## Does This Predict That?

This week your student will work with scatter plots. Scatter plots show us how two different variables are related. In the example below, each plotted point corresponds to a dog, and its coordinates tell us the height and weight of that dog. The point on the lower left of the graph, for example, might represent a dog that is 8 inches tall and weighs about 5 pounds. The plot shows that, generally speaking, taller dogs weigh more than shorter dogs. Since a larger value for one characteristic (height) generally means a larger value for the other characteristic (weight), we say that there is a positive association between dog height and dog weight.

In the next example, each point corresponds to a car, and its coordinates tell us the weight and fuel efficiency of the car. This time, we see that larger values for one characteristic (car weight) generally have lower values for the other characteristic (fuel efficiency), and so we say that there is a negative association between car weight and fuel efficiency.

The following scatter plot shows the relationship between average temperature and gas usage in a buildings.

1. How many points in the graph describe the building on 70-degree days? Approximately how much gas was used on each of these days?
2. Do the variables in the gas usage for a building scatter plot show a positive association or a negative association?
3. On a 78-degree day, would the building be most likely to use (a) 1,800 therms of gas, (b) 4,200 therms of gas, or (c) 5,800 terms of gas?

Solution:

1. There are two points that describe gas usage for 70-degree days. On one of those days, the building used a little less than 4,000 therms of gas. On the other, the building used a little more than 6,000 therms.
2. Since less gas is used on warmer days, there is a negative association.
3. Following the trend in the graph, the building would likely use about 1,800 therms on a 78-degree day. You may draw in a line as in the dog and car scatter plots to help see this.

## Associations in Categorical Data

This week your student will use two-way tables. Two-way tables are a way of comparing two variables. For example, this table shows the results of a study of the relation between meditation and state of mind of athletes before a track meet.

meditated did not meditate total
calm 45 8 53
agitated 23 21 44
total 68 29 97

23 of the people who meditated were agitated, while 21 of the people who did not meditate were agitated. Does this mean that meditation has no impact or even a slight negative association with mood? Probably not. When we look for associations between variables it can be more informative to know the percentages in each category, like this:

meditated did not meditate
calm 66% 28%
agitated 34% 72%
total 100% 100%

Of the people who meditated, 66% were calm, and 34% were agitated. When we compare that to the percentages for people who did not meditate, we can now see more easily that the group of people who meditated has a lower percentage of athletes who are agitated. The percentages in this table are called relative frequencies.

The following table contains data about whether people in various age groups use their cell phone as their main alarm clock.

use cell phone as alarm do not use cell phone as alarm total
18 to 29 years old 47 16 63
30 to 49 years old 66 23 87
50+ years old 31 39 70
total 144 78 220
1. Fill in the blanks in the table below with the relative frequencies for each row. These will tell us the percentage of people in each age group who use their phone as an alarm.
use cell phone as alarm do not use cell phone as alarm total
18 to 29 years old $75\%$, since $\frac{47}{63}=0.75$   100%
30 to 49 years old
50+ years old
2. Comparing just the 18 to 29-year olds and the 30 to 49-year olds, is there an association between cell phone alarm use and age?
3. Comparing the two youngest age brackets with the 50+ age bracket, is there an association between cell phone alarm use and age?

Solution:

1.   use cell phone as alarm do not use cell phone as alarm total
18 to 29 years old $75\%$, since $\frac{47}{63}=0.75$ $25\%$, since $\frac{16}{63}=0.25$ 100%
30 to 49 years old $76\%$, since $\frac{66}{87}=0.76$ $24\%$, since $\frac{23}{87}=0.24$ 100%
50+ years old  $44\%$, since $\frac{31}{70}=0.44$ $56\%$, since $\frac{39}{70}=0.56$ 100%
2. No: the relative frequencies are very similar.
3. Yes: using a cell phone as an alarm is associated with being in the younger age brackets. About 75% of 18 to 29 and 30 to 49-year olds use their cell phone as an alarm, but only 44% of people 50 years or older do.