MATH-164 - Chapter 1

Determine whether the quantitative variable is discrete or continuous. Volume of water in a reservoir

The variable is continuous because it is not countable

Determine whether the quantitative variable is discrete or continuous. Population of a city

The variable is discrete because it is countable.

Determine whether the variable is qualitative or quantitative. Favorite rock group

The variable is qualitative because it is an attribute characteristic.

Determine whether the variable is qualitative or quantitative. Number of pets

The variable is quantitative because it is a numerical measure

(a) What is an observational​ study? (b) What is a designed​ experiment? (c) Which allows the researcher to claim causation between an explanatory variable and a response​ variable?

(a) An observational study measures the value of the response variable without attempting to influence the value of either the response or explanatory variables. (b) A designed experiment is when a researcher assigns individuals to a certain​ group, intentionally changing the value of an explanatory​ variable, and then recording the value of the response variable for each group. (c) designed experiment.

Each of the following surveys has bias. Identify the type of bias: (a) A television survey that gives phone numbers for viewers to call with their vote. Each call costs $2.00. (b) An employer distributes a survey to her 450 employees asking them how many hours each week, on average, they surf the Internet during business hours. Three of the employees complete the survey. (c) A question on a survey asks, "Do you favor or oppose a minor increase in property tax to ensure fair salaries for teachers and properly equipped school buildings?" (d) A researcher conducting a poll about national politics sends a survey to a random sample of subscribers to Time magazine.

(a) Sampling bias (voluntary response) (b) Nonresponse bias (c) Response bias (poorly worded question) (d) Sampling bias (undercoverage)

The human resource department at a certain company wants to conduct a survey regarding worker benefits. The department has an alphabetical list of all 3545 employees at the company and wants to conduct a systematic sample of size 30.

(a) What is k​? To find​ k, compute​ N/n and round down. The population is the employees in the​ company, and hence N equals 3545.The desired sample size n is 30.To find k​, we compute N​/n and round down.​N/n equals 3545​/30 equals 118.2333333Thus, we take k to be 118.(b) Determine the individuals who will be administered the survey. Randomly select a number from 1 to k. Suppose that we randomly select 6. Starting with the first individual​ selected, the individuals in the survey will be6​,124​,242​...,3428 NOTE: 6=k6+k=6+118=124 6+2(11)=6+236=242 6+118(30-1)=6+118(29)=6+3422=3428

cluster sampling

A probability sampling technique in which clusters of participants within the population of interest are selected at random, followed by data collection from all individuals in each cluster. Example: students siting in a row. Each row is a cluster. cluster sample is selecting ALL students in one row, than selecting All students in a different row

Researchers wanted to test the effectiveness of a new drug therapy for treating patients with dementia. To do​ this, they identified 150 patients with a diagnosis of dementia. Patients were randomly assigned to one of three treatment groups. Fifty patients were randomly assigned to receive the new drug​ therapy, another 50 received the older drug​ therapy and the final 50received a placebo therapy. To measure the effectiveness of the​ treatment, researchers scored each patient on a standardized rating scale for dementia. After collecting and comparing the scores for the three treatment​ groups, the researchers concluded that the new drug therapy is significantly more effective than both the older drug therapy and the placebo therapy in the treatment of dementia. Complete parts​ (a) through​ (f).

(a) What type of experimental design is​ this? Completely randomized design ​(b) What is the population being​ studied? All patients with a diagnosis of dementia ​(c) What is the response variable in this​ study? The score on the standardized rating scale for dementia ​(d) What are the​ treatment(s)? The new drug​ therapy, the older drug​ therapy, and the placebo therapy ​(e) Identify the experimental units. Choose the correct answer below. The 150 patients with a diagnosis of dementia Figure one. The 3 different groups

To determine if topiramate is an effective treatment for alcohol​ dependence, researchers conducted a​ 14-week trial of 371 men and women aged 18 to 65 years diagnosed with alcohol dependence. In this​ doubleblind, randomized,​ placebo-controlled experiment, subjects were randomly given either 300 milligrams​ (mg) of topiramate​ (183 subjects) or a placebo​ (188 subjects)​ daily, along with a weekly compliance enhancement intervention. The variable used to determine the effectiveness of the treatment was​ self-reported percentage of heavy drinking days. Results indicated that topiramate was more effective than placebo at reducing the percentage of heavy drinking days. The researchers concluded that topiramate is a promising treatment for alcohol dependence. Complete parts​ (a) through​ (f).

(a)What does it mean for the experiment to be​ placebo-controlled? The experiment will have a control group that takes a​ placebo, which is a innocuous​ medication, such as a sugar tablet. This control group serves as a baseline treatment that can be used to compare to the group that is actually taking the medication. ​(b) What does it mean for the experiment to be​ double-blind? Why do you think it is necessary for the experiment to be​ double-blind? Neither the subject nor the researcher knows which treatment the subject is receiving. Why do you think it is necessary for the experiment to be​ double-blind? The experiment is​ double-blind so that the subjects receiving the medication do not behave differently and so the individual monitoring the subjects does not treat those receiving medication differently from those receiving a placebo. ​(c) What does it mean for the experiment to be​ randomized? It means that the subjects are randomly assigned to take either the topiramate or the placebo. (d) What is the population for which this study​ applies? What is the​ sample? The population is all​ 18-65 year olds with alcohol dependence. The sample is 371 men and women aged 18 to 65 years diagnosed with alcohol dependence. ​(e) What are the​ treatments? 300 mg of topiramate or a placebo​ daily, and a weekly compliance enhancement intervention (f) What is the response​ variable? Percentage of heavy drinking days

Define simple random sampling

A sample of size n from a population of size N is obtained through simple random sampling if every possible sample of size n has an equally likely chance of occurring. The sample is then called a simple random sample.

What is the process the methods of statistics follow?

1. Identify the research objective 2. Collect the info. needed to answer the question posed in 1. 3. Describe the data 4. Perform inference

Suppose a marketing firm wants to gather information about a new product. The firm divided a city into blocks as illustrated.​ (Note: the numbers shown represent house address​ numbers.) Which of the following represents a possible cluster sample of the homes in the​ city?

7480, 7483, 7484, 7487, 7488, 7491, 7492, 7495 look for patters within groups

Cluster sample

A cluster sample is obtained by selecting all individuals within a randomly selected collection or group of individuals.

What is a​ frame?

A frame is a list of the individuals in the population being studied.

Bias

A particular preference or point of view that is personal, rather than scientific. There are three sources of bias in sampling: Sampling bias Nonresponse bias Response bias

Explain the difference between a population and a sample.

A population is the entire group that is being studied while a sample is a subset of the population that is being studied.

Determine whether the quantitative variable is discrete or continuous. The number of notes in a song

The variable is because it countable.

Define factor.

A variable whose effect on the response variable is to be assessed by the experimenter

A news service conducted a survey of 1019 adults ages 18 years or older in a certain​ country, August 31 minus September ​2, 2015. The respondents were​ asked, "Of every tax dollar that goes to the federal​ government, how many cents of each dollar would you say are​ wasted?" Of the 1019 individuals​ surveyed, 38​% indicated that 51 cents or more is wasted. The news service reported that 38​% of all adults in the country 18 years or older believe the federal government wastes at least 51 cents of each dollar​ spent, with a margin of error of 5​% and a 90​% level of confidence. List the descriptive statistics. (a) What is the research objectives? (b) What is the population? (c) What is the sample? (d) List the descriptive statistics

A. To determine the percent of adults in the country who believe the federal government wastes 51 cents or more of every dollar B. Adults in the country aged 18 years or older C. the 1019 adults in the country that were surveyed D. 38​% of the individuals surveyed indicated that 51 cents or more is wasted. The news service is 95​% confident that the percentage of all adults in the country who believe the federal government wastes 51 cents or more of every dollar received is between 35​% and 41​%.

Which sampling method does not require a​ frame?

A. Systematic

Determine whether the following statements are true or false. ​(a) When a factor is controlled by setting it to three​ levels, the particular factor is of no interest to the researcher. ​(b) Randomization is used so that those factors not controlled in the experiment​ "average out" their effect on the response variable.

A. The statement is false because a factor that is controlled and set at various levels is a factor of interest to the researcher. B. The statement is true.

Define treatment.

Any combination of the values of the factors​ (explanatory variables)

A polling organization contacts 1037 adult men who are 40 to 60 years of age and live in the United States and asks whether or not they had seen their family doctor within the past 6 months. What is the population in the​ study? What is the sample in the​ study?

Adult men who are 40 to 60 of age and live in the United States. The 1,037 adult men who are 40 to 60 years of age and live in the United States.

The characteristics of the individuals in a study are variables.

An individual is a person or an object that is a member of the population being studied.

Define placebo.

An innocuous​ medication, such as a sugar​ tablet, that​ looks, tastes, and smells like the experimental medication

The owner of a shopping mall wishes to expand the number of shops available in the food court. She has a market researcher survey the first 80 customers who come into the food court during weekday mornings to determine what types of food the shoppers would like to see added to the food court. Which of the following is the best way to remedy this​ problem?

Ask customers throughout the day on both weekdays and weekends

Define experimental unit

A​ person, object, or some other​ well-defined item upon which a treatment is applied

are observations corresponding to a continuous variable.

Continuous Data

A(n) ___________ is obtained by dividing the population into groups and selecting all individuals from within a random sample of the groups.

C. cluster sample

A(n) _____________ is obtained by dividing the population into homogeneous groups and randomly selecting individuals from each group. A. cluster sample B. systematic sample C. stratified sample D. convenience sample E. simple random sample

C. stratified sample

A _______ _______ is a quantitative variable that has an infinite number of possible values that are not countable. A continuous variable may take on every possible value between any two values.

Continuous Variable

A club wants to sponsor a panel discussion on an upcoming election. The club wants to have four of its members lead the panel discussion. To be​ fair, however, the panel should consist of two members of each party. Below is a list of members in each party. Obtain a stratified sample of two people from party 1 and two from party 2.

Carter, Pawlak, Weber, May

The list of observed values for a variable is?

Data

___________statistics consists of organizing and summarizing information​ collected, while _______ statistics uses methods that generalize results obtained from a sample to the population and measure the reliability of the results.

Descriptive, Inferential

Determine whether the following statement is true or false. Explain. When obtaining a stratified​ sample, the number of individuals included within each stratum must be equal.

Determine whether the following statement is true or false. Explain.When obtaining a stratified​ sample, the number of individuals included within each stratum must be equal.

are observations corresponding to a discrete variable.

Discrete Data

A _________ ________ is a quantitative variable that has either a finite number of possible values or a countable number of possible values. A discrete variable cannot take on every possible value between any two possible values.

Discrete Variable

In​ statistics, results are always reported with​ 100% certainty.

False. In statistics, results are not reported with​ 100% certainty. Because statistical studies draw on​ samples, and because there is variation within​ groups, results cannot be reported with​ 100% certainty.

Statistical studies are not concerned with understanding the sources of variability in​ data, only with describing the variability in the data. Choose the correct answer below.

False. Statistical studies are concerned with both describing the variability in the data and understanding the sources of variability in data. Understanding the sources allows researchers to control it and reach better conclusions.

A(n) _________ is a person or object that is a member of the population being studied.

Individual

A(n) _______ is a numerical summary of a population.

Parameter

Determine whether the quantitative variable is discrete or continuous. Amount of water in a dog's bowl

The variable is continuous because it is not countable

The list of observed values for a variable is data.

Gender is a variable; the observations male and female are data.

Explain the difference between a​ single-blind and a​ double-blind experiment

In a single-blind experiment, the subject does not know which treatment is received. In a double-blind experiment, neither the subject nor the researchers in contact with the subject knows which received treatment

Setup Size (in) Screen Type Channels A 50 Projector 299 B 52 Plasma 120 C 54 Plasma 423 D 47 Projector 270 E 41 Projector 290 What are the individuals being studied? What are the variables and their corresponding data being studied? Size is a? Screen type is a? Number of channels is?

Individuals: The set ups A through E of widescreen high- definition televisions. Variables: Size, screen type, channels. Size: Continuous variable Screen Type: Qualitative variable Number of channels: Discrete variable

A variable is at the _______________ if it has the properties of the ordinal level of measurement and the differences in the values of the variable have meaning. A value of zero does not mean the absence of the quantity. Arithmetic operations such as addition and subtraction can be performed on the values of the variable.

Interval Level of Measurement

Population

N

Determine the level of measurement of the variable below. Dress color

Nominal

It is extremely important for a researcher to clearly define the variables in a study because this helps to determine the type of analysis that can be performed on the data. For​ example, if a researcher wanted to describe people based on Social Security number​, what level of measurement would the variable ​"Social Security number​" ​be? Now suppose the researcher felt that certain people who lived farther east received higher numbers. Does the level of measurement of the variable​ change? If​ so, how? **What is the level of measurement of the variable ​"Social Security number​" in the original​ scenario? **Does the level of measurement of the variable change in the second​ scenario?

Nominal Yes, it changes to Ordinal

A variable is at the _______________if the values of the variable name, label, or categorize. In addition, the naming scheme does not allow for the values of the variable to be arranged in a ranked or specific order.

Nominal Level of Measurement

A survey mailed to residents of a town has a response rate of less than 2%

Nonresponse bias

Distinguish between nonsampling error and sampling error.

Nonsampling error is the error that results from​ undercoverage, nonresponse​ bias, response​ bias, or​ data-entry errors. Sampling error is the error that results because a sample is being used to estimate information about a population.

1. A congresswoman wants to survey her constituency regarding public policy. She asks on of her staff members to obtain a sample of residents of the district. The frame she has available lists 9012 Democrats, 8302 Republicans, and 3012 Independents. Obtain a stratified random sample of 8 Democrats, 7 Republicans, and 3 Independents. Discuss the procedure used.

Obtain a simple random sample for each stratum. Be sure to use a different starting point in Table I or a different seed for each stratum. Results will vary.

Determine the level of measurement of the variable. Positions of persons in a line

Ordinal

Determine the level of measurement of the variable. Positions of runners in a race

Ordinal

A variable is at the _________________ if it has the properties of the nominal level of measurement. However, the naming scheme allows for the values of the variable to be arranged in a ranked or specific order.

Ordinal Level of Measurement

In a championship football game, a quarterback completed 59% of his passes for a total of 265 yards and 2 touch downs. Is this value a parameter or statistic?

The value is a parameter because the quarterback's passes are a population.

Variables that are nominal or ordinal are _______ variables.

Qualitative

are observations corresponding to a qualitative variable.

Qualitative Data

Variables that are interval or ratio are ________ variables.

Quantitative

are observations corresponding to a quantitative variable.

Quantitative Data

A variable is at the ___________ if it has the properties of the interval level of measurement and the ratios of the values of the variable have meaning. A value of zero means the absence of the quantity. Arithmetic operations such as multiplication and division can be performed on the values of the variable.

Ratio Level of Measurement

Define response variable

The quantitative or qualitative variable for which the experimenter wishes to determine how its value is affected by the explanatory variable

A survey conducted by revenue collection agents asks taxpayers to identify sources of fraudulent deductions

Response bias

In a survey, college students are asked to disclose the number of hours they study

Response bias

A survey that pertains to feelings about federal income tax does not include high-income earners

Sampling bias

The owner of a shopping mall wishes to expand the number of shops available in the food court. She has a market researcher survey the first 80 customers who come into the food court during weekday mornings to determine what types of food the shoppers would like to see added to the food court. What is the cause of the​ bias?

Sampling bias

​A(n) _______ is a numerical summary of a sample.

Statistic

Define statistics.

Statistics is the science of​ collecting, organizing,​ summarizing, and analyzing information to draw a conclusion and answer questions. In​ addition, statistics is about providing a measure of confidence in any conclusions.

Determine whether the underlined numerical value is a parameter or a statistic. Explain your reasoning. In a poll of a sample of 12 comma 000 adults in a certain city comma Modifying 12 % with underline said they left for work before 6 a.m.

Statistic​, because the data set of a sample of 12,000 adults in a city is a sample

Define confounding

The effect of two factors​ (explanatory variables on the response​ variable) cannot be distinguished.

Determine whether the study depicts an observational study or an experiment. Fifty patients with brain cancer are divided into two groups. one group receives an experimental drug to fight cancer, the other a placebo. After two years, the spread of cancer is measured. Does the description correspond to an observational study or an experiment

The study is an experiment because the researchers control one variable to determine the effect on the response variable

A discrete variable is a quantitative variable that has either a finite number of possible values or a countable number of possible values. A discrete variable cannot take on every possible value between any two possible values.

The term countable means that the values result from counting, such as 0,1,2,3, and so on.

Determine whether the underlined value is a parameter or a statistic. Following the election, 18% of the governors of all 50 areas of a country were female.

The value is a parameter because the governors of all 50 area of a country are a population.

The 115th United States Congress is comprised of members of the Senate and House of Representative. The 115th Congress begins January 3, 2017. In the 115th Congress, the United States Senate had 52 Republicans, 46 Democrats, and 2 Independents. The president wants to have a luncheon with 4 Republicans, 4 Democrats and 1 Independent. Obtain a stratified sample in order to select members who will attend the luncheon.

To obtain the stratified sample, conduct a simple random sample within each group. That is, obtain a simple random sample of 4 Republicans (from the 52), a simple random sample of 4 Democrats (from the 46), and a simple random sample of 1 Independent from the 2. Be sure to use a different seed for each stratum.

Suppose three different individuals conduct the same statistical​ study, such as estimating the average commute time of students at a college. It is possible that all three studies end up with different results.

True. Statistical studies typically look at samples rather than entire populations. Since each study is likely to draw different​ samples, it is quite possible that each study ends up with different​ results, due to variability in the data.

​_________ are the characteristics of the individuals of the population being studied.

Variables

A gambler wanted to compare two types of poker strategies. One type is playing fight and the other is a loose style of play. It is a common belief that playing tight will win more pots. This belief is tested by having 10 poker players use each type of strategy in a game and comparing the number of pots won under each method of play. A coin flip was used to determine which type of poker strategy each player would follow first. Results indicated that there was no difference in the first types of strategy. Complete parts (a) through (i) below.

What type of experimental design is this? match pairs design What is the response variable in this study? the number of pots won. What is the factor that is set to predetermined levels? What is the treatment? The factor is the type of strategy. The treatments are playing tight and a loose style of play Identify the experimental units. Choose the correct answer below. the poker players Why is a coin used to decide the poker strategy each player would follow first? to eliminate bias as to which strategy was used first the diagram would be the 3 boxes

placebo

a fake drug used in the testing of medication

convenience sample

a form of nonprobability sample using respondents who are convenient or readily accessible to the researcher—for example, employees, friends, or relatives. the individuals are easily obtained and not based on randomness.

simple random sample

every member of the population has a known and equal chance of selection

matched pairs design

is an experimental design in which the experimental units are paired up. The pairs are selected so that they are related in some way (that is, the same person before and after a treatment, twins, husband and wife, same geographical location, and so on). There are only two levels of treatment in a matched-pairs design.

stratified sample

is obtained by dividing the population into nonoverlapping groups called strata and then obtaining a simple random sample from each stratum. The individuals within each stratum should be homogenous (similar) in some way.

systematic sample

is obtained by selecting every kth individual from the population. The first individual selected corresponds to a number between 1 and k.

completely randomized design

is one in which each experimental unit is randomly assigned to a treatment.

Random sampling

is the process of using chance to select individuals from a population to be included in the sample.

Suppose the size of a population is N=3,000 and the sample size desired is n=18. What value of k should be used to obtain a systematic sample from this​ population?

k = N/n = 3,000/18 = 166

sample size

n

A professor wants to randomly select 4 students to go to the board. She decides to randomly select the 4 student who enters the classroom and every 9 student after that. Determine the students who will be going to the board. Write down the student numbers.

n, (n+9), (n+18), (n+27) = 4,13,22,31 In this case n=4, and 9 because its the number of students after that. so, first number is 4 for the randomly selected first student, then add 9 to ever number after that till you reach 4 students

Nonsampling errors

result from undercoverage, nonresponse bias, response bias, or data-entry error. Such errors could also be present in a census.

Sampling error

results from using a sample to estimate information about a population. This type of error occurs because a sample gives incomplete information about a population.

systematic sampling

select some starting point and then select every kth element in the population

Stratefied sampling

subdivisions of a population with similar characteristics, called strata. Example: On campus students/Off campus students

Researchers wanted to know if there is a link between proximity to​ high-tension wires and the rate of leukemia in children. To conduct the​ study, researchers compared the rate of leukemia for children who lived within​ 1/2 mile of​ high-tension wires to the rate of leukemia for children who did not live within​ 1/2 mile of​ high-tension wires. The researchers found that the rate of leukemia for children near​ high-tension wires was higher than the rate for those not near​ high-tension wires. Can the researchers conclude that proximity with​ high-tension wires causes leukemia in​ children?

​No, because this is an observational study.