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NC Math 3
Statistics
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What is the Difference Between a Population and a Sample? Population: the entire group of individuals that we want information about Census: A complete count of the population Sample: A part of the population that we actually examine in order to gather information. Use the sample to generalize to population
Why would we not use a census all the time? 1. Not accurate- Look at the U.S. census: it has a huge amount of error, and it takes a long time to compile the data making the data obsolete by the time we get it!
2. Very expensive- taking a census of any population takes time is VERY costly to do!
3. Perhaps impossible- Suppose you wanted to know the average weight of the white-tail deer population in Texas, would it be feasible to do a census?
4. If using destructive sampling, you would destroy population Examples: breaking strength of soda bottles, lifetime of flashlight batteries
What is the Difference Between a Statistic and a Parameter? A statistic and a parameter are very similar. They are both descriptions of groups, like “50% of dog owners prefer X Brand dog food.” The difference between a statistic and a parameter is that a statistic describes a sample, while a parameter describes an entire population. Statistics vary because samples vary. A parameter never changes, because the whole population is surveyed to find the parameter. Suppose you randomly poll voters in an election. You find that 55% of the population plans to vote for candidate A. That is a statistic. Why? You only asked a sample of the population who they are voting for. Inferential statistics enables you to make an educated guess about a population parameter based on a statistic computed from a sample randomly drawn from that population. You could ask a class of third graders who likes vanilla ice cream. 90% raise their hands. You have a parameter: 90% of that class likes vanilla ice cream. You know this because you asked everyone in the class. Steps to tell the difference between a statistic and a parameter: Step 1: Ask yourself, is this a fact about the whole population? Sometimes that’s easy to figure out. For example, with small populations, you usually have a parameter because the groups are small enough to measure:
10% of US senators voted for a particular measure. There are only 100 US Senators, you can count what every single one of them voted, so this is a parameter.
40% of 1,211 students at a particular elementary school got below a 3 on a standardized test. You know this because you have each and every students’ test score, so this is a parameter.
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33% of 120 workers at a particular bike factory were paid less than $20,000 per year. You have the payroll data for all of the workers, so this is a parameter.
Step 2: Ask yourself, is this a fact about a sample of a population? If in doubt, think about the time and cost involved in surveying an entire population. If you can’t imagine anyone wanting to spend the time or the money to survey a large number (or impossible number) in a certain group, then you almost certainly are looking at a statistic.
60% of US residents agree with the latest health care proposal. It’s not possible to actually ask hundreds of millions of people whether they agree. Researchers have to just take samples and calculate the rest, so this is a statistic.
45% of Jacksonville, Florida residents report that they have been to at least one Jaguars game. It’s very doubtful that anyone polled in excess of a million people for this data. They took a sample, so this is a statistic.
30% of dog owners scoop poop after their dog. It’s impossible to survey all dog owners—no one keeps an accurate track of exactly how many people own dogs. This data had to be from a sample, so this is a statistic.
A Tale of a Population and Two Samples Take the quiz at tinyurl.com/ncm23quiz Your Score: ____________
Class Mean: ____________ We will randomly select 5 people for sample 1. Write in the scores and calculate the mean. _______ _______ _______ _______ _______ Mean ________ We will randomly select 5 people for sample 2. Write in the scores and calculate the mean. _______ _______ _______ _______ _______ Mean ________ Was there a difference in the means for the samples? Compared to the population? Why? What can you do to adjust for this in an experiment?
Law of Large Numbers - as sample size grows, the mean of the sample (the statistic) will get closer and closer to the mean of the whole population (the parameter).
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PARAMETER OR STATISTIC? 1. For the studies described, identify the population, sample, population parameters, and sample statistics:
a) The Gallup Organization conducted a poll of 1003 Americans in its household panel to determine what percentage of people plan to cancel their summer vacation because of the increase in gasoline prices.
b) Harris Interactive surveyed 2435 U.S. adults nationwide and asked them to rate quality of
American public schools. c) The American Institute of Education conducts an annual study of attitudes of incoming
college students by surveying approximately 261,000 first-year students at 462 colleges and universities. There are approximately 1.6 million first-year college students in this country.
d) In a USA Today Internet poll, readers responded voluntarily to the question “Do you
consume at least one caffeinated beverage every day?” e) Astronomers typically determine the distance to galaxy by measuring the distances to just
a few stars within it and taking the mean (average) of these distance measurements. 2. Determine whether the numerical value is a parameter or a statistic (and explain):
a) A survey of 1103 students was taken from the university with 19,500 students. b) The 2006 team payroll of the New York Mets was $101,084,963. c) In a recent study of physics majors at the university, 15 students were double majoring in
math. d) A recent survey by the alumni of a major university indicated that the average salary of
10,000 of its 300,000 graduates was $125,000. e) The average salary of all assembly-line employees at a certain car manufacturer is
$33,000. f) The average late fee for 360 credit card holders was found to be $56.75.
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Population Parameters with M&M’S®
Student Activity Sheet Background: A population parameter is a number that describes some characteristic of a given
population. In statistics, the population is the entire group of individuals about which we want
information. The population parameter is a constant value that does not change. Many times it is
impractical or even impossible to calculate the population parameter of interest, the most common
reason being that populations are often composed of very large numbers of individuals. When we cannot
calculate the population parameter directly, we use a sample, which is a part of the population from
which we actually collect information. From the sample we calculate a statistic, which is a number that
describes some characteristic of the sample. A statistic will vary depending on the sample from which it
was calculated. How are the sample and the population related? We use information from a sample (a statistic) to draw
conclusions about a population parameter. We will use this relationship while performing the following
investigation. Claim: Mars, Incorporated claims that when producing milk chocolate M&M’S®, 16% of the total
number produced is green in color. Question of Interest: Is the claim that 16% of milk chocolate M&M’S® produced by Mars, Inc. are
green a valid claim? Instructions: Your class will be divided into pairs or small groups to perform the following
investigation in order to answer the question above. STEP 1: Preliminary questions. 1. What is the specific question(s) that needs to be addressed in your investigation? 2. How does this question relate to the background information at the beginning of the activity? 3. What is the population of interest in the investigation? 4. What is the population parameter of interest in the investigation? 5. Why can we not realistically calculate the population parameter of interest directly? 6. How could the concept of random sampling be used to investigate the population parameter of
interest?
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Your group will now be presented with a large classroom bin full of M&M’S®. Groups will then come up
to the bucket one-by-one and, without looking, grab 30 M&M’S®. The group will make note of the
number of M&M’S® of each color, place the sampled M&M’S® back in the bucket, and repeat the
process. Once two samples are recorded, the group will return to their seats to answer the questions in
Steps 2 through 4. STEP 2: Answer the following questions before using the samples of M&M’S® in your investigation. 1. What are we interested in finding out about each of the samples of M&M’S®? 2. What information do we need in order to answer our question or investigate the claim that 16% of milk
chocolate M&M’S® produced by Mars, Inc. are green? How can we use the samples of M&M’S® to obtain
this information? STEP 3: Each pair or group of students will perform the following investigation using one sample of
M&M’S®. The total number of M&M’S® in one sample is 30. 1. Calculate the proportion of green M&M’S® in the sample. Proportion of green M&M’S® __________________________ 2. Give an estimate for the proportion of green M&M’S® that Mars, Inc. makes based on your sample. 3. Write the proportion you found in (1) on the sticky note you were given and place it in the appropriate
position on the dot plot your teacher has drawn on the board. Did every sample have the same proportion of
green M&M’S®? 4. What value is at the center of the dot plot constructed by using one sample per group? Is it a
coincidence that the value is close to/far from the 16% claimed by Mars, Incorporated? 5. Mars, Inc. claimed that 16% of milk chocolate M&M’S® produced are green. Does the information
on the dot plot seem to support this claim? Why or why not? 6. Define each of the following in the context of the investigation you are performing with one sample
of M&M’S®. Population of interest ______________________________________________________________ Population parameter of interest______________________________________________________ Sample (drawn from population of interest) ____________________________________________ Statistic (used to estimate population parameter of interest) ________________________________
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STEP 4: Each pair or group of students will perform the following investigation using two samples
of M&M’S®. The total number of M&M’S® in two samples is 60. 1. Calculate the proportion of green M&M’S® in the overall sample (two samples with 30 M&M’S® in
each sample). Proportion of green M&M’S® in two samples __________________________ 2. Give an estimate for the proportion of green M&M’S® that Mars, Inc. makes based on your overall
sample. 3. Write the proportion you found in (1) on the sticky note you were given and place it in the appropriate
position on the dot plot your teacher has drawn on the board. Did every group have the same proportion
of green M&M’S® in two samples? 4. What value is at the center of the dot plot constructed by using the overall samples of each group? Is it
a coincidence that the value is close to/far from the 16% claimed by Mars, Inc.? 5. Mars, Inc. claimed that 16% of milk chocolate M&M’S® produced are green. Does the information
on the dot plot for two samples seem to support this claim more or less than the information obtained
from one sample? Why or why not? 6. What type of changes occurred in the dot plot when you used two samples of M&M’S® instead of
one? 7. If you only had one sample on which to base your estimate, how far off might your estimate be? What
would the worst case scenario be with one sample shown on the dot plot? What would the worst case
scenario be if you had two samples shown on the dot plot?
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Experimental Design, Sampling Design, and Bias
Experimental Design: methods to gather data Surveys: use a questionnaire to measure the characteristics and/or attitudes of people.
Observational studies: individuals are observed and outcomes are measured, but no attempt is made to affect the outcome.
Experiments: treatments are imposed prior to observation. The treatment can by anything that might affect the outcome.
Identify the experimental design:
a. A researcher gathered a group of people and asked them whether or not they took vitamin supplements. The researcher then compared the health of those who took vitamins with those who did not. b. A researcher gathered a group of people and divided them into two groups. One group got daily vitamin supplements and the other did not. After 1 year the researcher compared the health of the two groups.
c. A random sample of 30 digital cameras is selected and divided into 2 groups. One group is given a brand-name battery, while the other is given a generic battery. Pictures are taken under identical conditions and the battery life of the two groups is compared.
d. Using SAT scores to predict a student’s college success has created some controversy. A study was conducted to examine the SAT and college GPA information of 105 students who graduated from a state university with a B.S. in computer science.
e. Before viewing a lecture, students were given a summary of the instructors' prior teaching evaluations. These evaluations were rated either: Charismatic instructor or Punitive instructor. All the students then watched the same twenty-minute lecture given by the exact same lecturer. Following the lecture, students answered three questions about the leadership qualities of the lecturer.
f. How much television is too much for children? Parents answered a questionnaire about their children's age, characteristic behavior, and television viewing habits.
g. A study investigated the impact of participating in a school garden program. Subjects were 320 sixth-grade students at 3 schools. At 2 of the schools, garden-based learning activities were incorporated into the regular science class. The 3rd school did not include a garden program as part of its science class. Two questionnaires – Garden Vegetable Frequency Questionnaire and a taste test – were given to assess the outcome.
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Observational Study or Experiment?
For each situation, determine whether the research conducted is an observational study or an experiment. Explain your reasoning. 1. In an attempt to study the health effects of air pollution, a group of researchers selected 6 cities in very different environments; some from an urban setting (e.g. greater Boston), some from a heavy industrial setting (e.g. eastern Ohio), some from a rural setting (e.g. Wisconsin). Altogether they selected 8000 subjects from the 6 cities, and followed their health for the next 20 years. At this time their health prognoses were compared with measurements of air pollution in the 6 cities. 2. Among a group of women aged 65 and older who were tracked for several years, those who had a vitamin B12 deficiency were twice as likely to suffer severe depression as those who did not. 3. Forty volunteers suffering from insomnia were divided into two groups. The first group was assigned to a special no-desserts diet while the other continued desserts as usual. Half of the people in these groups were randomly assigned to an exercise program, while the others did not exercise. Those who ate no desserts and engaged in exercise showed the most improvement. 4. A study in California showed that students who study a musical instrument have higher GPAs than students who do not, 3.59 to 2.91. Of the music students, 16% had all A’s, compared with only 5% among the students who did not study a musical instrument. 5. Scientists at a major pharmaceutical firm investigated the effectiveness of an herbal compound to treat the common cold. They exposed each subject to a cold virus, and then gave him or her either the herbal compound or a sugar solution known to have no effect. Several days later, they assessed the patient’s condition, using a cold severity scale of 0 to 5. 6. In 2001, a report in the Journal of the American Cancer Institute indicated that women who work nights have a 60% greater risk of developing breast cancer. Researchers based these findings on the work histories of 763 women with breast cancer and 741 women without the disease. 7. To research the effects of dietary patterns on blood pressure in 459 subjects, subjects were randomly assigned to three groups and had their meals prepared by dietitians. Those who were fed a diet low in fat and cholesterol lowered their systolic blood pressure by an average of 6.7 points when compared with subjects fed a control diet. 8. Some people who race greyhounds give the dogs large doses of vitamin C in the belief that the dogs will run faster. Investigators at the University of Florida tried three different diets in random order on each of five racing greyhounds. They were surprised to find that when the dogs ate high amounts of vitamin C, they ran more slowly.
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Population Parameters and Sample Statistics/Observational Study vs. Experiment HW 1. The National Endowment of the Arts conducted a survey — titled "Reading at Risk" — on the
reading habits of approximately 17,000 adults. Of those surveyed, only 57% read a book in 2002.
a. What is the population under investigation?
b. What is the parameter of interest?
c. What is the sample?
d. What is the sample statistic?
2. A recent survey by the alumni of a major university indicated that the average salary of 10,000 of its 300,000 graduates was $125,000.
a. What is the population under investigation?
b. What is the parameter of interest?
c. What is the sample?
d. What is the sample statistic?
3. A major metropolitan newspaper selected a simple random sample of 1,600 readers from their list of 100,000 subscribers. They asked whether the paper should increase its coverage of local news. Forty percent of the sample wanted more local news.
a. What is the population under investigation?
b. What is the parameter of interest?
c. What is the sample?
d. What is the sample statistic?
Determine whether the numerical value is a parameter or a statistic (and explain):
4. The average salary of all assembly-line employees at a certain car manufacturer is $33,000.
5. The average late fee for credit card holders was found to be $56.75.
6. 10% of physics majors at NC State University were double majoring in math.
7. The average IQ score of college students is 115 with a standard deviation of 10.
8. Acme Corporation manufactures light bulbs. The CEO claims that an Acme light bulb lasts 300 days.
9. The owner of a fish market determined that the average weight for a catfish is 3.2 pounds.
10. The Pew Research Center reported that 24 percent of adults did not read a single book last year.
11. For incoming freshmen in 2013 at NC State University, the average SAT Math score is 625.
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For each scenario, describe the population and the sample.
12. The leaders of a large company want to know whether on-site day care would be considered a valuable employee benefit. They randomly selected 200 employees and asked their opinion about on-site day care.
13. A farmer saw some bollworms in his cotton field. Before deciding whether or not to reduce the
number of bollworms by spraying his field, he selected 50 plants and checked each carefully for bollworms.
14. A manufacturer received a large shipment of bolts. The bolts must meet certain specifications
to be useful. Before accepting shipment, 100 bolts were selected, and it was determined whether or not each met specifications.
Observational Study or Experiment?
For each situation, determine whether the research conducted is an observational study or an experiment. Explain your reasoning. 1. The muscles of men aged 40 - 50 were 40% to 50% stronger after they participated in a 10 week, high-intensity, resistance training program twice a week. 2. Among a group of women aged 65 and older who were tracked for several years, those who had a vitamin B12 deficiency were twice as likely to suffer severe depression as those who did not. 3. Forty volunteers suffering from insomnia were divided into two groups. The first group was assigned to a special no-desserts diet while the other continued desserts as usual. Half of the people in these groups were randomly assigned to an exercise program, while the others did not exercise. Those who ate no desserts and engaged in exercise showed the most improvement. 4. Some gardens prefer to use nonchemical methods to control insect pests in their gardens. Researchers have designed two kinds of traps and want to know which design will be more effective. They randomly choose 10 locations in a large garden and place one of each kind of trap at each location. After a week, they count the number of bugs in each trap. 5. In 2001, a report in the Journal of the American Cancer Institute indicated that women who work nights have a 60% greater risk of developing breast cancer. Researchers based these findings on the work histories of 763 women with breast cancer and 741 women without the disease.
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6. Scientists at a major pharmaceutical firm investigated the effectiveness of an herbal compound to treat the common cold. They exposed each subject to a cold virus, and then gave him or her either the herbal compound or a sugar solution known to have no effect. Several days later, they assessed the patient’s condition, using a cold severity scale of 0 to 5. 7. To research the effects of dietary patterns on blood pressure in 459 subjects, subjects were randomly assigned to three groups and had their meals prepared by dietitians. Those who were fed a diet low in fat and cholesterol lowered their systolic blood pressure by an average of 6.7 points when compared with subjects fed a control diet. 8. Some people who race greyhounds give the dogs large doses of vitamin C in the belief that the dogs will run faster. Investigators at the University of Florida tried three different diets in random order on each of five racing greyhounds. They were surprised to find that when the dogs ate high amounts of vitamin C, they ran more slowly. 9. An educational software company wants to compare the effectiveness of its computer
animation for teaching biology with that of a textbook presentation. The company gives a biology pretest to each of a group of high school juniors, and then divides them into two groups. One group uses the animation, and the other studies the test. The company retests all students and compares the increase in biology test scores in the two groups.
a. Is this an observational study or an experiment? Justify your answer. b. If the group using the computer animation has a much higher average increase in test scores
than the group using the textbook, what conclusions, if any, could the company draw? 10. What is the best way to answer each of the questions below: a survey, an experiment, or an
observational study? Explain your choices. For each, write a few sentences about how such a study might be carried out.
a. Are people generally satisfied with how things are going in the country right now? b. Do college students learn basic accounting better in a classroom or using an online course? c. How long do your teachers wait on average after they ask the class a question? 11. Can special study courses actually help raise SAT scores? One organization says that the 30
students they tutored achieved an average gain of 60 points when they retook the test. a. Explain why this does not necessarily prove that the special course caused the scores
to go up. b. Propose a design for an experiment that could test the effectiveness of the tutorial
course.
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Statistical Sampling Methods One of the main purposes of statistics is to be able to find information about a small part of the
population and then apply what we learn to the entire group. This is called sampling, and how a sample
is gathered will make a big difference on whether the information you gather is accurate or not. The goal
of sampling is to produce a representative sample, one that has the essential characteristics of the
population that is being studied, and is free of any type of bias. For example, if you were to conduct an
exit poll on Election Day and only interview voters in Raleigh about whether they voted for Kay Hagan
or Thom Tillis, you would have expected a very different result in the senatorial election. Why? Urban
voters tend to vote differently than rural voters – often they are more liberal – so an exit poll of this type
would be biased toward the Democratic candidate. So how do you keep your sample from becoming biased? The best way to do this is to use some
sort of random process to obtain it. There are two main types of samples: probability samples and
nonprobability samples. In a probability sample, each member of the population has a known probability
for being selected. There are several different types of probability samples: Probability Samples
Simple Random Sample: individuals are chosen randomly, every individual (or groups of the same size as the sample) in the population has an equal chance of being selected
o Requires a list of every individual in the population, which may be difficult or impossible to obtain.
Example:
Stratified Random Sample: divide the population into subgroups (strata) based on a characteristic of the group members. Randomly select individuals from each group.
Examples: Systematic random sample: from a randomly selected staring point choose every “nth“
individual.
Examples:
Cluster Sample: sample all the individuals in a randomly selected location. Examples:
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Non-Probability Samples
Of course, not all samples will be probability samples. Occasionally, people gather information
through other methods in hope that their results will be representative of the population. The danger in
using a nonprobability method is that the there is a higher likelihood that some bias will be present in your
results. Some examples of nonprobability sampling methods are:
Convenience sample: select individuals from a group that is conveniently accessible to the researcher.
o REMEMBER - A Cluster Sample uses a random location but a Convenience Sample uses a convenient location.
Examples:
Voluntary Response sample: individuals select themselves. People decide whether to
respond to the survey.
o The main problem with this method is that the people who respond typically have very
strong opinions one way or the other.
Examples:
Sampling Activity
Describe each of the sampling techniques that were used:
Voluntary Response:
Random Sample:
Systematic Sample:
Stratified Sample:
Cluster Sample:
How many times were you chosen? Does this happen in the real world? Where?
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Identify the sampling design:
a. Put the names of all APEX High students in a hat. Select 100 names from the hat.
b. Divide APEX High students by grade level. Randomly select 50 students from each grade level.
c. At Apex High, randomly select 10 classrooms during 2nd period. Sample all students in those rooms
d. Assign each APEX High student a number from 1 to total # of students. Choose a random number as the starting point, and then select every 25th student (assuming 2500 students, this gives a sample size of ~100 students)
e. Stand at the main entrance to APEX High one morning and pick the first 100 students to enter the school.
f. Post an on-line survey on the APEX High website.
g. The Educational Testing Service (ETS) needed a sample of colleges. ETS first divided all colleges into groups of similar types (small public, small private, etc.) Then they randomly selected 3 colleges from each group.
h. A county commissioner wants to survey people in her district to determine their opinions on a particular law up for adoption. She decides to randomly select blocks in her district and then survey all who live on those blocks.
i. A local restaurant manager wants to survey customers about the service they receive. Each night the manager randomly chooses a number between 1 & 10. He then gives a survey to that customer, and to every 10th customer after them, to fill it out before they leave.
j. A professor at UNC is interested in studying drinking behaviors among college students. The professor decides to stand at the main entrance to the Student Union and surveys the students who enter. k. A building contractor has a chance to buy an odd lot of 5000 used bricks at an auction. She is interested in determining the proportion of bricks in the lot that are cracked and therefore unusable for her current project, but she does not have enough time to inspect all 5000 bricks. Instead, she randomly selects a portion of the stack and checks the 100 bricks in that portion to determine whether each is cracked. l. A professor at a NCSU is interested in studying drinking behaviors among college students. The professor teaches a Sociology 101 class to mostly college freshmen and decides to use his or her class as the study sample. He or she passes out surveys during class for the students to complete and hand in.
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Sampling Designs- Random Rectangles Suppose we wish to know something about a population. For example, we might want to know the average height of a 17-year-old male, the proportion of Americans over 70 who send text messages, or the typical number of kittens in a litter. It is often not possible or practical to collect data from the entire population, so instead, we collect data from a sample of the population. If our sample is representative of the population, we can make inferences, or in other words, draw conclusions about the population. The Rectangles sheet has 100 random rectangles on it. They are already numbered so that you can easily sample them by randomly generating numbers. They are of varying size (area). Each box is 1 unit of area. Example: 3 units2: 6 units2: 8 units2: You will be doing 5 different TYPES of samples. Each time you will be sampling a total of 5 rectangles. 1. Judgmental Sample Study the rectangles on the front side of the sheet (side #1). After studying the 100 rectangles, select
ANY 5 that you believe are representative of the whole population. Let Z be the size (area) of the rectangle. Write the rectangle # that you chose, and the size (area) of your 5 below, then find the mean size
Rectangle #
Area (Z)
2. Simple Random Sample (SRS) Using the calculator, generate 5 random numbers from 1 to 100. (Randint (1,100). This will generate
one random number that corresponds to the rectangles on the sheet. Repeat until you have 5 separate
rectangles. Record the info below, then find the mean.
Rectangle #
Area (Z)
3. Stratified Sample On the back side of the rectangles, the rectangles have been separated into 5 groups (strata) based on
size. Each strata now has 20 rectangles of similar size. We want to take a sample in each strata, then
combine these to make our total sample.
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Follow the same procedure in Step #2 to generate a random number between 1 and 20 (randInt(1,20,1)),
and use that rectangle in Strata #1. Record the # you generated and rectangle size below. Then do the
same thing for each of the other strata (generate a new random number between 1 & 20 each time).
Record the sizes and find the mean.
Strata 1 Strata 2 Strata 3 Strata 4 Strata 5
Rectangle #
Area (Z)
4. Cluster Sample Here we want to take a sample of a group of rectangles that are near each other. We still want a sample
of five rectangles total, and we also still want it to be a random sample. Look back at the original sheet of rectangles (side #1). We can put the rectangles into clusters (groups)
of five based on their assigned number. So the first cluster would be rectangles #1-5, the second cluster
would be rectangles #6-10, and so on giving us 20 clusters to sample. Let’s choose a cluster for our sample. To do this, choose a random number, r, between 1 and 20. This is
the cluster you will use for your sample. Now calculate 5r – 4, and then 5r. The rectangles with
numbers from 5r – 4 to 5r are your cluster. This should be 5 rectangles. Ex: You get the random
number 6. 5(6) – 4 = 26 and 5(6) = 30. So this means you are looking at rectangles #26 – 30. Record the 5 rectangle numbers below, then their size, then the mean.
Random Number
Rectangle #
Area (Z)
5. Systematic Sample Use the original sheet of rectangles (side #1). Randomly generate a number between 1 and 20. This is
the first rectangle you will sample. Write this rectangle # in the first box below. Add 20 to the random
number to get the second rectangle you will sample. Continue to add 20 to get the next three rectangles
for your sample. This is like having to sample every 20th person who passes you. Ex: You get the
random number 6. Then you would inspect rectangles 6, 26, 46, 66, and 86. Record the rectangle numbers, sizes, and then find the mean size.
Random Number
Rectangle #
Area (Z)
Record your means for each of the 5 methods on the board.
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Compare sampling distributions 1. Sketch a dot plot of the class means from Judgmental sampling, SRS, Stratified sampling,
Systematic sampling, and Cluster sampling.
2. Discuss similarities and differences regarding shapes and spreads of each dot plot above.
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3. Calculate the mean of the sample averages for the guess, judgmental sample and all of the other
sampling techniques. Mark this value on each of the dot plots with a star symbol. How do these
CENTERS of the distributions of the means compare?
4. Which method do you believe is the least accurate? Why?
5. Do you think one method is doing a better job? Why?
6. The actual mean is 7.42. a. Do any of the plots have a center that is very close to the true average? If so, which
one(s)?
b. Do any of the plots have a center that is larger than the true average? If so, which one(s)? c. Which of the sampling strategies are most biased? Explain your answer.
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Random Rectangles stratified by size
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Survey Method Homework
In the following problems, identify the type of sampling used. Explain your choice.
1. In order to estimate the percentage of defects in a recent manufacturing batch, a quality control manager at Intel selects every 8th chip that comes off the assembly line starting with the 3rd, until she obtains a sample of 140 chips.
2. In order to determine the average IQ of ninth-grade students, a school psychologist obtains a list of all schools in the local public school system. She randomly selects five of these schools and administers an IQ test to all ninth-grade students at the selected schools.
3. In an effort to determine customer satisfaction, United Airlines randomly selects 50 flights during a certain week and surveys all passengers on the flights.
4. A member of Congress wishes to determine her constituency’s opinion regarding estate taxes. She divides her constituency into three income classes: low-income households, middle-income households, and upper-income households. She then takes a random sample of households from each income class.
5. In an effort to identify whether an advertising campaign has been effective, a marketing firm conducts a nationwide poll by randomly selecting individuals from a list of known users of the product.
6. A radio station asks its listeners to call in their opinion regarding the use of American forces in peacekeeping missions.
7. A farmer divides his orchard into 50 subsections, randomly selects 4 and samples all of the trees within the 4 subsections in order to approximate the yield of his orchard.
8. A school official divides the student population into five classes: freshman, sophomore, junior, senior, and graduate student. The official takes a random sample from each class and asks the members’ opinions regarding student services.
9. A survey regarding download time on a certain Web site is administered on the Internet by a market research firm to anyone who would like to take it.
10. A lobby has a list of 100 senators of the United States. In order to determine the Senate’s position regarding farm subsidies, they decide to talk with every seventh senator on the list starting with the third.
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For these problems, choose the best sampling method to obtain the individuals in the sample. Explain your choice. Explain how you would perform the sampling and be prepared to defend your choice. 11. The city of Raleigh is considering the construction of a new commuter rail station. The city wishes to survey the residents of the city to obtain their opinion regarding the use of tax dollars for this purpose. 12. The county sheriff wishes to determine whether a certain highway has a high proportion of speeders traveling on it. 13. Target wants to open a new store in the town of Apex. Before construction, they want to obtain some demographic information regarding the area under construction.
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Data Analysis The data appearing in the following charts was obtained from Lewis High School using
582 students from the Freshman class who carried backpacks. The data pertains to the
color of backpacks the students carry to school. Groups of students conducted surveys to
obtain the data charts that appear below.
Data Charts for the Backpack Color Study
Chart 1: Data collected from the entire population (collected through a survey of Freshmen carrying backpacks - 582 students) Backpack color black brown blue red yellow purple pink other
Response from
boys (292) 91 58 66 32 14 9 2 20
Response from
girls (290) 40 15 72 55 20 31 32 25
Totals 131 73 138 87 34 40 34 45
Chart 2: Data collected from a random sample population (collected through a survey of Freshmen carrying backpacks attending Study Hall - 54 students) Backpack color black brown blue red yellow purple pink other
Response from
boys (32) 12 2 10 6 0 1 0 1
Response from
girls (22) 2 0 8 6 2 2 2 0
Totals 14 2 18 14 2 3 2 1
Chart 3: Data collected from a biased sample population (collected through a survey of Freshmen girls carrying backpacks attending gym class - 42 students) Backpack color black brown blue red yellow purple pink other
Response from
girls (42) 7 0 8 12 4 4 6 1
Chart 4: Data collected from a biased sample population (collected through a survey of Freshmen boys carrying backpacks attending gym class - 36 students) Backpack color black brown blue red yellow purple pink other
Response from
boys (36) 16 2 8 7 1 2 0 0
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Use the information in the charts on the previous page to answer the following questions:
1. Are the results from the entire population what you would have expected them to be? Explain.
2. What factors might influence a student's choice of color for a backpack?
3. Why do you think that blue was the most popular color for the entire population?
4. 4. Examine the data from the random sample population. How similar is this data to the entire
population? List your observations of similarities and differences.
5. Would the random sample population data in this study be a good predictor for the entire
population? Explain.
6. Examine the results from both of the biased sample populations. Why is this data listed as
"biased"?
7. What are the similarities and differences between the two sets of biased data?
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8. Would the biased sample population data in this study be good predictors for the entire population?
Explain for both sets of biased data.
9. Prepare a chart that combines the two biased data charts. Prepare a graph of the new chart.
Backpack color black brown blue red yellow purple pink other
Response from
boys
Response from
girls
Totals
Would this new chart of the combined sample populations be a good predictor for the entire population?
Explain.
10.Prepare a chart that combines the three sample population charts (the random and the two biased)
to form a larger sample population. Prepare a graph of the new chart.
Backpack color black brown blue red yellow purple pink other
Response from
boys
Response from
girls
Totals
Would this new chart of a larger sample population be a good predictor for the entire population? Explain.
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Designing Simulations
Simulation is a way to model random events, such that simulated outcomes closely match real-world outcomes. By observing simulated outcomes, researchers gain insight on the real world. When designing a simulation, you need to make sure you understand and answer the following questions: What is the problem being simulated?
What are the possible outcomes? What is the probability of each outcome? Are there any assumptions? What question is the simulation trying to answer?
What random device will be used to do the simulation, and how will it be used?
Will it be a coin, spinner, playing cards, number cube, random digit table, or the random number generator?
o Coin: What does each side of the coin represent? o Spinner: What does each section represent? o Playing Cards: What does each color, number, or suit represent? o Dice: What does each face represent? o Random Digit Table: What does each number represent? o Random Number Generator: What does each number represent?
What is one trial in this simulation?
Coin: How many times do you need to flip? Spinner: How many times do you need to spin? Playing Cards: How many cards do you need to choose? Dice: How many times do you need to throw? Random Digit Table: How many digits do you need to look at? Random Number Generator: How many digits do you need to look at?
How many trials will be conducted?
Will the process be repeated 10, 20, 30 times?
What are the results of the simulated trials?
State the specific results of YOUR trial – you should get a fraction.
My results show that _________ out of _________ were _____________.
What predictions can be made based on these results?
This should include your conclusion based on the simulation you conducted.
Based on my results the chances of ______________ are _____________.
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Design a Simulation
Read the following situations and determine if/how each of random devices could be used to conduct a simulation. 1. A tennis player gets 50% of her serves in the service box. If she serves an average of 12 time per game, predict the number of serves that are playable.
Random Device Can it be used? How?
Coin
Spinner
Playing Cards
Dice
Random Number Table
Random Number Generator
2. If 5 families have triplets, predict the number of boys and girls at the play group.
Random Device Can it be used? How?
Coin
Spinner
Playing Cards
Dice
Random Number Table
Random Number Generator
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2. 67% of math students love trigonometry. Find the probability of selecting a student that loves trig.
Random Device Can it be used? How?
Coin
Spinner
Playing Cards
Dice
Random Number Table
Random Number Generator
Simulation Practice 1. Orders of frozen yogurt flavors (based on sales) have the following relative frequencies: 30%
chocolate, 50% vanilla, 20% strawberry. We want to simulate customers entering the store and
ordering yogurt. How would you simulate frozen yogurt sales based on recent history using table?
2. Design a simulation for the following situation: A couple plans to have children until they have a girl
or until they have 4 children, whichever comes first. What are the chances that they will have a girl
among their children? What tool would you use? Can you think of more than one way to do this?
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Simulation Problems Homework Describe how you would conduct one trial of a simulation model for each of the following situations. 1. Based on his history, Leon has an 80% chance of making a foul shot in a basketball game. Suppose Leon attempts 18 foul shots in a game. Describe one trial of a simulation model for Leon’s foul shot results in a game. 2. Based on her history, Mindy scores on 60% of her shots on goal in a field hockey game. Suppose she attempts 8 shots on goal in a game. Describe one trial of a simulation model for Mindy’s shots on goal results in a game. 3. The Bumble Bees’ chance of winning a football game is 20%. Suppose they play 15 football games in a season. Describe one trial of a simulation model for their 15 game season. 4. Carlos has two chances to get the correct answer on a multiple-choice question with three
choices. Suppose he guesses. He will answer correctly on the first try 2
3 of the time. If
he has to try again, he has a 50% chance of getting the correct answer. Describe one trial of a simulation model for Carlos getting the correct answer on a multiple-choice question. 5. Based on his history, Anthony has a 75% chance of making a foul shot in a basketball game. Suppose he makes 16 shots in a game. Describe one trial of a simulation model for Anthony’s foul shot results in a game. 6. A goalie saves half of the shots on goal. Suppose there were twelve shots in a game.
a) Describe how you would conduct one trial of a simulation that models the results of the shots on goal. b) Suppose the goalie saved 2/3 of the shots on goal. Describe how you would conduct one trial of a simulation that models the results of the shots on goal. c) How many trials should be conducted to obtain reasonable results? Use mathematics to justify your answer.
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Margin of Error A survey of a sample population gathers information from a few people and then the results are used to reflect the opinions of a larger population. The reason that researchers and pollsters use sample population is that it is cheaper and easier to poll a few people rather than everybody. One key to successful surveys of sample populations is finding the appropriate size for the sample that will give accurate results without spending too much time or money. Determining a margin of error depends on whether you’re working with a proportion or a mean. Proportions: Suppose that 900 American teens were surveyed about their favorite ski category of the 2002 Winter Olympics in Park City, Utah. Ski jumping was the favorite for 20% of those surveyed. What is the true population proportion of teens who enjoy ski jumping?
First, find the standard deviation: 𝜎 = √𝑝(1−𝑝)
𝑛
For our example, p = 0.2, so 1 – p = 0.8. 𝜎 = √0.2(0.8)
900 = 0.013
Since about 95% of the data will fall within almost two standard deviations, we will use the formula
ME = 1.96√𝑝(1−𝑝)
𝑛
The margin of error will be 1.96(0.013), or 0.02548. Let’s round that to 0.025. Because our survey did not ask every single teenager in America, we are basically making a guess here, so our margin of error provides a cushion around our guess. We believe that the true proportion will fall inside the interval created when we add and subtract the margin of error from our sample proportion:
0.20 ± 0.025 Our interval is 0.175 to 0.225. We believe the true proportion will lie inside that interval.
Find the margin of error for each of the following and create an interval for the true population proportion. 1. A sample of 550 people leaving a shopping mall showed that 64% of shoppers claim to have
spent over $25.
2. In a random sample of machine parts, 18 out of 225 were found to have been damaged in shipment.
3. A telephone survey of 1000 adults was taken shortly after the U.S. began bombing Iraq found
that 832 adults voiced their support for this action.
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4. An assembly line does a quality check by sampling 50 of its products. It finds that 16% of the parts are defective.
Now let’s look at how the sample size will affect the margin of error. Use the ski jumping example for the following: 1. Find the margin of error for a survey of 90 American teens. 2. Find the margin of error for a survey of 9,000 teens. 3. Find the margin of error for a survey of 90,000 teens. 4. Draw a conclusion about the margin of error based on the size of the sample. Why do you think this is so? 5. If you want to cut your margin of error in half, what would you have to do to the sample size? Why? Margins of error can also be used to estimate population means. Let’s see how this will work. A company that produces white bread is concerned about the distribution of the amount of sodium in its bread. The company takes a simple random sample of 100 slices of bread and computes the sample mean to be 103 milligrams of sodium per slice and the sample standard deviation is 12 milligrams.
The margin of error formula for means is 1.96𝑠
√𝑛, so in this situation, our margin of error will be:
1.9612
√100, or 2.352.
That means that we will expect our true population mean to fall between 103 ± 2.352, or 100.648 – 105.352. What if the bread label states that the sodium content of the bread is 100 milligrams? Should the company be concerned? Why or why not?
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Hershey’s Kisses and Confidence Intervals In this activity, we will estimate a confidence interval for how often a Hershey’s kiss lands on its base as opposed to its side. To do this, we will drop Hershey’s kisses, count how many land on their base, and calculate the confidence interval. What is the population? __________________________________ What is the sample? ______________________________________ To take your sample, gather ten Hershey’s kisses in your hands, shake them up, and drop them from about six inches above your desk. Count the number that land on their base. Repeat five times to get a sample of size 50. Compare with the others in your group. Your results: p = ____________ The results of others in your group: Did you all get the same answer? Why or why not? Follow the steps to get the 95% confidence interval:
Calculate the standard deviation of your sample proportion: 𝜎 = √𝑝(1−𝑝)
𝑛
Calculate the Margin of Error: ME = 1.96
Calculate the upper bound: _______________________ lower bound: ______________________ Compare your interpretation with your group members’ interpretations and come to an agreement on an appropriate interpretation. Write it below.
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Now merge your individual samples into one big sample, and make a new confidence interval. Remember that n is changing! Group confidence interval: Compare this confidence interval with the confidence intervals from your individual samples. • Are the means different? If so, how? • Are the standard deviations different? If so, how? • Are the widths of the confidence intervals different? If so, how? How did increasing the sample size affect the width of the confidence interval?
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Find the margin of error for the following and an interval that could contain the true mean: 1. You want to rent an unfurnished one-bedroom apartment for next semester. The mean monthly rent for a random sample of 10 apartments advertised in the local newspaper is $540 with a standard deviation of $80. 2. Your company sells exercise clothing and equipment on the Internet. To design the clothing, you collect data on the physical characteristics of your different types of customers. Here are the weights (in kilograms) for a sample of 24 male runners: 67.8 61.9 63.0 53.1 62.3 59.7 55.4 58.9 60.9 69.2 63.7 68.3 64.7 65.6 56.0 57.8 66.0 62.9 53.6 65.0 55.8 60.4 69.3 61.7 (Use your calculator – 1 Variable statistics) 3. A hardware manufacturer produces bolts used to assemble various machines. Suppose the average diameter of a simple random sample of 50 bolts is 5.11 mm and the standard deviation is 0.1 mm. 4. We have IQ test scores of 31 seventh-grade girls in a Midwest school district. We have calculated that sample mean is 105.84 and the standard deviation is 14.27. 5. Let’s look at problem #4 again. How would the margin of error change if there were 90 girls instead of the 31? 6. What if there were 250 girls? 7. How does the sample size change the margin of error?
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Calculating Margin of Error and Confidence Intervals HW
1. The lifetimes (in months) of a set of 50 randomly selected automobile batteries of a particular brand gives a mean of 16 months with a standard deviation of 1.5 months. The manufacturer claims, however, that this particular make of battery has an 18-month lifetime.
a) Find the margin of error for this sample. b) Now add and subtract your margin of error to the sample mean to create a confidence interval. c) Compare the manufacturer’s claim to your interval. Do you believe the manufacturer’s claim? Explain.
2. The caffeine content of a random sample of 81 cups of black coffee dispensed by a new machine is measured. The mean and standard deviation for the sample are 110 mg and 5.0 mg, respectively. The manufacturer of the machine claims that the average caffeine content per cup is 109 mg.
a) Find the margin of error for this sample. b) Now add and subtract your margin of error to the sample mean to create a confidence interval. c) Do you believe that the manufacturer’s claim is valid or invalid? Explain.
3. You want to rent an unfurnished one-bedroom apartment for next semester. The mean monthly rent for a random sample of 10 apartments advertised in the local newspaper is $540. Assume that the standard deviation is $80.
a) Find the margin of error for the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community. b) Use your margin of error to find an estimate of the amount you should expect to pay for the one-bedroom apartment. c) Now recalculate your margin of error for a sample of 100 apartments. Let’s assume the mean and standard deviation do not change. What do you notice about your margin of error? d) Without calculating it, what do you think will happen to the margin of error if we increase the sample size to 500 apartments? Why?
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4. Your company sells exercise clothing and equipment on the Internet. To design the clothing, you collect data on the physical characteristics of your different types of customers. Here are the weights (in kilograms) for a sample of 24 male runners. Suppose the standard deviation of the population is known to be 4.5 kg.
67.8 61.9 63.0 53.1 62.3 59.7 55.4 58.9 60.9 69.2 63.7 68.3 64.7 65.6 56.0 57.8 66.0 62.9 53.6 65.0 55.8 60.4 69.3 61.7
a) Use your calculator to find the sample mean. Now calculate the margin of error for the weights of male runners. b) Provide an interval for an estimate of the weights of male runners for your company.
5. The paralyzed Veterans of America is a philanthropic organization that relies on contributions. They send free mailing labels and greeting cards to potential donors on their list and ask for voluntary contribution. To test a new campaign, they recently sent letters to a random sample of 100,000 potential donors and received 4781 donations.
a) Find the sample proportion of donors. b) Now calculate the margin of error and an interval which contains the true proportion of donors. c) A staff member thinks that the true rate is 5%. Given the confidence interval you found, do you find that percentage plausible?
6. A national health organization warns that 30% of the middle school students nationwide have been drunk. Concerned, a local health agency randomly and anonymously surveys 110 of the middle 1212 middle school students in its city. Only 21 of them report having been drunk.
a) What proportion of the sample reported having been drunk? b) Does this mean that this city’s youth are not drinking as much as the national data would indicate? c) Create a confidence interval for the proportion of the city’s middle school students who have been drunk. d) Is there any reason to believe that the national level of 30% is not true of the middle school students in the city?
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Bias: A systematic error in measuring the estimate, favors certain outcomes. Anything that causes the data to be wrong! It might be attributed to the researchers, the respondent, or to the sampling method!
Sources of Bias: things that can cause bias in your sample
Voluntary response: people chose to respond. Usually only people with very strong opinions respond. The way to determine voluntary response is: Self-selection!
Convenience sampling: sampling people who are easy to ask can produce biased results. This method is often used for surveys & results reported in newspapers and magazines!
Undercoverage: when some groups of the population are left out of the sampling process.
Nonresponse: an individual chosen for the sample can’t be contacted or refuses to
cooperate. People are chosen by the researchers, BUT refuse to participate
Response bias: when the behavior of respondent or interviewer causes bias in the sample.
Lurking variables: variables that cause results in a study, that have been taken into consideration by the experimenters or surveyors.
Wording of the Questions: wording of the question can influence the answers that are
given. Questions must be worded as clearly and as neutrally as possible to avoid influencing the response.
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Identify the source of bias: a. Before the presidential election of 1936, FDR against Republican ALF Landon, the magazine Literary Digest predicted Landon winning the election in a 3-to-2 victory. The Digest’s survey of 2.8 million people came from magazine subscribers, car owners, telephone directories, etc. George Gallup surveyed only 50,000 people and predicted that Roosevelt would win.
b. A survey of high school students on drug abuse uses a uniformed police officer to interview each student in the sample.
c. Conduct a survey by stopping friendly-looking people in the mall.
d. A restaurant conducts a survey of its customers by leaving surveys on the tables.
e. A survey is conducted by calling randomly selected names from the phone book.
f. Suppose that you want to estimate the total amount of money spent by students on textbooks each semester at UNC. You collect register receipts for students as they leave the bookstore during lunch one day.
g. To find the average value of a home in Apex, one averages the price of homes that are listed for sale with a realtor.
h. A surveyor is conducting a survey outside of a local library. He is asking people to take the survey as they come into the library.
i. A magazine includes a questionnaire in one of its issues. The magazine is for guns and ammunition, and the questionnaire is asking about public opinion of gun control.
j. A theatre company hands out a questionnaire about the quality of their productions. They are only interested in the people who attend the theatre.
k. Susan and Walter are getting ready for their big summer barbecue party. Not sure how many hamburgers to buy, Susan sends out a survey to all of the guests. Here is Susan’s survey question: Do you want to have someone kill a defenseless animal, skin it, grab some of it, add preservatives to it, and force me to inhale the fumes while I cry silently because it reminds me of all my animal farm friends from when I was a child?
l. Walter reads Susan’s question and shakes his head and sighs. He writes a new survey question and sends it out to their friends: Do you want to support local American farmers in these troubled economic times by grilling up a traditional American juicy burger?
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Sampling Bias Activity Read through each scenario and determine whether or not bias exists in the sampling method. If so, identify the source of the bias. Then make suggestions on how this sampling method could be improved in order to select an unbiased sample.
Scenario #1: A company is interested in opening a gym on its premises for all employees. The company operates 3 shifts: morning, evening, and late night. They ask the first 25 people reporting to work during the morning and evening shifts if they would use the gym, and what hours they would like the gym to be open.
Scenario #2: A newspaper is interested in determining whether working women support federal aid for child care. A reporter attends a conference designed for women in professional careers and randomly selects 40 women attending the conference to complete the survey.
Scenario #3: A company makes products typically used by the elderly. The company manager provides samples of various products to residents of the retirement home where his mother lives. The residents use the products for several weeks and then complete a survey, giving their opinions of the company’s products.
Scenario #4: The school board is interested in taxpayers’ opinions on cutting funding for fine arts. They decide to ask parents since they would have an interest in school funding. A school board member attends a local high school chorus concert and interviews several parents as they are leaving the concert.
Scenario #5: The Republican Party sends out a survey to 500 registered Republicans in 3 states in the Northeast to determine the issues that the Republican Party should focus on for the next election.
Scenario #6: A television rating service is interested in finding out what shows are being watched most often during prime time viewing hours. To obtain a sample of households, the service dials numbers taken at random from telephone directories.
Scenario #7: WRAL 101.5 announces during the morning show that they are interested in listeners’ opinions on the new toll road between Holly Springs and I-40. Listeners are asked to call in and answer a few questions regarding the amount of tolls being charged on the road.
Scenario #8: A clothing company wants to know what color leggings teenagers will buy. The company decides to spend one day in the junior departments of five randomly selected stores in randomly selected cities and ask every teenager who enters what color leggings they buy.
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Analyzing Misleading Graphs
Look at each graph or set of data carefully with your partner. Discuss how each graph is misleading and what you could do to correct it. 1. 2. 3. 4. 5. 6.
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Biased or Unbiased? HW
Determine whether the survey sample is biased or unbiased. Explain your answers. 1. Question: What is your favorite sport? Sample is chosen from people attending a soccer game.
2. Question: What is your favorite soft drink? Sample is chosen by picking names out of a telephone book.
3. Question: Should more money be put into athletic programs or music programs at school? Sample is chosen from students in the band program.
4. Question: What is your favorite vacation destination? Sample is chosen by asking every student in the class.
Tell whether the following survey questions are potentially biased. Explain your answer. If the
question is potentially biased, rewrite it so that it is not. 5. “Don’t you agree that the voting age should be lowered to 16 because many 16-year-olds are
responsible and informed?“
6. “Do you think the city should risk an increase in pollution by allowing expansion of the Northern Industrial Park? “
7. “Don’t you agree that the school needs a new baseball field more than a new science lab? “
8. “Would you pay even higher concert ticket prices to finance a new arena? “
9. “The budget of the Wake County Public School System is short of funds. Should taxes be raised in order for this district to fund extra-curricular sports programs?“
10. “Due to diminishing resources, should a law be made to require people to recycle?“
Answer the following:
11. You want to determine whether to serve hamburgers or pizza at a soccer team party. Write a survey question that would likely produce biased results. Write a survey question that would likely produce unbiased results.
12. You want to find students’ opinions on the current attendance policy. Give two ways that your sample for the survey might be selected. The first must be an example of a biased sample and the second must be an example of an unbiased sample. Thoroughly explain your answers.
13. Two toothpaste manufacturers each claim that 4 out of every 5 dentists use their brand exclusively. Both manufacturers can support their claims with survey results. Explain how this is possible.
14. A survey about Americans’ interest in soccer, asks the question “How interested are you in the world’s most popular sport, soccer?” Give two ways that your sample for the survey might be selected. The first must be an example of a biased sample and the second must be an example of an unbiased sample.
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Review
I. Determine whether each study depicts an observational study, or experimental study.
1. A parent group randomly examines 25 randomly selected PG-13 movies and 25 randomly
selected PG movies and records the number of curse words that occur in each. They then
compare the number of curse words between the two movie ratings.
2. A sample of 504 patients in early stages of Alzheimer’s disease is divided into two groups. One
group receives an experimental drug; the other group receives a placebo. The advance of the
disease in the patients from the two groups is tracked at 1-month intervals over the next year.
II. Determine the type of sampling used.
3. On election day, a pollster from Fox News positions herself outside a polling place near her
home. She then asks the first 50 voters leaving the facility to complete a survey.
4. An internet service provider randomly selects 15 residential blocks from a large city. It then
surveys every household in these 15 block to determine the number that would use a high-speed
internet service it if were made available.
5. Thirty-five sophomores, 22 juniors, and 35 seniors are randomly selected to participate in a study
from 574 sophomores, 462 juniors, and 532 seniors at a certain high school.
6. A quality-control engineer wants to be sure that bolts coming off an assembly line are within
prescribed tolerances. The quality-control engineer randomly picks blots off of the assembly line
to test.
7. You have 30 subjects available for a comparative experiment. 15 subjects will receive the
treatment. Number the subjects 01 to 30. Using the random digit table below, what are the first
five subjects?
07511 88915 41267 10753 84569 79367 32337 03316
8. The Bumble Bees’ chance of winning a football game is 20%. Suppose they play 15 football
games in a season. Describe one trial of a simulation model for their 15 game season.
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9. Modern Managed Hospitals (MMH) is a national for-profit chain of hospitals. Management
wants to survey patients discharged this past year to obtain patient satisfaction profiles. They
wish to use a sample of such patients. Some sampling techniques are described below.
Categorize each technique as simple random sampling, stratified sampling, systematic sampling,
cluster sampling, or convenience sampling.
a) Obtain a list of patients discharged from all MMH facilities. Divide the patients according to
the length of hospital stay (3 days or less, 3 – 7 days, 8 – 14 days, more than 14 days). Draw
simple random samples from each group.
b) Obtain a list of patients discharged from all MMH facilities. Number these patients, and then
use a random-number table to obtain the sample.
c) Randomly select a few MMH facilities from each of five geographic regions, and then take all
patients on the discharge list of the selected hospitals.
d) At the beginning of the year, instruct all MMH facilities to survey every 500th patient
discharged.
e) Instruct each MMH facility to survey 10 discharged patients this week and send in the results.
10. 7. A set of animals is fed a certain type of grain for a ten-week period to estimate average
weight gain. A margin of error of 1.5 pounds was calculated. Past experience indicates that the
standard deviation is 7 pounds.
a) How large was the sample size?
b) If we double the sample size, what will happen to the margin of error?
c) If we quadruple the sample size, what will happen to the margin of error?
