Call Now to Set Up Tutoring:

(888) 888-0446

AP Statistics : Inference

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 : Confidence Intervals

300 hundred eggs were randomly chosen from a gravid female salmon and individually weighed. The mean weight was 0.978 g with a standard deviation of 0.042. Find the 95% confidence interval for the mean weight of the salmon eggs (because it is a large n, use the standard normal distribution).

Possible Answers:

(-1.96 g, 1.96 g)

(0.973 g, 0.9827 g)

(0.563 g, 0.294 g)

(1.042 g, 0.827 g)

(0.974 g, 0.982 g)

Correct answer:

(0.973 g, 0.9827 g)

Explanation:

Because we have such a large sample size, we are using the standard normal or z-distribution to calculate the confidence interval.

Formula:

$\bar{x} \pm z \cdot \frac{\sigma}{\sqrt{n}}$

$\bar{x}= 0.978 g$

$\sigma= 0.042$

We must find the appropriate z-value based on the given $\alpha$ for 95% confidence:

$1-\frac{\alpha}{2}= 1-\left(\frac{0.05}{2}\right)= 0.975$

Then, find the associated z-score using the z-table for

$z _{0.975} = 1.96$

Now we fill in the formula with our values from the problem to find the 95% CI.

$0.978 \pm 1.96 \left( \frac{0.042}{\sqrt{300}}\right)= (0.973 g, 0.9827 g)$

Report an Error

Example Question #1 : Confidence Intervals

A sample of n=56 observations of 0₂ consumption by adult western fence lizards gave the following statistics:

Mean= 249 mm^3/g/hr

s= 24 mm^3/g/hr

Find the $90\%$ confidence limit for the mean 0₂ consumption by adult western fence lizards.

Possible Answers:

(119.23 mm^3/g/hr, 219.23 mm^3/g/hr)

(187.34 mm^3/g/hr, 225.73 mm^3/g/hr)

(242.57 mm^3/g/hr, 255.42 mm^3/g/hr)

(273 mm^3/g/hr, 225 mm^3/g/hr)

(243.63 mm^3/g/hr, 254.36 mm^3/g/hr)

Correct answer:

(243.63 mm^3/g/hr, 254.36 mm^3/g/hr)

Explanation:

Because we are only given the sample standard deviation we will use the t-distribution to calculate the confidence interval.

Appropriate Formula:

$= \bar{x} \pm t \cdot \frac{s}{\sqrt{n}}$

Now we must identify our variables:

$\bar{x}=249$

s=24

n=56

We must find the appropriate t-value based on the given $\alpha$

t-value at 90% confidence:

$t_{\frac{\alpha }{2}, n-1}= t _{\frac{0.10}{2}}= t_{0.05, 55}$

Look up t-value for 0.05, 55 , so t-value= ~ 1.6735

90% CI becomes:

$249 \pm 1.6735\left(\frac{24}{\sqrt{56}}\right)$

= (243.63 mm^3/g/hr, 254.36 mm^3/g/hr)

Report an Error

Example Question #1 : How To Find Confidence Intervals For A Mean

Subject	Horn Length (in)	Subject	Horn Length (in)
1	19.1	11	11.6
2	14.7	12	18.5
3	10.2	13	28.7
4	16.1	14	15.3
5	13.9	15	13.5
6	12.0	16	7.7
7	20.7	17	17.2
8	8.6	18	19.0
9	24.2	19	20.9
10	17.3	20	21.3

The data above represents measurements of the horn lengths of African water buffalo that were raised on calcium supplements. Construct a 95% confidence interval for the population mean for horn length after supplments.

Possible Answers:

(14.47 in, 18.56 in)

(21.81 in, 11.23 in)

(13.99 in, 19.07 in)

(14.57 in, 18.28 in)

(18.84 in, 16.21 in)

Correct answer:

(13.99 in, 19.07 in)

Explanation:

First you must calculate the sample mean and sample standard deviation of the sample.

$\bar{x}=16.53$

$\sigma=5.29$

Because we do not know the population standard deviation we will use the t-distribution to calculate the confidence intervals. We must use standard error in this formula because we are working with the standard deviation of the sampling distribution.

Formula:

$\bar{x} \pm t\cdot \frac{s}{\sqrt{n}}$

To find the appropriate t-value for 95% confidence interval:

$\alpha =0.05, \frac{0.05}{2}=0.025$

df=n-1= 20-1 = 19

Look up $t_{\frac{\alpha }{2}, 19}$ in t-table and the corresponding t-value = 2.093.

Thus the 95% confidence interval is:

$16.53 \pm 2.093\left(\frac{5.29}{\sqrt{19}}\right)= (13.99 mm Hg, 19.07 mm Hg)$

Report an Error

Example Question #1 : Confidence Intervals

The population standard deviation is 7. Our sample size is 36.

What is the 95% margin of error for:

1) the population mean

2) the sample mean

Possible Answers:

1) 13.720

2) 2.287

1) 11

2) 3

1) 12.266

2) 3.711

1) 14.567

2) 4.445

1) 15.554

2) 3.656

Correct answer:

1) 13.720

2) 2.287

Explanation:

For 95% confidence, Z = 1.96.

1) The population M.O.E. =

$1.96 \times 7 = 13.720$

2) The sample standard deviation =

$7\div\sqrt{36}=1.167$

The sample M.O.E. =

$1.96\times1.167=2.287$

Report an Error

Example Question #1 : Confidence Intervals

The $95\%$ confidence interval created for the difference in means between two training programs for middle distance college runners is $\small \left ( -4.5, 6.5\right )$ . The variable being measured is the improvement in seconds of mile times over the course of a season. One program has more speed work and intervals, while the other focuses more on distance training.

What does the confidence interval tell us about the difference in the two programs?

Possible Answers:

Zero is in the interval, so do not reject the null. There is no evidence that one program is better than the other.

The confidence interval is large, so one program is clearly better at reducing mile times than the other.

The mean improvement of $\small 1$ second $\small \left ( (6.5-4.5)\times 0.5 = 1\right )$ is too small to matter, so reject the null. There is no evidence that one program is better at reducing miles times.

Zero is not in the interval, so reject the null. This is evidence that one program is significantly better at reducing mile times.

$\small 6.5$ is greater than $\small \left | -4.5\right |$ , so reject the null. This is evidence that one program is better at reducing mile times than the other.

Correct answer:

Zero is in the interval, so do not reject the null. There is no evidence that one program is better than the other.

Explanation:

For there to be a statistically significant difference in the training programs, the 95% confidence interval cannot include zero. $\small \left ( -4.5, 6.5\right )$ includes zero, so we can't say that one program is significantly better than the other.

Report an Error

Example Question #4 : Confidence Intervals

Possible Answers:

Correct answer:

Explanation:

Report an Error

Example Question #2 : Confidence Intervals

Possible Answers:

Correct answer:

Explanation:

Report an Error

Example Question #1 : Confidence Intervals

In a simple random sample of 1000 people, 261 are left-handed. Find a $95\%$ confidence interval for the true proportion of left-handed people in the entire population.

Possible Answers:

$0.261\pm \sqrt{\frac{(0.261)(0.739)}{1000}}$

$0.261\pm1.96 \sqrt{\frac{(0.261)(0.739)}{1000}}$

The sample is not sufficiently large. A confidence interval cannot be used.

$0.261\pm 1.64\sqrt{\frac{(0.25)(0.75)}{1000}}$

$0.261\pm 1.64\sqrt{\frac{(0.261)(0.739)}{1000}}$

Correct answer:

$0.261\pm1.96 \sqrt{\frac{(0.261)(0.739)}{1000}}$

Explanation:

Step 1: Determine the approporiate formula. In this case, the statistic is a proportion because the question says 261 people out of 1000 are left-handed. We want to use a confidence interval for proportions. The equation is

$\hat{p}\pm z^{*} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Step 2: $\hat{p}$ , or the sample proportion is equal to $\frac{261}{1000} = 0.261$

Step 3: The questions asks for a 95% confidence interval. You can assume a normal distribution because of the large sample size. Therefore, in a normal distribution, 95% of the data is contained within $z=\pm 1.96$ .

$z^{*}=1.96$

Step 4: Substitue all the values into the equation to get the answer.

$0.261\pm1.96 \sqrt{\frac{(0.261)(0.739)}{1000}}$

Report an Error

Example Question #11 : Confidence Intervals

A health insurance executive suspects that 50% of insurance applications are incomplete (i.e. critical data is missing from the application). He wants to gather a sample to estimate the true proportion. He wants to be within 5% of the true mean with 95% confidence.

What sample size does he need?

Possible Answers:

395

375

385

405

365

Correct answer:

385

Explanation:

$n=pq\times(Z/E)^{2}$

$n=.5\times.5\times(1.96/.05)^{2} = 384.16 = 385$

Report an Error

Example Question #1 : How To Find Confidence Intervals For The Slope Of A Least Squares Regression Line

You estimate a regression model with $\beta = 3.4$ and SE = .024 , where $\beta$ is the beta coefficient and is the standard error. Construct 95% confidence intervals for $\beta$ .

Possible Answers:

$\left [ 3.4,3.45 \right ]$

$\left [ 3.35,3.45 \right ]$

$\left [ 3.1,3.45 \right ]$

Correct answer:

$\left [ 3.35,3.45 \right ]$

Explanation:

To construct 95% confidence intervals for $\beta$ , we simply take the coefficient and add/subtract $1.96\times the\ standard\ error\ of\ \beta$ . This is because $\beta$ is assumed to follow a symmetrical distribution (the normal), and 95% of the values in the sampling distribution are contained within 1.96 standard errors of $\beta$ .

Report an Error

View AP Statistics Tutors

Rajpal
Certified Tutor

Central University of Rajasthan, Bachelor of Science, Statistics. Delhi University, Master of Science, Statistics.

View AP Statistics Tutors

Muhammad
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Mechanical Engineering.

View AP Statistics Tutors

Michelle
Certified Tutor

University of Rhode Island, Bachelor of Science, Marine Biology. University of Rhode Island, Master of Arts, High School Teac...

All AP Statistics Resources

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

Popular Subjects

Math Tutors in Houston, English Tutors in Washington DC, English Tutors in New York City, GMAT Tutors in Seattle, Math Tutors in Denver, MCAT Tutors in San Diego, Computer Science Tutors in San Diego, LSAT Tutors in Atlanta, GMAT Tutors in San Diego, Chemistry Tutors in Dallas Fort Worth

Popular Courses & Classes

Spanish Courses & Classes in Phoenix, Spanish Courses & Classes in Miami, LSAT Courses & Classes in Los Angeles, ACT Courses & Classes in Miami, GMAT Courses & Classes in Boston, GMAT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Washington DC, MCAT Courses & Classes in New York City, SSAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Seattle

Popular Test Prep

GRE Test Prep in Chicago, ACT Test Prep in Seattle, LSAT Test Prep in Philadelphia, GMAT Test Prep in San Francisco-Bay Area, SAT Test Prep in Phoenix, SSAT Test Prep in Phoenix, ISEE Test Prep in Philadelphia, ISEE Test Prep in Washington DC, SAT Test Prep in Seattle, LSAT 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 : Inference

Study concepts, example questions & explanations for AP Statistics

All AP Statistics Resources

Example Questions

Example Question #1 : Confidence Intervals

Example Question #1 : Confidence Intervals

Example Question #1 : How To Find Confidence Intervals For A Mean

Example Question #1 : Confidence Intervals

Example Question #1 : Confidence Intervals

Example Question #4 : Confidence Intervals

Example Question #2 : Confidence Intervals

Example Question #1 : Confidence Intervals

Example Question #11 : Confidence Intervals

Example Question #1 : How To Find Confidence Intervals For The Slope Of A Least Squares Regression Line

All AP Statistics Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: