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# Interpreting Graphical Displays

## Margin of error

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Interpreting Data

Have you ever had a question about how to interpret different results? Take a look at this dilemma.

“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 Concept, you will learn all about margin of error. By the end of it, you will understand how a margin of error can impact results.

### Guidance

Drawing conclusions based on data is a major goal in gathering data. 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.

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.

Worst Problem with Telephone
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: .39×125000=48,750\begin{align*}.39 \times 125000=48,750\end{align*}

Worst Problem with Telephone
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.

Let's think about margin of error. Now we are are looking for information that is 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 ±3%\begin{align*}\pm 3 \%\end{align*}. This means that the measurements for the entire population may vary either up or down by 3%.

Take a look at this situation.

A survey company reports that 51% of people surveyed said they will vote for Candidate X (with a margin of error of ±3%\begin{align*}\pm 3 \%\end{align*}).

Can we be sure that Candidate X will win? No, since there is a margin of error of ±3%\begin{align*}\pm 3 \%\end{align*}, it could be as high as 51+3\begin{align*}51 + 3\end{align*} or 54% or it could be as low as 513\begin{align*}51 - 3\end{align*} or 48%.

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.

Take a look at this situation.

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 ±3%\begin{align*}\pm 3 \%\end{align*}). 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 ±3%\begin{align*}\pm 3 \%\end{align*}.

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.

All 450 students in the graduating class at Springstead High School were surveyed about homework. 27% said they wanted more homework, while 73% said they wanted less homework.

#### Example A

If the margin of error was 3%, how would this impact the percentages?

Solution: More homework could be as low as 24% or as high as 30%. Less homework could be as low as 70% or as high as 76%.

#### Example B

How many students wanted more homework?

Solution: 121.5 or 122 students

#### Example C

How many students wanted less homework?

Solution: 328.5 or 329 students

Now let's go back to the dilemma from the beginning of the Concept.

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

Margin of Error
A percentage by which the true number for the entire population may differ.

### Guided Practice

Here is one for you to try on your own.

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:

Which foreign language class would you like to take?
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.

Which foreign language class would you like to take?
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?

Solution

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.

### 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:

Job Satisfaction Survey(margin of error
Very Dissatisfied Dissatisfied Satisfied Very Satisfied
16% 21% 41% 22%
1. Based on the margin of error, find the percent range of responses in each category.
2. Assuming that there are 340,000 people in the industry, how many people are dissatisfied?
3. Assuming that there are 340,000 people in the industry, how many people are satisfied?
4. Assuming that there are 340,000 people in the industry, how many people are very satisfied?
5. Assuming that there are 340,000 people in the industry, how many people are very dissatisfied?
6. What conclusions can you draw based upon this data?

Directions: Look at the chart regarding the Prom Location. Assume a margin of error of ±5%\begin{align*}\pm 5 \%\end{align*}.

1. If the school has 420 students planning to attend, what is the percent range of students who will prefer each of the locations?
2. How many students want the Holiday Inn?
3. How many students want the Hilton?
4. How many students want the Marriott?
5. How many students want the Four Seasons?
6. What conclusions can you draw from the data

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!”

1. What conclusions have been drawn by the writer of this article?
2. Do you agree with them? Why or why not?
3. What bias might the writer have had?

### Vocabulary Language: English

Margin of Error

Margin of Error

A percentage by which the true number for the entire population may differ.