Lesson 6
| Site: | MoodleHUB.ca 🍁 |
| Course: | Math 20-2 SS |
| Book: | Lesson 6 |
| Printed by: | Guest user |
| Date: | Saturday, 6 September 2025, 2:36 AM |
Description
Created by IMSreader
1. Lesson 6
Module 4: Statistical Reasoning
Lesson 6: Confidence in Data
Focus
In 2010, Ipsos-Reid did a survey for Agriculture and Agri-Food Canada to determine how consumers felt about the safety and the quality of food in Canada.
One part of the survey looked at how confident consumers were that the food produced in Canada is safe. According to the survey results, approximately 96% of Canadians are at least somewhat confident in the safety of food produced in Canada, while 4% are not very confident that Canadian food is safe.
Adapted from © Agriculture and Agri-Food Canada
The percentage of people who have confidence in the Canadian food supply varies based on the respondent’s province, age, gender, and whether the respondent was from a rural or urban area.
The people that were not completely confident in the safety of the Canadian food supply gave various reasons for their decreased confidence. For instance, 28% said that food products will never be 100% safe, while 15% mentioned that the 2008 listeriosis crisis affected their confidence. (Listeriosis is a dangerous infection caused by eating food contaminated with Listeria, which is a type of bacteria.) Others mentioned that they did not trust the food inspection system or company’s quality control systems.
Think about how confident you need to be in the products and services that you use in your life. For instance, do you think that you need to have the same level of confidence in your doctor that you have in the person making you lunch at a restaurant?
In this lesson you will investigate the level of confidence given to statistical data in a variety of situations.
This lesson will help you answer the following critical question:
- How can the level of confidence be used to show how certain statisticians are that their data accurately represents the true population?
Assessment
- Try This 2
All assessment items you encounter need to be placed in your course folder.
Materials and Equipment
- calculator
1.1. Discover
Module 4: Statistical Reasoning
Discover
© caroline letrange/11236192/Fotolia
The level of confidence you require may vary for different situations. For example, the level of confidence you need in your doctor is probably different than the level of confidence you need in the person designing the video games you play.
Level of confidence can be affected by a variety of factors. For instance, a video game company may require a higher level of confidence in the video game designers it hires than the average consumer requires. The company’s success is dependent to some degree on the ability of the video game creator to create interesting and fun games.
Another factor affecting level of confidence could be the company’s history. You would probably have more confidence in a bungee jumping company that has had over 10 000 successful jumps and a flawless safety record than a bungee jumping company where you are the first customer to jump.
Try This 1
Use the My Level of Confidence Survey questionnaire to indicate how much confidence you feel you need to have in various situations.
Keep a record of the answers you chose for each question in this survey, and save your results to your course folder. You may choose to record your answers in a document, or you may wish to take screen captures of each of your answers.
Share 1
With a partner, compare your required level of confidence for the various scenarios in Try This 1. In your discussion, provide reasons why you chose the range of confidences that you did in each situation. Also, consider whether your level of confidence changed towards a certain situation or a person if you were a consumer instead of the employer or a government inspector.
Did you revise your answers based on this discussion with a partner? Why or why not? Place your notes in your course folder.
1.2. Explore
Module 4: Statistical Reasoning
Explore
In statistics, level of confidence is used to show how certain statiticians are that the data captures the true meaning of whatever is being tested. Level of confidence also gives an indication of whether or not the results are repeatable. In other words, if the survey or test was repeated, how certain could you be that you would get the same results over and over again?
Health Canada
Consider the following survey results.
In January 2011, a Vision Critical/Angus Reid poll of 1022 Canadian adults found that 82% of repondents support the use of health warnings on cigarette packages. The results of the poll were accuate within plus or minus 3.1% 19 times out of 20.
margin of error: the measurement of the accuracy of the results of a survey
The larger the margin of error, the less accurate the estimated value. Margin of error is usually expressed as plus or minus a percent. Example: ±5%
confidence interval: interval in which the true value you are trying to determine would be expected to lie, to a stated degree of accuracy
Confidence interval can be expressed using plus or minus notation or as a range of values. Example: 55% ± 5% or 50% to 60%
confidence level: likelihood that the result for the “true” population lies within the confidence interval
A confidence level of 95% is usually used for surveys, but 90% or 99% is sometimes used.
— From CANAVAN-MCGRATH ET AL. Principles of Mathematics 11, © 2012 Nelson Education Limited. Reproduced by permission.
The poll tells us that the companies doing the study know that the survey results are not 100% accurate. The companies are confident that the actual percentage of people who share this belief is unlikely to be 3.1% higher than 82% or 3.1% lower than 82%; this is known as a margin of error. The pollsters can accurately say that if you repeated this survey on the entire adult population, you would get results that were within 3.1% of their results 19 times out of 20.
So, the pollsters can say with some confidence that between 78.9% (82% − 3.1%) and 85.1% (82% + 3.1%) of the population would support the use of health warnings on cigarette packages. This range between 78.9% and 85.1% is called the confidence interval. The confidence interval is determined by taking the survey or poll result (82%) plus or minus the margin of error (3.1%).
The confidence level is 19 out of 20 or 95%. For this survey, you can be 95% certain that between 78.9% and 85.1% of the “true” population would be in support of the health labels.
The confidence interval can be used to estimate the range of the mean for the population. In other words, it can help you estimate how many people in the population would support health labels on cigarettes. Suppose the total population of adults is 50 000. According to the survey results, between 78.9% and 85.1% of the population would support health labels.
50 000 × 78.9% = 39 450
50 000 × 85.1% = 42 550
So, you can say with 95% confidence that between 39 450 to 42 550 adults, in a population of 50 000 adults, would support the use of health labels on cigarettes.
Read “Example 1: Analyzing and applying survey results” on page 295 of your textbook to see another example of how confidence intervals can be used to determine the certainty of a survey’s results.
Try This 2
You will submit your work from Try This 2 to your teacher for marking.
The results of the Vision Critical/Angus Reid poll also found that 60% of Canadians believe the images currently used for health warnings on cigarette packages are about right, while 24% wanted even more graphic imagery.
Calculate the range of people that would want more graphic images in a population of 30 000 adults. Recall that the margin of error for the survey was ±3.1% and the confidence level was 95% (19/20). (3 marks)
1.3. Explore 2
Module 4: Statistical Reasoning
Self-Check 1
Complete “Practising” questions 5 and 6 on page 303 of your textbook. Note: You will need some of the data that is in question 1 on page 302 in order to complete question 5. Answer
How do confidence intervals, confidence levels, and margins of error relate to the normal distribution?
In Lesson 3 you learned that you can determine the percentage of data that falls between any two whole number standard deviations on the normal curve using the rule of 68-95-99.7. That is, 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations of the mean, and 99.7% of data falls within three standard deviations of the mean.
To interpret this differently, there is a 68% confidence level that a piece of data lies within one standard deviation of the mean, there is a 95% confidence level that a piece of data lies within two standard deviations of the mean, and so on.
The animation titled Normal Curve will help you understand the connection between confidence intervals and the normal curve.
Consider the OJ Juice Company example from Lesson 5. One of the company’s best sellers is a 500-mL bottle of orange juice. When the production process is running smoothly, the volume of juice in the bottles is normally distributed with a mean of 500 mL.
Try This 3
Use the interactive applet titled Check on Quality Standard to determine a 95% confidence interval.
1.4. Explore 3
Module 4: Statistical Reasoning
If you perform a survey or experiment only once, you cannot have much confidence that you can repeat the survey or experiment and get the same results. The more you repeat the survey or experiment, the more confidence you will have in predicting the results of the next survey or experiment.
Consider a high school with 250 students in Grade 12. If you ask one student what theme song he or she wants for graduation, you can’t have any confidence that you will receive the same answer when you ask a second student. If, however, you have asked 100 students and 80 of them picked the same song, you can have increased confidence that 8 of the next 10 students you ask might pick the same song.
There is a demo posted online that shows the connection between levels of confidence and sample size. To explore the demo, put the following keywords into your favourite search engine: “The diagram below illustrates a series of tests—each unique in terms of test data—that are run on a system.”
Using those keywords, you should find a link to a website that explores “Level of Confidence” and its connection to sample size. At the bottom of the website page is a simple simulation. If you cannot find this website through your search, contact your teacher for assistance.
The simulation, which is provided near the bottom of the website, shows a line segment dropping down to the normal curve at different spots to replicate a series of tests to determine the level of confidence. To get a full explanation of how to use the demo, choose “Instructions For Using This Demo.”
Important features to note:
- To change the sample size, left-click once to increase the sample size by 50 each time. Double-click to get the sample size back to 50.
- To change the confidence level, right-click and the level will increase 5% each time. The default confidence level of the test is 75%, so right-click four times to get a 95% confidence interval.
- A catch occurs when the falling red bar includes the mean.
- A miss occurs when the falling red bar does not include the mean.
- The width of the red bar is the confidence interval.
- Each time you click, you reset the simulation.
Watch the simulation. Notice that the total number of tests, number of catches, number of misses, and “% of catches so far” changes the longer you let the demo run.
- When you first start the demo, how close are your experimental results (% catches so far) to your confidence level?
- As the simulation goes on, what do you notice about these two values?
- Does the number of trials affect the outcome for the level of confidence?
Self-Check 2
1.5. Connect
Module 4: Statistical Reasoning
Connect
If you have not done so already, submit your work from Try This 2 to your teacher for marking.
1.6. Lesson 6 Summary
Module 4: Statistical Reasoning
Lesson 6 Summary
In statistics, level of confidence is used to show how confident statiticians are that the data captures the true meaning of whatever is being tested. It also gives an indication of whether or not the results are repeatable. In other words, if the survey or test was repeated, level of confidence measures how certain you could be that you would get the same results over and over again.
BananaStock/Thinkstock
The level of confidence of the data or the certainty of the results can be determined by calculating the margin of error, confidence interval, and confidence level of the data.
When survey companies publish results, information that indicates the certainty of the results is usually included. The confidence interval is expressed as the survey or poll result (e.g., 60%) plus or minus the margin of error (e.g., 3.1%). The confidence interval is the range of the mean of a population for a given confidence level.
Confidence level is how confident you are that the test results would fall within the confidence interval if you repeated the test. A confidence level of 95% tells you that if the survey was repeated 20 times, the survey result would fall within the given confidence interval in 19 out of those 20 times.
When comparing the level of confidence to the normal curve, the area under the curve represents the level of confidence. The 68-95-99.7 rule relating to normal curves and standard deviations can help you to find a 95% confidence interval quickly on a normal curve. By finding the values of the boundaries for this region (within two standard deviations of the mean), you are identifying the confidence interval. Since 95% of the data will fall within two standard deviations of the mean, you can state that you are 95% confident that the results are between μ + 2σ and μ – 2σ.
In Lesson 7 you will look at what types of factors affect level of confidence and what types of things you need to consider when making inferences from statistical data.