Introduction To Statistics And Data Analysis
Introduction To Statistics And Data Analysis
6th Edition
ISBN: 9781337793612
Author: PECK, Roxy.
Publisher: Cengage Learning,
bartleby

Videos

Textbook Question
Book Icon
Chapter 10, Problem 92CR

According to a large national survey conducted by the Pew Research Center (“What Americans Think About NSA Surveillance, National Security and Privacy,” May 2, 2015, pewresearch.org, retrieved December 1, 2016), 54% of adult Americans disapprove of the National Security Agency collecting records of phone and Internet data. Suppose that this estimate was based on a random sample of 1000 adult Americans.

  1. a. Is there convincing evidence that a majority of adult Americans feel this way? Test the relevant hypotheses using a 0.05 significance level.
  2. b. The actual sample size was much larger than 1000. If you had used the actual sample size when doing the calculations for the test in Part (a), would the P-value have been larger than, the same as, or smaller than the P-value you obtained in Part (a)? Provide a justification for your answer.

a.

Expert Solution
Check Mark
To determine

Check whether it can be concluded that the proportion of adults of Country A who disapprove the National Security Agency collecting records of phone and internet data is greater than 0.5.

Answer to Problem 92CR

Yes, the sample data provide the evidence that the proportion of adults of Country A who disapprove the National Security Agency collecting records of phone and internet data is greater than 0.5.

Explanation of Solution

Calculation:

In a sample of 1,000 adults of Country A, 54% of adults were founded to disapprove the National Security Agency collecting records of phone and internet data.

Step 1:

Population characteristic of interest:

Let p be the population characteristic of interest.

In this context, p denotes the population proportion of adults of Country A who disapprove the National Security Agency collecting records of phone and internet data.

Step 2:

Null hypothesis: H0:p=0.5

That is, the proportion of adults of Country A who disapprove the National Security Agency collecting records of phone and internet data is 0.5.

Step 3:

Alternative hypothesis: Ha:p>0.5

That is, the the proportion of adults of Country A who disapprove the National Security Agency collecting records of phone and internet data is greater than 0.5.

Step 4:

Significance level, α:

It is assumed that the significance level, α=0.05.

Step 5:

Test statistic, z:

z=p^pp(1p)n,

Where, p^ be the sample proportion, p is the hypothesized proportion, and n is the sample size.

Substitute the hypothesized proportion, p=0.5, in the test statistic.

z=p^0.50.5(10.5)n

Here, the sample proportion p^ is not known.

Step 6:

Assumptions:

  • Let p^ be the sample proportion from a random sample.
  • The large sample z test can be used if the sample size n satisfies the conditions: n(hypothesized value)10 and n(1hypothesized value)10.
  • The sample size should not be greater than 10% of the population size.

Requirement check:

  • It is given that the sample is from a random sample.
  • Check the conditions: n(hypothesized value)10 and n(1hypothesized value)10.

n(hypothesized value)=np=1,000(0.5)=500>10

n(1hypothesized value)=n(1p)=1,000(10.5)=1,000(0.5)=500>10

Since n(hypothesized value) and n(1hypothesized value) are greater than 10, the sample size requirement is met.

Therefore, the large sample z test is appropriate.

  • Although, the population size is not known, it is reasonable to assume that the sample size of 1,000 acts as the representative for all adults of Country A. It is also definite that the sample size is less than 10% of the population of the adults of Country A.

Step 7:

The value of the test statistic is obtained as follows:

z=p^0.50.5(10.5)n

By substituting the value of sample proportion, the test statistic is obtained as follows:

z=0.540.50.5(10.5)1,000=0.040.01582.53

Thus, the value of test statistic is 2.53.

Step 8:

P-value:

In this context, the alternative hypothesis denotes that the test carried out is right-tailed test. Therefore, the P-value is the area under the z curve and to the right of the calculated z value.

The P-value for the test statistic value of 2.53 is obtained as follows:

P-value=Area to the Right of 2.53=P(z>2.53)=1P(z<2.53)

Use Standard Normal Probabilities (Cumulative z curve areas) table to find the z-value.

Procedure:

For z at 2.53:

  • Locate 2.5 in the left column of the table.
  • Obtain the value in the corresponding row below .03.

That is, P(z<2.53)=0.9943.

The probability of the event is as given below:

P(z>2.53)=1P(z<2.53)=10.9943=0.0057

Thus, the P-value for the test statistic of 2.53 is 0.0057.

Step 9:

Decision rule:

If P-valueSignificance level, then reject the null hypothesis H0.

If P-value>Significance level, then fail to reject the null hypothesis H0.

Here, the P-value of 0.0057 is less than the significance level 0.05.

That is, P-value(=0.0057)<Significance level(=0.05).

The decision is that the null hypothesis is rejected.

Conclusion:

Hence, the sample data provide the evidence that the proportion of adults of Country A who disapprove the National Security Agency collecting records of phone and internet data is greater than 0.5.

b.

Expert Solution
Check Mark
To determine

Justify whether the P-value is larger, same as, or smaller than the P-value obtained in Part (a) when the actual sample is used.

Answer to Problem 92CR

The P-value would be smaller than that of the P-value obtained in Part (a).

Explanation of Solution

Greater sample size has the chance to increase the test statistic and also reduces the P-value.

If the actual sample size is much higher than that used in Part (a), then there is more chance to have greater test statistic. In such a way, the P-value would also have been less than that obtained in Part (a).

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Chapter 10 Solutions

Introduction To Statistics And Data Analysis

Ch. 10.1 - The article Facebook Use and Academic Performance...Ch. 10.2 - Researchers at Bostons Childrens Hospital and...Ch. 10.2 - The paper Breast MRI as an Adjunct to Mammography...Ch. 10.2 - Medical personnel are required to report suspected...Ch. 10.2 - How accurate are DNA paternity tests? By comparing...Ch. 10.2 - A television manufacturer claims that (at least)...Ch. 10.2 - A manufacturer of hand-held calculators receives...Ch. 10.2 - Water specimens are taken from water used for...Ch. 10.2 - Suppose that for a particular hypothesis test, the...Ch. 10.2 - Suppose that you arc an inspector for the Fish and...Ch. 10.2 - The paper Living Near Nuclear Power Plants and...Ch. 10.2 - An automobile manufacturer is considering using...Ch. 10.3 - Use the definition of the P-value to explain the...Ch. 10.3 - For which of the following P-values will the null...Ch. 10.3 - Pairs of P-values and significance levels, , are...Ch. 10.3 - Pairs of P-values and significance levels are...Ch. 10.3 - Let p denote the proportion of students at a...Ch. 10.3 - Assuming a random sample from a large population,...Ch. 10.3 - In a survey conducted by CareerBuilder.com,...Ch. 10.3 - In a survey of 1000 women age 22 to 35 who work...Ch. 10.3 - The paper Debt Literacy, Financial Experiences and...Ch. 10.3 - Most Like it Hot is the title of a press release...Ch. 10.3 - In a survey of 1005 adult Americans, 46% indicated...Ch. 10.3 - The article Euthanasia Still Acceptable to Solid...Ch. 10.3 - The report Digital Democracy Survey (Deloitte...Ch. 10.3 - The article How to Block Nuisance Calls (The...Ch. 10.3 - The article Facebook Use and Academic Performance...Ch. 10.3 - The article Public Acceptability in the UK and the...Ch. 10.3 - The article Cops Get Screened for Digital Dirt...Ch. 10.3 - Refer back to the previous exercise. The actual...Ch. 10.3 - The report 2007 Electronic Monitoring ...Ch. 10.3 - The United States Elections Project...Ch. 10.4 - Give as much information as you can about the...Ch. 10.4 - Give as much information as you can about the...Ch. 10.4 - Give as much information as you can about the...Ch. 10.4 - Paint used to paint lines on roads must reflect...Ch. 10.4 - A certain pen has been designed so that actual...Ch. 10.4 - Prob. 48ECh. 10.4 - The paper Playing Active Video Games Increases...Ch. 10.4 - A study of fast-food intake is described in the...Ch. 10.4 - Refer to the study and hypothesis test of the...Ch. 10.4 - The report 2016 Salary Survey Executive Summary...Ch. 10.4 - The Economist collects data each year on the price...Ch. 10.4 - The report Majoring in Money: How American College...Ch. 10.4 - The authors of the paper Changes in Quantity,...Ch. 10.4 - Medical research has shown that repeated wrist...Ch. 10.4 - A comprehensive study conducted by the National...Ch. 10.4 - In a study of media use, each person in a large...Ch. 10.4 - Refer to the study description and sample...Ch. 10.4 - The paper titled Music for Pain Relief (The...Ch. 10.4 - Many consumers pay careful attention to stated...Ch. 10.4 - Much concern has been expressed regarding the...Ch. 10.5 - The power of a test is influenced by the sample...Ch. 10.5 - Water specimens are taken from water used for...Ch. 10.5 - Prob. 65ECh. 10.5 - Let denote the mean lifetime (in hours) for a...Ch. 10.5 - The city council in a large city has become...Ch. 10.5 - The amount of shaft wear after a fixed mileage was...Ch. 10.5 - Optical fibers are used in telecommunications to...Ch. 10.5 - Let denote the mean diameter for bearings of a...Ch. 10.5 - Use the information given in the previous exercise...Ch. 10.6 - In 2006, Boston Scientific sought approval for a...Ch. 10.6 - The article Boy or Girl: Which Gender Baby Would...Ch. 10.7 - We are only beginning to learn about the long-term...Ch. 10.7 - A study of hospitalized patients who develop...Ch. 10.7 - Prob. 76ECh. 10.7 - Prob. 77ECh. 10.7 - Prob. 78ECh. 10.7 - Prob. 79ECh. 10.8 - Prob. 80ECh. 10.8 - Prob. 81ECh. 10.8 - Prob. 83ECh. 10.8 - Prob. 84ECh. 10 - The report A Crisis in Civic Education (American...Ch. 10 - In a national survey of 2013 adults, 1590...Ch. 10 - Students at the Akademia Podlaka conducted an...Ch. 10 - Prob. 89CRCh. 10 - People in a random sample of 236 students enrolled...Ch. 10 - According to a large national survey conducted by...Ch. 10 - In a representative sample of adult Americans age...Ch. 10 - The article Facebook Use and Academic Performance...Ch. 10 - A number of initiatives on the topic of legalized...Ch. 10 - Duck hunting in populated areas faces opposition...Ch. 10 - Past experience has indicated that the response...Ch. 10 - An automobile manufacturer who wishes to advertise...Ch. 10 - A student organization uses the proceeds from a...Ch. 10 - A hot tub manufacturer advertises that with its...Ch. 10 - Prob. 1CRECh. 10 - Prob. 2CRECh. 10 - Prob. 3CRECh. 10 - Prob. 4CRECh. 10 - Prob. 5CRECh. 10 - The article Should Canada Allow Direct-to-Consumer...Ch. 10 - Prob. 7CRECh. 10 - It probably wouldnt surprise you to know that...Ch. 10 - The article Doctors Cite Burnout in Mistakes (San...Ch. 10 - The National Geographic Society conducted a study...Ch. 10 - Heinz Plays Catch-up After Under-Filling Ketchup...Ch. 10 - In a survey conducted by Yahoo Small Business,...Ch. 10 - In an AP-AOL sports poll (Associated Press,...Ch. 10 - The article titled 13% of Americans Dont Use the...Ch. 10 - A survey of teenagers and parents in Canada...Ch. 10 - The same survey referenced in the previous...
Knowledge Booster
Background pattern image
Statistics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Text book image
Holt Mcdougal Larson Pre-algebra: Student Edition...
Algebra
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Text book image
Glencoe Algebra 1, Student Edition, 9780079039897...
Algebra
ISBN:9780079039897
Author:Carter
Publisher:McGraw Hill
Text book image
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Text book image
College Algebra
Algebra
ISBN:9781938168383
Author:Jay Abramson
Publisher:OpenStax
Hypothesis Testing using Confidence Interval Approach; Author: BUM2413 Applied Statistics UMP;https://www.youtube.com/watch?v=Hq1l3e9pLyY;License: Standard YouTube License, CC-BY
Hypothesis Testing - Difference of Two Means - Student's -Distribution & Normal Distribution; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=UcZwyzwWU7o;License: Standard Youtube License