Interpreting Linear Relationships Using Data: Practice Problems

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  • 0:02 Linear Relationships
  • 0:59 Practice Problem 1
  • 2:30 Practice Problem 2
  • 4:13 Practice Problem 3
  • 5:24 Lesson Summary
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Understanding linear relationships is an important part of understanding statistics. This lesson will help you review linear relationships and will go through three practice problems to help you retain your knowledge. When you are finished, test out your knowledge with a short quiz!

Interpreting Linear Relationships

A linear model is a comparison of two values, usually x and y, and the consistent change between those values. The easiest way to understand and interpret slope and intercept in linear models is to first understand the slope-intercept formula: y = mx + b, where m is the slope, or the consistent change between x and y, and b is the y-intercept. Often, the y-intercept represents the starting point of the equation. In statistics, you will also see the formula: y = ax + b. It's the same as the slope-intercept formula, except a = slope instead of m.

The line in the center is known as a regression line, a straight line that attempts to predict the relationship between two points. This relationship is the same thing as the slope, and you may also hear the terms 'consistent change' or 'interval.' These three words are used interchangeably and mean the same thing in this case.

A regression line
Graph with regression line

Practice Problems

Jaime is the owner of a pet grooming shop. He is working on increasing the efficiency of his employees. Let's take a look at some of the problems he is looking to resolve.

Problem One

Jamie wants to calculate how long it takes to heat up water to the perfect temperature so his employees don't have to constantly check the temperature of the water. He has created a graph based on the data he has collected.

Graph of data for problem 1

This is the graph that Jaime created. Can you determine the relationship between the temperature and the time it takes to heat the water? How long will it take to heat the water to 99 degrees? Can you interpret the meaning of the slope and the intercept? Pause the video here to find the answer on your own.

Notice that Jaime checks the temperature of the water every 5 seconds, and the water heats up approximately 2 degrees every time, or .4 degrees per second. We can see from the last point that the water will take 65 seconds to heat up to 90 degrees. If we create an equation based on this data, we would have y =.4x + 72. We've already interpreted the slope here. For every second, the water heats up by .4 degrees, but what about the intercept? The y-intercept in this equation is (0,72). This means that when the water is turned on, it's already 72 degrees. Jaime will install an alarm that alerts his employees to turn the water off or to cool at 65 seconds.

Problem Two

Next, Jamie wants to determine how long it takes to fill up the pet tubs, so his employees can set a timer that will automatically shut off the water when the tub is filled. Jaime has collected the following data from his employees:

Employee 1

Time (mins) Gallons
5 21
5.5 22
5.5 23
6 25
5 20
5.5 22

Employee 2

Time (mins) Gallons
4.5 21
4.5 20
5 22
5 21
4.5 20
5.5 23

Employee 3

Time (mins) Gallons
6 24
5.5 23
5.5 24
6 25
5 22
5.5 23

Can you use this data to determine the relationship between the length of time the water is running and the amount of water in the tub? Can you interpret the meaning of the slope and the intercept? Pause the video here to find the answer on your own.

How did you do? First, I've created a graph and found the equation for the data that Jaime collected.

Graph of data from problem 2 tables

Notice that the data is clustered in one central area. That tells us two things. One, the water pressure had very little variety; it pretty much produced the same flow of water each time. Two, the amount of water in the tub couldn't be any more than 25 gallons, since no employee filled the tubs past this point. The equation tells us two things; first, the rate at which the water flows is about three gallons per minute. Second, theoretically, the intercept tells us that before we turn the water on, there is already 6.7 gallons in the tub. We know that this isn't correct, but we can hypothesize that the water starts pouring out more quickly at the beginning and then slows down as more water is added and the pressure decreases.

Jaime decides he wants to fill the tubs up to approximately 20 gallons of water. How long do you think he should set his timer for automatic shut off? Right, five minutes will be perfect.

Problem Three

Jaime is trying to determine how long each appointment time should last for a typical grooming. He knows that the size of the dog will often determine how long it takes for a grooming. He asks his employees to collect data based on the weight of the dog and the time it takes to groom the dog. He gets this information:

Employee 1

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