8.4.5: Comparing Populations With Friends
Lesson
Let's ask important questions to compare groups.
Exercise \(\PageIndex{1}\): Features of Graphic Representations
Dot plots, histograms, and box plots are different ways to represent a data set graphically.
Which of those displays would be the easiest to use to find each feature of the data?
- the mean
- the median
- the mean absolute deviation
- the interquartile range
- the symmetry
Exercise \(\PageIndex{2}\): Info Gap: Comparing Populations
Your teacher will give you either a problem card or a data card . Do not show or read your card to your partner.
If your teacher gives you the problem card :
- Silently read your card and think about what information you need to be able to answer the question.
- Ask your partner for the specific information that you need.
-
Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem. - Share the problem card and solve the problem independently.
- Read the data card and discuss your reasoning.
If your teacher gives you the data card :
- Silently read your card.
-
Ask your partner
“What specific information do you need?”
and wait for them to
ask
for information.
If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information. - Before sharing the information, ask “ Why do you need that information? ” Listen to your partner’s reasoning and ask clarifying questions.
- Read the problem card and solve the problem independently.
- Share the data card and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Are you ready for more?
Is there a meaningful difference between top sports performance in two different decades? Choose a variable from your favorite sport (for example, home runs in baseball, kills in volleyball, aces in tennis, saves in soccer, etc.) and compare the leaders for each year of two different decades. Is the performance in one decade meaningfully different from the other?
Exercise \(\PageIndex{3}\): Comparing to Known Characteristics
-
A college graduate is considering two different companies to apply to for a job. Acme Corp lists this sample of salaries on their website:
\($45,000\quad $55,000\quad $140,000\quad $70,000\quad $60,000\quad $50,000\)
What typical salary would Summit Systems need to have to be meaningfully different from Acme Corp? Explain your reasoning. -
A factory manager is wondering whether they should upgrade their equipment. The manager keeps track of how many faulty products are created each day for a week.
\(6\quad 7\quad 8\quad 6\quad 7\quad 5\quad 7\)
The new equipment guarantees an average of 4 or fewer faulty products per day. Is there a meaningful difference between the new and old equipment? Explain your reasoning.
Summary
When using samples to comparing two populations, there are a lot of factors to consider.
- Are the samples representative of their populations? If the sample is biased, then it may not have the same center and variability as the population.
- Which characteristic of the populations makes sense to compare—the mean, the median, or a proportion?
- How variable is the data? If the data is very spread out, it can be more difficult to make conclusions with certainty.
Knowing the correct questions to ask when trying to compare groups is important to correctly interpret the results.
Glossary Entries
Definition: Interquartile Range (IQR)
The interquartile range is one way to measure how spread out a data set is. We sometimes call this the IQR. To find the interquartile range we subtract the first quartile from the third quartile.
For example, the IQR of this data set is 20 because \(50-30=20\).
| \(22\) | \(29\) | \(30\) | \(31\) | \(32\) | \(43\) | \(44\) | \(45\) | \(50\) | \(50\) | \(59\) |
| Q1 | Q2 | Q3 |
Definition: Proportion
A proportion of a data set is the fraction of the data in a given category.
For example, a class has 18 students. There are 2 left-handed students and 16 right-handed students in the class. The proportion of students who are left-handed is \(\frac{2}{20}\), or 0.1.
Practice
Exercise \(\PageIndex{4}\)
An agent at an advertising agency asks a random sample of people how many episodes of a TV show they watch each day. The results are shown in the dot plot.
The agency currently advertises on a different show, but wants to change to this one as long as the typical number of episodes is not meaningfully less.
- What measure of center and measure of variation would the agent need to find for their current show to determine if there is a meaningful difference? Explain your reasoning.
- What are the values for these same characteristics for the data in the dot plot?
- What numbers for these characteristics would be meaningfully different if the measure of variability for the current show is similar? Explain your reasoning.
Exercise \(\PageIndex{5}\)
Jada wants to know if there is a meaningful difference in the mean number of friends on social media for teens and adults. She looks at the friend count for the 10 most popular of her friends and the friend count for 10 of her parents’ friends. She then computes the mean and MAD of each sample and determines there is a meaningful difference.
Jada’s dad later tells her he thinks she has not come to the right conclusion. Jada checks her calculations and everything is right. Do you agree with her dad? Explain your reasoning.
Exercise \(\PageIndex{6}\)
The mean weight for a sample of a certain kind of ring made from platinum is 8.21 grams. The mean weight for a sample of a certain kind of ring made from gold is 8.61 grams. Is there a meaningful difference in the weights of the two types of rings? Explain your reasoning.
Exercise \(\PageIndex{7}\)
The lengths in feet of a random sample of 20 male and 20 female humpback whales were measured and used to create the box plot.
Estimate the median lengths of male and female humpback whales based on these samples.
(From Unit 8.4.1)