1$ , what is the distribution of the average of the random variables, $\small \small \frac{X_1+X_2+...+X_{10}}{10}?$

Possible Answers:

Chi-square distribution with $\small 10$ degrees of freedom.

Normal distribution with mean $\small 0$ and variance $\small 10$ .

Normal distribution with mean $\small 0$ and variance $\small 1$ .

Normal distribution with with mean $\small 0$ and variance $\small \frac{1}{10}$ .

Correct answer:

Normal distribution with with mean $\small 0$ and variance $\small \frac{1}{10}$ .

Explanation:

Any linear combination of independent random variables is also normally distributed with the mean and variance depending on the weights on the random variables. The mean is additive in the sense that

$\small mean\left(\frac{X_1+...+X_{10}}{10}\right)=\frac{1}{10}\left(mean(X_1)+...+mean(X_{10})\right)$

Each $\small mean(X_i)$ is $\small 0$ , so the sum is equal to zero.

This means the sum of the average

$\small \small \small \frac{X_1+X_2+...+X_{10}}{10}$ is $\small 0$ .

The variance satisfies

$Var\left(\frac{X_1+X_2+...+X_{10}}{10}\right)=\frac{1}{10^2}Var(X_1+...+X_{10})$

$\small =\frac{1}{100}(Var(X_1)+...Var(X_{10}))=\frac{10}{100}=\frac{1}{10}$ because of independence.

This means that the average is normally distributed with mean $\small 0$ and variance $\small \frac{1}{10}$ .

Report an Error

Example Question #1 : Measures Of Independent Random Variables

Suppose you have three independent normally distributed random variables, $\small X_1,X_2,X_3$ , such that

$\small X_1$ has mean $\small \frac{1}{4}$ and variance $\small \frac{1}{2}$ ,

$\small X_2$ has mean $\small -1$ and variance $\small \frac{3}{8}$ ,

$\small X_3$ has mean $\small \frac{3}{4}$ and variance $\small \frac{1}{8}$ .

What is the probability that the sum, $\small X_1+X_2+X_3$ , is less than $\small -1$ ?

Possible Answers:

$\small 0.3431$

$\small 0.8413$

$\small 0.6827$

$\small 0.1587$

Correct answer:

$\small 0.1587$

Explanation:

There is a relatively simple way of doing this problem. The sum of any set of independent normal random variables is also distributed normally. So $\small X_1+X_2+X_3$ has a normal distribution. Now we can compute the mean and variance. The mean is additive:

$\small mean(X_1+X_2+X_3)=mean(X_1)+mean(X_2)+mean(X_3)$

$\small \small =\frac{1}{4}-1+\frac{3}{4}=0$

Variance is also additive in some sense, when the random variables are independent:

$\small \small Var(X_1+X_2+X_3)=Var(X_1)+Var(X_2)+Var(X_3)$

$\small =\frac{1}{2}+\frac{3}{8}+\frac{1}{8}=1$

Thus, $\small X_1+X_2+X_3$ is normally distributed with mean $\small 0$ and variance $\small 1$ .

This sum is a standard normal distribution.

The chance that $\small X_1+X_2+X_3<-1$ is thus $\small 0.1587$ , if we use a normal table.

Report an Error

Example Question #1 : How To Find The Mean Of The Sum Of Independent Random Variables

An experiment is conducted on the watermelons that were grown on a small farm. They want to compare the average weight of the melons grown this year to the average weight of last year's melons. Find the mean of this year's watermelons using the following weights:

4 lbs, 5 lbs, 5 lbs, 7 lbs, 10 lbs, 3.5 lbs, 6 lbs, 5 lbs, 6 lbs, 9 lbs

Possible Answers:

3lbs

5lbs

6.05 lbs

5.5lbs

10lbs

Correct answer:

6.05 lbs

Explanation:

To find the mean you sum up all of your values then divide by the total amount of values. The total sum of the weights is 60.5lbs and there are 10 melons. $60.5\div10=60.5lbs$

Report an Error

Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

A high school calculus exam is administered to a group of students. Upon grading the exam, it was found that the mean score was 95 with a standard deviation of 12. If one student's z score is 1.10, what is the score that she received on her test?

Possible Answers:

109.2

105.3

108.2

107.2

110.1

Correct answer:

108.2

Explanation:

The z-score equation is given as: z = (X - μ) / σ, where X is the value of the element, μ is the mean of the population, and σ is the standard deviation. To solve for the student's test score (X):

X = ( z * σ) + 95 = ( 1.10 * 12) + 95 = 108.2.

Report an Error

Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

and are independent random variables. If has a mean of and standard deviation of while variable has a mean of and a standard deviation of , what are the mean and standard deviation of 2A + B ?

Possible Answers:

$\mu = 19, \sigma = 20$

$\mu = 15, \sigma = 15$

$\mu = 19, \sigma = 10$

$\mu = 29, \sigma = 20$

$\mu = 9, \sigma = 10$

Correct answer:

$\mu = 19, \sigma = 10$

Explanation:

First, find that has $\mu=2\cdot 5 = 10$ and standard deviation $\sigma = 2 \cdot 4 = 8$ .

Then find the mean and standard deviation of 2A + B .

$\mu = 10 + 9 = 19$

$\sigma = \sqrt{8^2 + 6^2} = \sqrt{100} = 10$

Report an Error

Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

Consider the discrete random variable $\small X$ that takes the following values with the corresponding probabilities:

with $\small \small P(X=1)=\frac{1}{4}$
$\small X=2$ with $\small \small P(X=2)=\frac{1}{4}$
$\small X=3$ with $\small \small P(X=3)=\frac{1}{4}$
$\small X=4$ with $\small P(X=4)=\frac{1}{4}$

Compute the variance of the distribution.

Possible Answers:

$\small .625$

$\small 1.25$

$\small 1.75$

$\small 0.3125$

Correct answer:

$\small 1.25$

Explanation:

The variance of a discrete random variable is computed as

$\small Var(X)=\sum_x (x-\mathbb{E}[x])^2P(X=x)$

for all the values of $\small x$ that the random variable $\small X$ can take.

First, we compute $\small \mathbb{E}[X]$ , which is the expected value. In this case, it is $\small 2.5$ .

So we have

$\small \small Var(X)=\frac{1}{4}(1-2.5)^2+\frac{1}{4}(2-2.5)^2+\frac{1}{4}(3-2.5)^2+\frac{1}{4}(4-2.5)^2$

$\small =\frac{1}{4}(2(1.5^2)+2(0.5^2))=\frac{1}{4}(2(2.25)+2(0.25))=\frac{1}{4}(4.5+0.5)$

$\small \small =\frac{5}{4}=1.25$

Report an Error

Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

Clothes 4 Kids uses standard boxes to ship their clothing orders and the mean weight of the clothing packed in the boxes is pounds. The standard deviation is pounds. The mean weight of the boxes is pound with a standard deviation of 0.15 pounds. The mean weight of the plastic packaging is pounds per box, with a 0.25 pound standard deviation. What is the standard deviation of the weights of the packed boxes?

Possible Answers:

$2.021\:\mbox{pounds}$

$24.00\:\mbox{pounds}$

$20.01\:\mbox{pounds}$

$4.085\:\mbox{pounds}$

$21.01\:\mbox{pounds}$

Correct answer:

$2.021\:\mbox{pounds}$

Explanation:

Note that the weight of a packed box = weight of books + weight of box + weight of packing material used.

It is given that $\sigma_{books} = 2\: \mbox{pounds}, \sigma_{box} = 0.15\: \mbox{pounds}, \mbox{and}\: \sigma_{plastic packaging} = 0.25\: \mbox{pounds}$ .

The calculation of the standard deviation of the weights of the packed boxes is

$\\ \sigma_{packed boxes} = \sqrt{\sigma_{clothes} ^{2} + \sigma_{clothes} ^{2} + \sigma _{plastic packaging}^{2}} \\ \\ = \sqrt{2^{2}+ 0.15^{2} + 0.25^{2}} \\ = 2.021\: \mbox{pounds}$

Report an Error

View AP Statistics Tutors

Odysseas
Certified Tutor

University of New Hampshire at Manchester, Master of Science, Civil Engineering. National Technical University of Athens, Doc...

View AP Statistics Tutors

Kayley
Certified Tutor

New Mexico State University-Main Campus, Bachelor of Science, Biology, General.

View AP Statistics Tutors

Nishunk
Certified Tutor

University of Delhi, Bachelor of Science, Mathematics and Statistics. University of Delhi, Master of Science, Applied Mathema...

All AP Statistics Resources

4 Diagnostic Tests 140 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

English Tutors in Seattle, Computer Science Tutors in Seattle, Biology Tutors in Miami, GRE Tutors in Chicago, Statistics Tutors in Dallas Fort Worth, Calculus Tutors in Houston, SAT Tutors in Los Angeles, Math Tutors in Los Angeles, Reading Tutors in Atlanta, GRE Tutors in Los Angeles

Popular Courses & Classes

ISEE Courses & Classes in Miami, SSAT Courses & Classes in Chicago, GRE Courses & Classes in Houston, GMAT Courses & Classes in Denver, GMAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Washington DC, SSAT Courses & Classes in Washington DC, SAT Courses & Classes in Philadelphia, GMAT Courses & Classes in Houston, ISEE Courses & Classes in Boston

Popular Test Prep

LSAT Test Prep in Houston, GRE Test Prep in Philadelphia, SAT Test Prep in Washington DC, ISEE Test Prep in Houston, LSAT Test Prep in Washington DC, GRE Test Prep in Miami, SAT Test Prep in Seattle, ACT Test Prep in Denver, GRE Test Prep in Chicago, GRE Test Prep in Atlanta

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:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

AP Statistics : Measures of Independent Random Variables

Study concepts, example questions & explanations for AP Statistics

All AP Statistics Resources

Example Questions

Example Question #1 : How To Find The Mean Of The Sum Of Independent Random Variables

Example Question #1 : How To Find The Mean Of The Sum Of Independent Random Variables

Example Question #1 : Measures Of Independent Random Variables

Example Question #1 : How To Find The Mean Of The Sum Of Independent Random Variables

Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

Example Question #1 : How To Find The Standard Deviation Of The Sum Of Independent Random Variables

All AP Statistics Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: