22-23 APS Unit 4 Packet - KEY

pdf

School

Case Western Reserve University *

*We aren’t endorsed by this school

Course

601

Subject

Statistics

Date

Jan 9, 2024

Type

pdf

Pages

Uploaded by CaptainCoyotePerson973

Page 10 Example of Explanation Paragraph for selecting a simple random sample from a random digit table Let’s say there are 20 subjects in the population, and you want to choose 8 for your sample. Gaffney Engelmann Chase Petulante Kelly Kim Csulak McDonald Jessell Bozzo Liu Ostrowski Chang Chaney Catita Goetz Davis Gruenspecht Taylor Melis 1. Start by labeling each student. Show those labels ሺ01, 02, etcሻ 01 Gaffney 06 Engelmann 11 Chase 16 Petulante 02 Kelly 07 Kim 12 Csulak 17 McDonald 03 Jessell 08 Bozzo 13 Liu 18 Ostrowski 04 Chang 09 Chaney 14 Catita 19 Goetz 05 Davis 10 Gruenspecht 15 Taylor 20 Melis 2. Then, write the following paragraph: Label each student with a 2-digit number from 01 – 20, where each 2-digit number represents one student. Starting at the given line of the random digit table, look at two digits at a time, reading from left to right, ignoring 00, numbers above 20, and any repeats. Stop when you have selected 8 distinct students for the sample. 3. Show your work on the line from the random digit table provided, by circling numbers that will end up in your sample and crossing out those not in your sample ሺsee example belowሻ 25630 30845 46204 00347 21353 66727 09342 52458 09113 68890 63910 73205 01456 66910 00535 56819 16589 31568 16546 54572 4. Then, list those members of the population in your sample ሺwrite the names separately from the population list givenሻ Jessell Chase Bozzo Kim Melis Davis Chaney Gaffney

Page 11 1. Method with technology: Label each student with a 2-digit number from 1 – 28 as shown above. Use a random number generator to generate 5 distinct numbers between 1-28. The students corresponding to those numbers will be interviewed about the quality of the course. Method with digit table: Label each student with a 2-digit number from 01 – 28 as shown above. Starting at line 136 of the random digit table ሺshown aboveሻ, look at two digits at a time, reading from left to right, ignoring 00, numbers above 28, and any repeats. Stop when you have selected 5 numbers. The numbers we obtained are 08, 14, 20, 09 and 24. This corresponds to Dewald, Fernandez, Hicks, Petrucelli and Rubin. These students will be interviewed about the quality of the course. 2. Identify the population and the sample in each of the following situation. a. A realtor is interested in the median selling price of homes in Worcester County, Massachusetts. She collects data on the selling price of 50 homes. Population: homes in Worcester County Sample: 50 homes in Worchester County b. A psychologist is concerned about the health of veterans who served in combat. She examines 25 veterans to assess whether or not they are showing signs of PTSD. Population: veterans who served in combat Sample: 25 combat veterans c. An educator asks 20 seniors from Eastern Connecticut State University whether or not they had taken an online course while at the university. Population: seniors from ECSU Sample: 20 seniors from ECSU

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Page 12 AP Statistics – Sampling Practice AP Problem Sample Answers: Scoring Rubric:

Page 13 1. Identify which of the following methods is used in the following sampling scenarios: SRS, stratified, cluster, convenience, systematic, voluntary response. a. From a class of 25 students, the teacher selects the last 5 to enter the room. Convenience sample b. A professor numbers his students from 1 to 200, places those numbers in a hat, mixes thoroughly and chooses 6 numbers without looking. Simple Random Sample ሺSRSሻ c. From a group of 100 employees, the manager randomly selects 12 of the 60 women and 8 of the 40 men to form a sample of 20. Stratified random sample d. In a class of 12 boys and 12 girls, a teacher selects 5 students by numbering the boys 01 to 12 and the girls 13 to 24 and uses a random number table to choose 5 numbers between 01 and 24. Simple Random Sample ሺSRSሻ e. In a movie queue, an interviewer selects the fifth person at exactly 1:00 PM to ask questions about their movie preferences. She then selects every 20 th person in the queue for questioning. Systematic sample f. The director of high school education for a school district randomly selects two elementary schools from 20 and sends a survey to the parents of each student in the school. Cluster sample 2. Explain why the following method does not produce an SRS: A large elementary school has 15 classes with 24 children in each classroom. A sample of 30 is chosen by the following procedure: Each of the 15 teachers selects 2 children from his or her classroom to be in the sample by numbering the children from 01 to 24 and using a random number generator to select two different distinct numbers between 01 and 24. The two children who correspond to those numbers are in the sample. Not every possible combination of 30 students is possible with this method. This procedure instead produces a stratified random sample. 3. A high school principal decides to conduct a survey of the senior class by selecting one government class and interviewing each student in that class. Why does this method suffer from bias? Seniors in other classes don’t have any chance of being interviewed. The principal is using a convenience sample.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Page 18 Identify what is wrong in each of these surveys. Be sure to explain. 1. The mayor of Springfield is interested in finding out the average age of people in the city. He obtains a list of all of the landline telephones in the city, and then contacts a simple random sample of 300 people. He uses the data from the sample to estimate the average age of all the people in the city. a. What is wrong with this survey? Anyone without a landline phone has no chance of participating in the survey. b. Do you think the Mayor will over or underestimate the true mean age of people in Springfield? Why? The mayor will probably overestimate the true mean age, because we would expect that younger people would be less likely to have a landline phone, especially since younger people may not have a permanent residence. 2. The administration at a school wants to know the proportion of students that did all of their homework last night. They select a simple random sample of 100 students and send an email to each of them asking if they did all of their homework last night. Of the 40 responses, 36 of the students said that they did all of their homework last night ሺ90%ሻ. a. What is wrong with this survey? Most students would not want to admit that they did not do their homework, especially not to their administrators. This likely is why 60% of the sample did not respond. It also means that some students out of the 40 that did respond may not have been truthful. b. Do you think the administration will over or underestimate the true proportion of students who did all of their homework last night? Why? The administration will probably overestimate the true proportion of students who did all of their homework last night, because the group of students who did not do their homework would be much more likely to not respond or to be dishonest in their response than the group of students who did do their homework. 3. Peter wants to know the proportion of people in his neighborhood who support the Boy Scouts. He takes a random sample of 30 homes and visits them dressed in his uniform. a. What is wrong with this survey? People may not want to respond negatively to an actual boy scout. b. Do you think Peter will over or underestimate the true proportion of his neighbors who support the Boy Scouts? Why? Peter will probably overestimate the true proportion who support the Boy Scouts, because the appearance of the interviewer probably caused some people to respond dishonestly.

Page 19

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Page 20 1. In June 2008 Parade magazine posed the following question: “Should drivers be banned from using all cell phones?” Readers were encouraged to vote online at www.parade.com. The July 13, 2008, issue of Parade reported the results: 2407 ሺ85%ሻ said “Yes” and 410 ሺ15%ሻ said “No.” a. What type of sample did the Parade survey obtain? Voluntary Response Sample b. Explain why this sampling method is biased. People who do use their phone while driving may be less likely to respond. c. Is 85% likely to be greater than or less than the percentage of all adults who believe that cell- phone use while driving should be banned? Why? It is likely to be greater than the percentage of all adults who believe that cell phone use while driving should be banned, because responding “No” would be an unpopular opinion. 2. To help eliminate bias, a reporter from Parade decides she will go out and ask people in person if they think drivers should be banned from using cell phones. She lives close to the local high school so she goes to the parking lot at 3:00 pm and asks the first 100 people she sees. a. What type of sample did the reporter obtain? Convenience sample b. Explain why this sampling method is biased. Anyone who is not at the local high school at 3:00 PM has no chance of being surveyed. c. How could Parade magazine avoid the bias described above? Some element of random sampling is required. The specific methodology for gathering the data depends on the population of interest, for example, car owners, cell phone owners, registered voters, etc. 3. A survey paid for by makers of disposable diapers found that 84% of the sample opposed banning disposable diapers. Here is the actual question: “It is estimated that disposable diapers account for less than 2% of the trash in today’s landfills. In contrast, beverage containers, third-class mail, and yard wastes are estimated to account for about 21% of the trash in landfills. Given this, in your opinion, would it be fair to ban disposable diapers?” Do you think the estimate of 84% is less than, greater than, or about equal to the percent of all people in the population who would oppose banning disposable diapers? Explain your reasoning. This is probably an overestimate of the percent that oppose banning disposable diapers, because the question is worded to make disposable diapers seem less impactful on the environment.

Page 21 4. A factory runs 24 hours a day, producing wood pencils on three 8-hour shifts— day, evening, and overnight. In the last stage of manufacturing, the pencils are packaged in boxes of 10 pencils each. Each day a sample of 300 pencils is selected and inspected for quality. a. Describe how to select a stratified random sample of 300 pencils. Explain your choice of strata. For each shift ሺday, evening, overnightሻ, label all pencils using a unique serial number ሺor just a number from 1 to n where n is the number of pencils in that shiftሻ. Use a random number generator to select 100 distinct serial numbers and then inspect the pencils corresponding to those serial numbers. We use the three shifts as strata because we would expect different employees to be supervising the factory for each shift, so we therefore expect the pencils in each shift to be similar to each other in quality level, and we may want to check whether there are differences between shifts. b. Describe how to select a cluster sample of 300 pencils. Explain your choice of clusters. We could use the boxes of pencils as clusters. We could label all boxes of pencils from 1 to n where n is the number of boxes of pencils. Randomly select 30 numbers from 1 to n and inspect all of the pencils in each of the boxes corresponding to the 30 randomly chosen numbers. c. Explain a benefit of using a stratified random sample and a benefit of using a cluster random sample in this context. A stratified sample could potentially give us a more precise estimate of the true mean pencil quality, whereas a cluster sample would be more practical to implement because we would not have to assemble 300 pencils from potentially different boxes. 5. Each of the following is a possible source of bias in a sample survey. Name the type of bias that could result. a. The sample is chosen at random from a telephone directory. Undercoverage bias, since anyone without a listed telephone number cannot participate. b. Some people cannot be contacted in five calls. Nonresponse bias, since the selected respondents are not participating. c. Interviewers choose people walking by on the sidewalk to interview. Undercoverage bias, since anyone not walking by the interviewer cannot participate.

Page 22 AP Statistics – Confounding Variables Last year a local high school offered an after school SAT prep class that students could volunteer to take. 44 students took the course and then took the SAT. The average SAT score for this group was 1220. The average SAT score for all students who did not take the prep class was 1050. 1. Is the situation described an observational study or an experiment? Explain. This is an observational study because the students are not forced to take the course or not take the course, so no treatments are being imposed. 2. Identify the explanatory variable and the response variable. The explanatory variable is whether they took the course. The response variable is the SAT score. 3. Can you conclude that taking the prep course will cause a student’s SAT score to increase? Why or why not? We cannot conclude that taking the prep course will cause a higher SAT score because we have not controlled for confounding variables . For example, amount of time studying outside of school is probably associated with higher SAT scores. It may be that the students who voluntarily take the SAT course spend more time studying outside of school than the students who don’t. 4. Identify as many other possible variables that you can that may explain why the SAT scores are higher for those who took the prep course than for those who did not.  Future college and career plans  Number of other extracurriculars the students are involved in  Whether the students have a job after school  Whether or not students have transportation after school  Level of motivation relative to education 5. Design an experiment that would allow us to determine if the SAT prep causes an increase in SAT scores. Be as thorough as possible. Randomly choose 100 juniors to be part of the experiment. Assign each junior a number from 1 to 100 and use a random number generator to select 50 distinct numbers between 1 and 100. The juniors who were assigned those numbers will take the SAT course. The remaining 50 juniors do not take the course. After the course is over, all the 100 subjects take the SAT. We compare the average SAT scores of the two groups.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Page 23 Write the definitions of the following vocabulary words. 1. Treatment – A specific condition applied to the individuals in an experiment. 2. Experimental Units – The smallest collection of individuals to which treatments are applied. 3. Subjects – Experimental units that are human beings. 4. Factor – Another name for an explanatory variable in an experiment. 5. Bias (in experiments) – Any element of an experiment that leads to results that systematically differ from the true effects of the treatments. 6. Confounding – When two variables are associated in such a way that their effects on a response variable cannot be distinguished from each other. 7. Lurking Variable – A variable that is not being measured in an experiment that is confounded with the explanatory variable(s) that are being measured. 8. Control Group – A treatment group within an experiment that receives no treatment, or receives an “inactive” treatment. 9. Placebo Effect – A phenomenon in which human subjects respond favorably to a “dummy” treatment. 10. Completely Randomized Design – Any experiment in which the experimental units are assigned to the treatments completely by chance. 11. Block – A group of experimental units that are known before the experiment to be similar in some way that is expected to affect the response to the treatments. 12. Randomized Block Design – Any experiment in which the random assignment of experimental units to treatments is carried out separately within each block. 13. Matched Pairs Design – A type of randomized block design in which only two treatments are being compared between “matched” experimental units. Blocks in matched pair experiments consist of only 1 or 2 experimental units per block. 14. Double-Blind Experiment – Any experiment in which neither the subjects nor those who interact with the subjects know which treatment a subject has received. 15. Statistically significant – An observed effect in an experiment so large that it would rarely occur just by chance.

List the four principles of experimental design. Briefly explain each. 1. Comparison, which refers to using a design that compares two or more treatments. 2. Random assignment, which refers to using chance to assign experimental units to treatments. 3. Control, which refers to keeping other variables that might affect the response variable consistent for all treatment groups. 4. Replication, which refers to using enough experimental units in each treatment group so that any differences in the effects of the treatments can be distinguished from chance differences between the groups.

Page 24 AP Statistics – Placebo Effect Example Similar to the video, Mr. Schuler wants to use a beverage to test the affect that caffeine can have on heart rate. Here is an initial plan:  measure initial pulse rate  give each student some caffeine ሺCoca-Colaሻ  wait for a specified time  measure final pulse rate  compare final and initial rates 2. What are some problems with this plan? What other variables will be sources of variability in pulse rates?  We have nothing to compare the results to in order to see if there was any significant change in the heart rate.  Similar to the video, since people know they are getting caffeine, any difference in pulse rate could be caused by the placebo effect.  Other ingredients in Coca-Cola could be causing variability in pulse rates, such as the amount of processed sugar. 3. Go back up to your list in #2 and propose a solution to each problem.  Create a second treatment group that gets a drink without caffeine.  Do not inform the test subjects whether they are getting a drink with caffeine or with no caffeine.  Make sure that the second treatment group gets a drink very similar to normal Coca-Cola. Also make sure that subjects are randomly assigned to the two treatment groups. 4. Design an experiment to test the effect that caffeine has on heart rate. Measure the initial pulse rate of all of your test subjects. Then, have all of your subjects write their name on a piece of paper and put it into a hat. Mix the pieces of paper thoroughly and draw names one at a time until half of the subjects have been drawn. Assign those subjects to drink normal Coca-Cola and assign the remaining subjects to drink caffeine-free Coca-Cola. Wait for a specified amount of time and measure the final pulse rate. Find the differences between the final pulse rates and initial pulse rates for the Coca-Cola drinkers and compare this to the differences between final and initial pulse rates for the caffeine-free Coca-Cola drinkers.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Page 25 A local high school has decided to offer an SAT prep class. It will be offered in two different formats: online or classroom teacher. The counselors want to know which teaching method will yield higher SAT scores so they have allowed us to set up an experiment. 50 students have signed up to take some form of the SAT prep class. ሺ20 seniors and 30 juniorsሻ 1. Outline a completely randomized design to compare the two treatments. Assign each student a number from 1 to 50. Use a random number generator to select 25 distinct numbers between 1 and 50. The students corresponding to those numbers will be assigned to take the course online. The remaining students will be assigned to take the course from a classroom teacher. All 50 students take the SAT and the scores for each group are compared. 2. The counselors hypothesize that the online vs. classroom results could be greatly affected by the grade level of students that were put into each treatment group. They know that seniors generally score better on the SAT than juniors. Describe an alternative design for this experiment that would better address this situation. Assign each junior a number from 1 to 30. Use a random number generator to select 15 distinct numbers between 1 and 30. The juniors corresponding to those numbers will be assigned to take the course online. The remaining juniors will be assigned to take the course from a classroom teacher. Then, assign each senior a number from 1 to 20. Use a random number generator to select 10 distinct numbers between 1 and 20. The seniors corresponding to those numbers will be assigned to take the course online. The remaining seniors will be assigned to take the course from a classroom teacher. All 50 students take the SAT and the scores for each of the two treatment groups are compared. 3. The counselors are now worried that a student’s GPA is certainly going to affect their SAT score. Let’s look only at the juniors. Describe an alternative design for this experiment that ensures that GPAs are evenly distributed into the two treatment groups. Put all of the 30 juniors in order from smallest GPA to greatest. Take the two students with the highest GPA and pair them up. Assign one student to the number 1 and the other student to the number 2 and use a random number generator to select one number between 1 and 2. The student corresponding to the generated number will take the course online and the other subject will take the course with a classroom teacher. Repeat this process until all of the juniors are paired and each junior is assigned to one of the two treatment groups. Compare the differences in SAT scores within each pair.

Page 26 AP Statistics – Matched Pairs Practice Within each block, assign one dog the number 1 and the other to number 2. Use a random number generator to select one number between 1 and 2. The dog corresponding to that number will be assigned the supplement and the other dog will be assigned the placebo.

Page 27 Here are four proposed studies for investigating the claim that listening to music while studying will help improve your GPA. Suppose we found that the mean GPA of students who listen to music is significantly lower than the mean GPA of students who didn’t listen to music. What conclusions could we make? Can we generalize and can we determine causation? 1. Get all the students in your statistics class to participate in a study. Ask them whether or not they study with music on and divide them into two groups based on their answer to this question. Random sample? No Random assignment? No Conclusion: For the students in this class, the students that listen to music while studying happen to have higher GPAs than students who don’t listen to music while studying. This represents a census instead of a random sample, and because we did not do random assignment, we have no evidence of a cause-and- effect relationship between listening to music and GPA. 2. Select a random sample of students from your school to participate in a study. Ask them whether or not they study with music on and divide them into two groups based on their answer to this question. Random sample? Yes Random assignment? No Conclusion: For the students in the school, the students that listen to music while studying happen to have higher GPAs than students who don’t listen to music while studying. This represents a random sample, but because we did not do random assignment, we have no evidence of a cause-and-effect relationship between listening to music and GPA. 3. Get all the students in your statistics class to participate in a study. Randomly assign half of the students to listen to music while studying for the entire semester and have the remaining half abstain from listening to music while studying. Random sample? No Random assignment? Yes Conclusion: This represents a census instead of a random sample. Since we did random assignment, we have evidence of a cause-and-effect relationship between listening to music and GPA for students in this class only. 4. Select a random sample of students from your school to participate in a study. Randomly assign half of the students to listen to music while studying for the entire semester and have the remaining half abstain from listening to music while studying. Random sample? Yes Random assignment? Yes Conclusion: This represents a random sample, and because we did random assignment, we have evidence of a cause-and-effect relationship between listening to music and GPA for students in this school.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Page 28 1. Does reducing screen brightness increase battery life in laptop computers? To find out, researchers obtained 30 new laptops of the same brand. They chose 15 of the computers at random and adjusted their screens to the brightest setting. The other 15 laptop screens were left at the default setting— moderate brightness. Researchers then measured how long each machine’s battery lasted. Was this an observational study or an experiment? Justify your answer. This is an experiment because the researchers imposed two treatments ሺdefault setting vs. maximum brightnessሻ. 2. Many utility companies have introduced programs to encourage energy conservation among their customers. An electric company considers placing small digital displays in households to show current electricity use and what the cost would be if this use continued for a month. Will the displays reduce electricity use? One cheaper approach is to give customers a chart and information about monitoring their electricity use from their outside meter. Would this method work almost as well? The company decides to conduct an experiment using 60 households to compare these two approaches ሺdisplay, chartሻ with a group of customers who receive information about energy consumption but no help in monitoring electricity use. a. Explain why it is important to have a control group that didn’t get the display or the chart. It allows us to see how much electricity customers normally use to be able to compare to the level of use with a digital display or with a chart. b. What is the purpose of randomly assigning treatments in this context? It reduces bias and allows us to conclude whether the results of the experiment are statistically significant. c. Create a completely randomized design for the experiment using a sample of 60 households. Make sure to describe how to randomly assign the treatments to the 60 households. Measure the average electricity use per month for an entire year for all 60 households. Then, label all households from 1 to 60. Use a random number generator to choose 20 distinct numbers between 1 and 60. Assign the households that correspond to those numbers a digital display. Generate another 20 distinct numbers between 1 and 60 and assign the households that correspond to those numbers a chart. The remaining households get no treatment. Measure the electricity output for all 60 houses for another year. Compare the average electricity conservation between the three groups.

Page 29 Questions 3–4 refer to the following setting: According to an ABC News article, “Teenagers who eat with their families at least five times a week are more likely to get better grades in school.” This finding was based on a sample survey conducted by researchers at Columbia University. 3. What are the explanatory and response variables? The explanatory variable is whether or not the teenagers ate with family at least 5 times a week. The response variable is grades ሺperhaps measured by GPAሻ. 4. Explain clearly why such a study cannot establish a cause-and-effect relationship. Suggest a variable that may be confounded with whether families eat dinner together. We can’t say that there is a cause-and-effect relationship because there may be a confounding variable that has a differential effect on the two groups ሺteens who eat with their families at least 5 times a week and teens who do notሻ that also has an effect on grades. For example, level of parental involvement in their children’s education likely leads to better grades. We might also expect that teens who eat with family at least 5 times a week would be more likely to have parents who are more involved in their children’s education than teens who do not eat with family at least 5 times a week.. Researchers would like to design an experiment to compare the effectiveness of three different advertisements for a new television series featuring the work of Jane Austen. There are 300 volunteers available for the experiment. 5. Describe a completely randomized design to compare the effectiveness of the three advertisements. Assign each volunteers a number from 1 to 300. Use a random number generator to select 100 distinct numbers between 1 and 300. The volunteers corresponding to those numbers will be shown the first advertisement. Repeat this process to select another 100 distinct numbers between 1 and 300. The volunteers corresponding to those numbers will be shown the second advertisement. The remaining volunteers will be shown the third advertisement. All 300 volunteers will take a survey after watching the advertisement that will ask them how likely they are to watch the series, and we will compare the responses between the three groups.

6. Describe a randomized block design for this experiment. Justify your choice of blocks. Give the volunteers a survey before the experiment to assess whether or not they are familiar with Jane Austen. Split the volunteers into two groups, one that is familiar with Jane Austen and one that is not. Randomly assign them to watch the three advertisements in a similar manner as the procedure described in ሺ5ሻ, so that there is approximately 1/3 of each of the two blocks in each treatment group. All 300 volunteers will take a survey after watching the advertisement that will ask them how likely they are to watch the series, and we will compare the responses between the six groups. 7. Why might a randomized block design be preferable in this context? We would expect people familiar with Jane Austen to be more likely to watch a series about her than people who are not familiar with Austen, so blocking by familiarity will lead to more precise estimate of the effectiveness of each advertisement.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

22-23 APS Unit 4 Packet - KEY

Related Documents