Introduction
The New Track
“Wow! It looks like we are going to get a new track at the school,” Sam said at breakfast as he read the newspaper over his Dad’s shoulder. “It must be because we won at regionals.”
Yes, the track team the Hawks had taken first place at regionals. It was the talk of the town and the newspaper article seemed to suggest that great things were going to happen as a result.
“Well, I don’t want to disappoint you Sam, but 53% are for the new track but there is a 5% margin of error,” his Dad explained.
“What does that have to do with anything?”
“A lot. A margin of error can mean that the survey results aren’t completely accurate, or you could say that you can’t take them at face value,” his Dad said.
Sam is perplexed. 53% is 53%, or is it?
In this lesson, you will learn all about margin of error. By the end of the lesson, you will understand how a margin of error can impact results.
What You Will Learn
In this lesson, you will learn how to complete the following skills.
- Make population predictions based on representative survey data.
- Use given survey margin of error to determine upper and lower bounds of the interval containing a sample percent in which a population percent is most likely to lie.
- Interpret and evaluate conclusions of newspaper or magazine articles containing sample survey results or statistical displays.
- Analyze and interpret statistical data organized and displayed in a variety of forms, justifying conclusions.
Teaching Time
I. Make Population Predictions Based on Representative Survey Data
Drawing conclusions based on data is a major goal in gathering data. As you have seen, some data can be misleading and some conclusions can be doubtful. For that reason, it is important to keep an eye open for public information that may not be entirely based on data. We must be critical about where information comes from because biases may occur in the sampling methods, survey questions, display methods, and conclusions. Nevertheless, good science and math can lead to valuable information. Let’s look at how we can make predictions based on survey data.
Example
A cellular phone company conducts a random survey in their service area regarding the problems encountered with phone service. They surveyed 700 people with cellular phones in a population of 125,000 people. They find the following results.
Too Expensive | poor Signal | Not Enough Features | Ugly Design |
---|---|---|---|
39% | 33% | 16% | 12% |
Based on their survey, they can extrapolate the data to the entire population. In other words, when the survey was carried out in a proper manner, they can assume that the survey is accurate for the entire population. If there are 125,000 people in the population, they can assume that the same results found in the survey will be true for all of the 125,000 people. So, how many people will find the phones too expensive? Signal quality poor? Too few features?
Calculate the percent found in the data of the entire population.
Like this:
Too Expensive | poor Signal | Not Enough Features | Ugly Design |
---|---|---|---|
39% | 33% | 16% | 12% |
48,750 | 41,250 | 20,000 | 15,000 |
We could also calculate several other features such as mean, median and mode.
Example
A high school conducts a survey for their foreign language program. They need to know how many sections of each class they need. There are 3,400 students in the high school and each class can have a maximum of 35 students in it. Of 200 students surveyed, they find the following number of student responses:
Class | Spanish | French | German | Chinese | None |
---|---|---|---|---|---|
# of Responses | 85 | 35 | 39 | 23 | 18 |
In order to apply this information to the entire student body, the percentages are calculated. Then, each percent is applied to the entire student population.
Class | Spanish | French | German | Chinese | None |
---|---|---|---|---|---|
Percent | 42.5% | 17.5% | 19.5% | 11.5% | 9% |
Total for population | 1445 | 595 | 468 | 391 | 306 |
Based on this data, how many classes of each should they offer? Divide the number of students for each course by 35 students in each section.
Class | Spanish | French | German | Chinese | None |
---|---|---|---|---|---|
# of classes Needed | 42 | 17 | 14 | 12 | 0 |
Because 35 is the maximum number of students, the number of classes must be rounded up to the next whole number. Classes cannot go over 35 and fractional classes cannot be offered.
II. Use Given Survey Margin of Error to Determine Upper and Lower Bounds of the Interval Containing a Sample Percent in which a Population Percent is Most Likely to Lie
First, notice that in this section we are looking for things that are not true. This is not the usual way that we think through things, so we must keep this fact in mind as we work.
As you know, the method of choosing samples is important to find data whose results you can trust. The better the sampling method is the better the data collected. When data is gathered well, its results will be truer for the entire population. Nevertheless, most research companies and survey takers understand that it is actually quite difficult to find a perfect sample. There will always be a margin of error, or a percentage by which the true numbers for the entire population may differ. In other words, a survey company may calculate a margin of error of . This means that the measurements for the entire population may vary either up or down by 3%.
Example
A survey company reports that 51% of people surveyed said they will vote for Candidate X (with a margin of error of ).
Can we be sure that Candidate X will win? No, since there is a margin of error of , it could be as high as or 54% or it could be as low as or 48%.
Example
A survey with a margin of error of shows that 42% of participants reported preferring a shopping trip to the Sun Valley Mall versus a shopping trip to the Newberry Mall. If the population of shoppers that the two malls serve is 272,000 people, what might be the least number of people that prefer Sun Valley Mall and what might be the greatest?
Reported Preference to Sun Valley | Margin of Error | Extremes | # of People in Population | |
---|---|---|---|---|
Least | 42% | -5% | 37% | at least 100,640 |
Greatest | 42% | +5% | 47% | at most 127,840 |
III. Interpret and Evaluate Conclusions of Newspaper or Magazine Articles Containing Sample Survey Results or Statistical Displays
In modern media, particularly newspapers and magazines, it is common to find reports based on survey results. Oftentimes a margin of error is mentioned. Keep in mind, however, that although magazines and newspapers may intend to report just the news, there is oftentimes a bias based on the author’s opinions, the beliefs of the owners or managers of the companies, or a sheer desire to report exciting or eye-opening news with the intent to sell more issues. For this reason, we must keep a critical point of view when we match what an article might say to what the data tells us.
Example
A local newspaper article entitled “More Beets than Meats” discusses more people switching over to vegetarianism. The article says, “In a hundred people surveyed in 1998, 8% reported being a vegetarian. In a similar survey taken in 2000, that number grew by 50% to 12% (margin of error in surveys ). It’s a great time to invest in produce companies because that number will just keep on growing!”
Now let’s examine the article itself. This article reports that the number of vegetarians grew by 50%. However, it also reports a margin of error of .
This means that the actual results may have been as high as 11% vegetarians in 1998 and as low as 9% vegetarians in 2000. It’s possible that the number of vegetarians actually declined. Also, even if the number of vegetarians did increase by 50% during those two years, it does not mean that the number will continue to increase at that rate. Finally, there is no mention of the sampling method. The method they used may not have been representative of the entire population.
IV. Analyze and Interpret Statistical Data Organized and Displayed in a Variety of Forms, Justifying Conclusions
“Don’t believe everything you read.” That is what a lot of people will tell you. What is more important is that you interpret statistical information and displays appropriately. Based on your own interpretations, you can then draw your own conclusions.
Example
In 1945, the world population was about 2 billion people. In 2005, it was about 6 billion. In 2020, the world population is expected to be 9 billion people.
The data is graphed below.
What conclusions can be drawn from this data?
- The population grew faster in more recent years.
- The population is 3 times higher today than it was 60 years ago.
- The trend implies that the population will continue to grow in the future at faster rates.
How might this affect the world?
- There being more people implies that more resources will be needed to sustain them.
- More housing and infrastructure will be needed to accommodate more people.
- Increased population may also increase the economy as people create wealth and spend money.
Now let’s go back and apply what we have learned to the problem from the introduction.
Real-Life Example Completed
The New Track
Here is the original problem once again. Reread it and then calculate the margin of error.
“Wow! It looks like we are going to get a new track at the school,” Sam said at breakfast as he read the newspaper over his Dad’s shoulder. “It must be because we won at regionals.”
Yes, the track team the Hawks had taken first place at regionals. It was the talk of the town and the newspaper article seemed to suggest that great things were going to happen as a result.
“Well, I don’t want to disappoint you Sam, but 53% are for the new track but there is a 5% margin of error,” his Dad explained.
“What does that have to do with anything?”
“A lot. A margin of error can mean that the survey results aren’t completely accurate, or you could say that you can’t take them at face value,” his Dad said.
Sam is perplexed. 53% is 53%, or is it?
Solution to Real – Life Example
First, we have to look at the given information and the margin of error. We know that the survey said that 53% were in favor of the new track. The margin of error is 5%. This means that it could be as high as 58% or as low as 48% in favor of the new track. Depending on the percentage that the school thinks it needs to allocate funds for a new track, this percentage could impact whether or not a new track is gotten.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Margin of Error
- A percentage by which the true number for the entire population may differ.
Time to Practice
Directions: Use what you have learned on each problem presented below.
A job satisfaction survey is taken of 500 people who work in the auto industry. The results are shown in the table below:
Very Dissatisfied | Dissatisfied | Satisfied | Very Satisfied |
---|---|---|---|
16% | 21% | 41% | 22% |
- Based on the margin of error, find the percent range of responses in each category.
- Assuming that there are 340,000 people in the industry, what is the range of responses for each category?
- What conclusions can you draw based upon this data?
Directions: Look at the chart regarding the Prom Location. Assume a margin of error of .
- If the school has 420 students planning to attend, what is the percent range and the numerical range of students who will prefer each of the locations?
- What conclusions can be drawn on this data?
- What other information do you think may be important in making the decision?
A college newspaper reports the following:
“Ever since the new volleyball stadium was built at the end of 2005, the interest in the Women’s Volleyball Team has grown. In turn, that has made the team go nearly perfect this year. Their current record is 14 wins and 3 losses. Last year at this time, they were only 13 and 4. Come and see the new stadium and encourage the college to build the new football stadium, too!”
- What conclusions have been drawn by the writer of this article?
- Do you agree with them? Why or why not?
- What bias might the writer have had?
Directions: Median house prices for 3 states are shown below.
- What conclusions can be drawn from this data?
- What would be an inappropriate conclusion drawn from this data? Why?
Extension – have students use a newspaper and find statistics. Then have them interpret the data and the different data displays. Have students present their findings to the class.