Blinding is when, in a study with a placebo, the participants do not know whether they receive the treatment or the placebo. Control Group is a group in an experiment study that does not receive the treatment. Confounding Variable occurs when an experimenter cannot determine which factor affected the study. Double-Blind Experiment is when, in a study with a placebo, the participants and the researcher do not know which group had the treatment or the placebo. Experimental Unit is a part of the control group that is given a placebo. Placebo is a fake treatment, participants in an experiment may receive something that they think is the treatment but it is really just a sugar pill or something else. Placebo Effect occurs when subjects are given a placebo and have a positive reaction because they convince themselves that it is the cure and so they get a positive effect from no treatment at all.
Lesson Objectives 1.3 Data Collection and Experimental Design Part 1 1.) How to design a Statistical Study. 2.) How to collect data by doing an Observational study, performing an Experiment, using a Simulation, or using a Survey. The goal of every statistical study is to collect data and then use the data to make a decision. Any decision you make using the results of a statistical study is only as good as the process used to obtain the data. If the process is flawed then the resulting decision will be questionable. Even though you may never design a statistical study, it is likely you will have to interpret the results of one, and before interpret one you should determine whether the results are valid or not. To be able to determine if the results of a study are valid you must be familiar with how to design a statistical study. Guidelines 1.) Identify the variable(s) of interest or focus and the population of the study 2.) Develop a detailed plan for collecting data. If you use a sample, make sure the sample is representative of the population. 3.) Collect the data. 4.) Describe the data using Descriptive Statistics techniques. 5.) Interpret the data and make decisions about the population using Inferential Statistics. 6.) Identify any possible errors.
Data Collection There are several ways to collect data but the best way to collect your data will depend upon the study you are doing. Four Methods to Collect Data 1.) Observational Study when a researcher observes and measures activities or behaviors of interest in a population but does not control any part of the study. Example: The behavior infants to three years old of placing nonfood objects in their mouths. 2.) Experiment when an experiment is performed a treatment is applied to a part of a population and the responses to it are observed. Another part of the population may be a control group in which no treatment is applied. A lot of times a part of the control group called experimental unit are given a placebo. When performing an experiment it is a good idea to have the same amount of participants in the treatment as in the control group. Example: An experiment was performed where diabetics took cinnamon extract daily for 40 days and a control group took nothing. 3.) Simulation is the use of a mathematical or physical model to reproduce the conditions of a situation or process. Normally simulations are used when study situations are impractical or even dangerous to create in real life and often save time and money. Example: Crash dummies are used to simulate the effect of vehicle accidents on the human body for studies to improve safety features. 4.) Survey are verbal or written questionnaires given to people to investigate a characteristic or behavior of a population.
Example A survey is conducted on a sample of female physicians to determine whether the main reason for their career choice is because of the money. Example 1 Pg.17, 1-4; Try It Yourself 1, 1-2. Which method of data collection would you use to collect data for each study? Explain. 1.) A study of the effect of changing flight patterns on the number of airplane accidents. A simulation would be best used here because it would be impractical to model the situation in real life. 2.) A study of the effect of eating oatmeal on lowering blood pressure. An experiment would be best used here because it is talking about a treatment given to a population. 3.) A study of how fourth grade students solve a puzzle. An observational study would be best because you do not want to influence the students or control the study, only to observe how the students solve the puzzle. 4.) A study of U.S. residents approval rating of the U.S. President. A survey would be best used here because you are asking for the opinions or questioning a population about a behavior or attitude. Try it Yourself 1 1.) A study of the effect on exercise on relieving depression. An experiment would be best used here because you are using a treatment for a population and there would be a control group. 2.) A study of the success of graduates of a large university in finding a job within one year of graduation. a. Identify the focus of the study. Success of graduates in finding a job within one year of graduation. b. Identify the population of the study.
Graduates of a large University that have been graduates for at least one year. c. Choose an appropriate method of data collection. A survey would be best utilized in this situation because you are questioning a population about something in their lives, not a treatment, not something that can simply be observed, and not something that can be simulated because you want real world results.
Lesson Objectives 1.3 Data Collection and Experimental Design Part 2 1.) How to design an experiment 2.) How to create a sample using random sampling, simple random sampling, stratified sampling, cluster sampling, and systematic sampling and how to identify a biased sample. In order to produce meaningful unbiased results, experiments should be carefully designed and executed. There are three key elements of a well-designed experiment, they are: control, randomization, and replication. Control Because experimental results can be ruined by a variety of factors, being able to control these factors is important. One factor is a confounding variable. Experimenters can help control confounding variables by being sure to only test ONE variable at a time. Experimenters can also have an issue with the placebo effect. This effect is when a participant has a positive reaction to a placebo when, in fact, the participant has not been given any treatment at all. Experimenters can control the placebo effect using blinding or double-blind studies. Randomization This is a process of randomly assigning subjects to different treatment groups.
In a completely randomized design, subjects are assigned to different treatment groups through random selection. Some experiments require blocks, which are groups of subjects with similar characteristics. When using blocks there is often use of randomized block design, where the subjects are put into groups with certain characteristics and then randomly assigned to treatments. Other than block and completely randomized design, there is also matched-pairs design, where subjects are paired up according to similarity and then one subject is given one treatment and the other subject receives another treatment, this helps to compare and contrast how each treatment affects a similar subject. Replication Replication is the repetition of an experiment under the same or similar conditions. Just like in science you must repeat and repeat an experiment because you want to see if the same results will happen over and over again, because maybe the first experiment s results were a fluke and you cannot replicate those results again. Also, the sample size is important because the size of the groups is important, if you have too small a group for a treatment or a placebo you will not be able to determine whether the results are conclusive. If you choose two groups of 10,000 people one for the treatment and one for the placebo, and do not choose the groups so that they are similary (according to age and gender), the results are of less value.
Examples: Pg.18, Example 2, #s 1-2; Try it Yourself 2 Example 2: A company wants to test the effectiveness of a new gum developed to help people quit smoking. Identify a potential problem with the given experimental design and suggest a way to improve it. 1.) The company identifies ten adults who are heavy smokers. Give of the subjects are given the new gum and the other give subjects are given a placebo. After two months, the subjects are evaluated and it is found that the five subjects using the new gum have quit smoking. The sample size is too small to validate the results of the experiment. Also, the experiment must be replicated to check the results. 2.) The company identifies one thousand adults who are heavy smokers. The subjects are divided into blocks according to gender. Females are given the new gum and males are given the placebo. After two months, a significant number of the female subjects have quit smoking. The groups are not similar. The new gum may have a greater effect on woman than on men, or vice versa. The subjects can be divided into block according to gender, but then, within each gender block there must be some that are in the treatment group and some in the control. Try it yourself 2 Using the information in Example 2, suppose the company identifies 240 adults who are heavy smokers. The subjects are randomly assigned to be in a treatment group or in a control group. Each subject is also given a DVD featuring the dangers of smoking. After four months, most of the subjects in the treatment group have quit smoking. a.) Identify a potential problem with the experimental design. There is no way to tell why people quit smoking. They could have quit from the gum or watching the DVD. They created a confounding variable.
b.) How could the design be improved? Two experiments could be done; one using the gum and the other using the DVD.
Sampling Techniques 1.3 Data Collection and Experimental Design Part 3 Recall that it is easier to get responses from a sample than it is a population because it is costly and difficult to get responses from an entire population. A census is a count or measure of an entire population; which can be done and provides complete information but is, as said, costly and difficult to perform. So, instead of taking a census we generally try something called sampling. *sampling is a count or measure of part of a population, and is more commonly used in statistical studies. To collect unbiased data, a researcher must ensure that the sample is representative of the (entire) population and not only a small part of it. Even using the best methods for sampling a sampling error may occur. *A sampling error is when there is a difference between the results of a sample and the results of a population. Sampling Methods and Techniques There are FIVE different sampling techniques. 1.) Random Sample is sample in which every member of a population has an equal chance of being selected for the sample. 2.) Simple Random Sample is a sample in which every possible sample of the same size has the same chance of being selected. This basically means that instead of choosing from individuals there will be a group of a certain size randomly generated. One way to collect a simple random sample is to assign a different number to each member of the population and then use a random number generator to choose a certain number of people for the population each time.
Example: There is a study of the number of people who live in West Ridge County. To use a simple random sample to count the number of people that live in West Ridge County households, you could assign a different number to each household, use a random number generator to generate a group of numbers and then count the number of people living in each selected household. 3.) Stratified Sample this type of sampling is used when you divide the population into strata or groups that are based on similar characteristics such as age, gender, ethnicity, or even political preference. A sample is then randomly selected from each strata. Using this type of sampling ensures that each segment of the population is represented. Example: To collect a stratified sample of the number of people who live in West Ridge County households, you could divide the households into socioeconomic levels (levels of household incomes), and then randomly select households from each different level. 4.) Cluster Sample When the population falls into naturally occurring groups, having similar characteristics, then a cluster sample may be the most appropriate. To select a cluster sample, divide the population into groups, called clusters, and select all of the members in one or more (but not all) of the clusters. Example: To collect a cluster sample of people who live in West Ridge County households, divide the households into groups according to zip codes, then select all the households in one or more, but not all, zip codes and count the number of people living in each household. You must be careful, though, the clusters you choose must all have similar characteristics. If you choose a zip code that has mostly rich people living there then the data might not be representative of the population because most likely the majority of people living in West Ridge County are not wealthy.
5.) Systematic Sample is a sample in which each member of the population is assigned a number. They are ordered in some way. A starting number is randomly chosen (meaning maybe the starting number will be 1 or maybe it will be 1,000) then sample members are selected at regular intervals. For example, every 3 rd person, every 5 th, every 100 th, and so on. This type of sampling is easy to use but if there is any regularly occurring pattern in the data, this type of sampling should be avoided. Example: To collect a systematic sample of the West Ridge County households, you could assign a different number to each household, randomly choose a starting number of 72, and then select every 100 th household from that number up and count the number of people living in each. Examples: Pg.20, Example 3, Try it Yourself 3, Pg.22, Example 4, Try it Yourself 4 Example 3 There are 731 students currently enrolled in a statistics course at your school. You wish to form a sample of eight students to answer some survey questions. Select the students who will belong to the simple random sample. Assign numbers 1 to 731 to the students in the course. In the table of random n numbers (Pg. 20), choose a starting place at random and read the digits in groups of three. (Because 731 is a three-digit number).
Try it Yourself 3 A company employs 79 people. Choose a simple random sample of five to survey. a.) In the table in Appendix B (Pg.A7), randomly choose a starting place. Many correct answers are possible, but for example, start with the first digits 92630782 b.) Read the digits in groups of two. 92 63 07 82 40 19 26 c.) Write the five random numbers. 63, 7, 40, 19, 26 Example 4 You are doing a study to determine the opinions of students at your school regarding stem cell research. Identify the sampling technique you are using if you select the samples listed. Discuss potential sources of bias (if any). Explain. 1.) You divide the student population with respect to majors and randomly select and question some students in each major. Because students are dividing into strata (majors) and a sample is selected from each major, this is a stratified sample. No bias known. 2.) You assign each student a number and generate random numbers. You then question each student whose number is randomly selected. Each sample of the same size has an equal chance of being selected, so this is a simple random sample. No bias known.
3.) You select students who are in your biology class. Because the sample is taken from students that are readily available, this is a convenience sample. The sample may be biased because biology students may be more familiar with stem cell research than other students and may have stronger opinions. Try it Yourself 4 You want to determine the opinions of students regarding stem cell research. Identify the sampling technique you are using if you select the samples listed. 1.) You select a class at random and question each student in the class. The sample was selected by using the students in a randomly chosen class. Cluster sampling. 2.) You assign each student a number, and after choose a starting number, question every 25 th student. a. Determine how the same is selected and identify the corresponding sampling technique. The sample was selected by numbering each student in the school, randomly choosing a starting number, and selecting students at regular intervals from the starting number. Systematic Sampling. b. Discuss potential sources of bias (if any). Explain The sample may be biased if there is any regularly occurring pattern in the data.
Chapter 2 Descriptive Statistics o 2.1 Frequency Distributions and Their Graphs o Frequency Distributions o Graphs of Frequency Distributions o 2.2 More Graphs and Displays o Graphing Quantitative Data Sets o Graphing Qualitative Data Sets o Graphing Paired Data Sets o 2.3 Measures of Central Tendency o Mean, Median, and Mode o Weighted Mean and Mean of Grouped Data o The Shapes of Distributions o 2.4 Measures of Variation o Range o Deviation, Variance, and Standard Deviation o Interpreting Standard Deviation o Standard Deviation for Grouped Data o 2.5 Measures of Position o Quartiles o Percentiles and Other Fractiles o The Standard Score
2.1 Frequency Distributions and Their Graphs Part 1 Frequency Distributions Vocabulary Center Shape Variability Frequency f Frequency Distribution Classes Intervals Lower Class Limit Upper Class Limit Class Width Range Midpoint Relative Frequency Cumulative Frequency Remember a data set is a set of qualitative or quantitative data. When organizing and describing a data set there are three important characteristics: its center, its variability (its spread), and its shape. Frequency Distribution is a table that shows the different classes or intervals that the data fit in and a count of the number of entries in each class. Frequency f - is the frequency of a class, and it is the number of data entries in that class. Lower Class Limit the least number that could be in a class. Upper Class Limit the largest number that could be in a class.
Class Width the lower limits of two consecutive classes subtracted, or the upper limits of two consecutive classes subtracted. Range the maximum entry minus the minimum entry. How to Construct a Frequency Distribution from a Data Set Examples Pg. 39-40, Example 1 and Try it Yourself 1
Example 1 The following sample data set lists the prices (in dollars) of 30 portable GPS navigators. Construct a frequency distribution that has seven class. Solution: 90 130 400 200 350 70 325 250 150 250 275 270 150 130 59 200 160 450 300 130 220 100 200 400 200 250 95 180 170 150 Following the guidelines. 1.) The problem states there should be 7 classes. 2.) The minimum data entry is 59 and the maximum data entry is 450, so the range is 450 59 = 391. Divide the range by the number of classes and round up to find the class width. 391/7 = 55.86 when rounded up, round to 56. The reason we do this is because we want to know how to evenly divide up the numbers from 59 450 evenly into 7 groups. Range 391 ; Class Width 56. 3.) Use the minimum data entry as the lower limit for the first class. To find the lower limits of the remaining six classes, add the class width of 56 to the lower limit of each previous class. 4.) Then tally the numbers that go into each interval. 5.) Add up your tallies and write the frequency as a number in your table. Classes Tallies Frequency 59-114 5 115-170 8 171-226 6 227-282 5
283-338 2 339-394 1 395-450 3 Try it Yourself 1, Pg. 40 Construct a frequency distribution using the ages of the 50 richest people data set listed below. Use eight class. Solution: 1.) 8 classes 2.) Range: 89-35 = 54 Class Width: 54/8 = 6.75 7 3.) 4.) and 5.) Classes Tallies Frequency 35-41 2 42-48 5 49-55 7 56-62 7 63-69 10 70-76 5 77-83 8 84-90 6
The above frequency distributions are called standard frequency distributions. You can also include other features such as midpoint, relative frequency, and cumulative frequency. Midpoint is the sum of the lower and upper limits of the class divided by two. Also called the class mark. Formula: lower limit+upper limit 2 Relative Frequency is the portion or percent of the data that falls in that particular class. To find the relative frequency, divide the frequency f, by the sample size n. Formula: frequency total number of data entries = f n Cumulative Frequency is the sum of the frequencies of that class and all previous classes. The cumulative frequency of the last class is equal to the sample size n. You only have to find class the first midpoint using the formula. Then you can just add the class width to the previous midpoint to find the next midpoint.
Examples Pg.41, Example 2 and Try it Yourself 2 Example 2: Using the frequency distributions you found in example 1, find the midpoint, relative frequency, and cumulative frequency of each class. Identify any patterns. Solution: Classes f Midpoint Relative f Cumulative f 59-114 5 (59+114)/2 5 = 86.5 30 =.166 = 17% 5 115-170 8 (115+170)/2 8 = 142.5 30 =.266 = 27% 5+8 = 13 171-226 6 (171+226)/2 6 = 198.5 30 =.2 = 20% 5+8+6 = 19 227-282 5 (227+282)/2 5 = 254.5 30 =.166 = 17% 5+8+6+5 = 24 283-338 2 (283+338)/2 2 = 310.5 30 =.066 = 7% 5+8+6+5+2 = 26 339-394 1 (339+394)/2 1 = 366.5 30 =.033 = 3% 5+8+6+5+1 = 27 395-450 3 (395+450)/2 3 = 422.5 30 =.1 = 10% 5+8+6+5+1+3 = 30
Try it yourself 2 Using the frequency distribution from try it yourself 1, find the midpoint, relative frequency, and cumulative frequency of each class. Identify any patterns. Solution: Classes f Midpoint Relative f Cumulative f 35-41 2 (35+41)/2 = 38 2 50 =.04 = 4% 5 42-48 5 (42+48)/2 = 45 5 50 =.1 = 10% 5+2 = 7 49-55 7 (49+55)/2 = 52 7 50 =.14 = 14% 7+5+2 = 14 56-62 7 (56+62)/2 = 59 7 50 =.14 = 14% 7+7+5+2 = 21 63-69 10 (63+69)/2 = 66 10 50 =.2 = 20% 10+7+7+5+2 = 31 70-76 5 (70+76)/2 = 73 5 50 =.1 = 10% 5+10+7+7+5+2 = 36 77-83 8 (77+83)/2 = 80 8 50 =.16 = 16% 8+5+10+7+7+5+2 = 44 84-90 6 (84+90)/2 = 87 6 50 =.12 = 12% 6+8+5+10+7+7+5+2 = 50
2.1 Part 2 Graphs of Frequency Distributions There are four different graphs to display frequency distributions. 1.) Frequency Histogram 2.) Frequency Polygon 3.) Relative Frequency Histogram 4.) Cumulative Frequency Ogive 1.) Frequency Histogram is a bar graph that represents the frequency distribution of a data set. It must have the following properties. The horizontal scale (across or the x-axis) is quantitative and measures the data values. The vertical scale (up and down or the y-axis) measures the frequencies of the classes. Consecutive bars (bars next to each other) much be touching. Because the bars of a histogram touch they must begin and end at class boundaries, which are the numbers that separate classes without gaps between them. If data entries are integers (,-3, -2, -1, 0, 1, 2, 3, ) subtract 0.5 from each lower limit to find the lower class boundary and add 0.5 to each upper limit to find the upper class boundary.
Steps to Creating a Histogram 1.) Draw a quarter plane, only positive x and positive y axis. 2.) Label x-axis with lower and upper limits of your frequency distribution or with the midpoints of each class. 3.) Label y-axis with numbers for the frequency. 4.) Make a bar for each interval that goes up to the frequency of that interval. *MAKE SURE ALL BARS TOUCH* Examples; Example 3 and Try it Yourself 3 on Pg.42-43 Example 3: Draw a frequency histogram for the frequency distribution in Example 2. Describe any patterns. Frequency distribution is show below. Classes f 59-114 5 115-170 8 171-226 6 227-282 5 283-338 2 339-394 1 395-450 3
Pattern: Over half the GPS navigators are priced below $226.50. Try it Yourself 3: Draw a frequency histogram for the frequency distribution in try it yourself 2. Describe any patterns. Frequency distribution is show below. Classes f Midpoint Relative f Cumulative f 35-41 2 38 4% 2 42-48 5 45 10% 7 49-55 7 52 14% 14 56-62 7 59 14% 21 63-69 10 66 20% 31 70-76 5 73 10% 36 77-83 8 80 16% 44 84-90 6 87 12% 50
Pattern: The most common age bracket for the 50 richest people is 63-69. Frequency Polygon is a line graph that emphasizes the continuous change in frequencies. A frequency polygon is another way to graph a frequency distribution. Steps to Constructing a Frequency Polygon 1.) Draw a quarter plane, only positive x and positive y axis. 2.) Label x-axis with the midpoint of your frequency distribution and subtract the class width from the first class midpoint and add the class width to the last midpoint to extend the graph to the left and right so that the beginning and ending points touch the x-axis so that it creates a polygon. 3.) Label y-axis with numbers for the frequency.
4.) Place points that correspond to the given values and connect with lines (making a line graph). Examples, Pg.43 Example 4 and Try it Yourself 4 Example 4 Classes f 59-114 5 115-170 8 171-226 6 227-282 5 283-338 2 339-394 1 395-450 3
Try it Yourself 4 Classes f Midpoint Relative f Cumulative f 35-41 2 38 4% 2 42-48 5 45 10% 7 49-55 7 52 14% 14 56-62 7 59 14% 21 63-69 10 66 20% 31 70-76 5 73 10% 36 77-83 8 80 16% 44 84-90 6 87 12% 50
There are two other types of graphs you can use to represent frequency distributions. Relative Frequency Histogram is a histogram which graphs the relative frequencies rather than the actual frequencies the y-axis is labeled with the decimal equivalent of the percentage, not the percentage itself. Steps to Constructing a Relative Frequency Histogram 1.) Draw a quarter plane, only positive x and positive y axis. 2.) Label x-axis with lower and upper limits of your frequency distribution or with the midpoints of each class. 3.) Label y-axis with the relative frequencies of each class (in decimal form). 4.) Make a bar for each interval that goes up to the frequency of that interval. *MAKE SURE ALL BARS TOUCH* Example
Ogive is a line graph which graphs the cumulative frequency of a frequency distribution. Should always go upwards. Steps to Constructing an Ogive (Cumulative Frequency Graph) 1.) Draw a quarter plane, only positive x and positive y axis. 2.) Label x-axis with lower and upper limits of your frequency distribution or with the midpoints of each class. 3.) Label y-axis using a scale that will contain all cumulative frequencies. 4.) Plot points that correspond to the data values and connect the points with lines (making a line graph).