9.05 Analyze bivariate data

Describe the relationship between the time since the ball was dropped and its distance from the ground. Is it quadratic or linear?

Construct a scatterplot to get a visual of the data.

Then consider the form, strength, and direction.

The data appears to fit a strong quadratic model.

Use technology to create a model and graph the model alongside a scatterplot of the data.

We can use technology to calculate the quadratic regression equation. Remember that a quadratic function is a polynomial of degree 2.

To find the equation using technology, we can follow these steps:

1. Enter the x-values and y-values in two separate columns.

2. Highlight the data and select Two Variable Regression Analysis.

3. Under the Regression Model drop down menu, choose Polynomial. The degree drop down menu defaults to 2, which is a quadratic function.

1. Enter the x-values and y-values in two separate columns.

   

2. Highlight the data and select Two Variable Regression Analysis.

   

3. Under the Regression Model drop down menu, choose Polynomial. The degree drop down menu defaults to 2, which is a quadratic function.

   

The equation of the curve of best fit is y = -0.8521x^2 + 1.4149x + 24.3536.

Based off your model, when would you predict the ball would hit the ground?

Looking at the graph, the ball would hit the ground when the distance from the groud is zero feet. Follow the pattern of the scatterplot or look at the model created using technology and predict when that would be. Alternatively, we can verify our solution by finding the x-intercept for our regression model.

Looking at the model made with technology, the ball would hit the ground after approximately 6.25 seconds, which is the x-intercept when the distance from the ground is 0 feet.

Remember, that this prediction is just an educated guess based on our model, and your answer may differ slightly based on the model you chose. According to this one, a more precise answer is about 6.24 seconds.

Identify the two variables that Ronaldo should collect data for in his investigation of potential apartments and then formulate a statistical question to investigate them.

Consider the factors that Ronaldo is interested in:

• A two-bedroom apartment

• A spacious apartment

• An affordable rental price

Then, determine which factors would require the collection of data.

Ronaldo should collect data that describes the size of the apartment, usually measured by square footage, and the rental price, usually given as a monthly rate. The apartments should all have two bedrooms, since that is the type (category) of apartment he is interested in.

One possible question is, "What is the price range of two-bedroom apartments with 1000–1200 square feet?"

In this context, the size of the apartment (in square feet) is the independent variable, and the monthly rental price (in dollars) is the dependent variable.

Other possible questions are:

• How does the monthly rental price of a two-bedroom apartment change with the size of the apartment?

• What size apartments are typically \$1500–\$1700 per month?

• How do the prices and sizes of two-bedroom apartments compare to those of two-bedroom houses?

Collect data that could be used to answer the statistical question you formulated.

Previously, we determined that data should be collected on the size of the apartment, usually measured by square footage, and the rental price, usually given as a monthly rate.

This information can be acquired online. Typically, rental properties in an area are advertised on websites such as Zillow.com or Apartments.com.

This is an example data set of current rental properties around Norfolk, VA:

Remember that the sample should be collected randomly, and there should be a decent amount of two-bedroom apartments in the sample to be representative of the population.

Determine whether a linear or quadratic function would represent the relationship best. Calculate the equation of the curve of best fit.

First, we can use technology to create a scatterplot and examine the shape of the data. After determining which function models the data best, we can find the equation of the curve of best fit with technology.

To find the equation using technology, we can follow these steps:

1. Enter the x-values and y-values in two separate columns.

2. Highlight the data and select Two Variable Regression Analysis. This will generate the scatterplot.

3. Under the Regression Model drop down menu, choose Linear or Polynomial, depending on the shape of the data.

Enter the data into the GeoGebra statistics calculator, and perform the Two Variable Regression Analysis.

The relationship between the variables is not strong, but the y-values tend to increase as the x-values increase. This indicates there is a moderate, linear relationship between the variables.

Now, we can find the equation of the line of best fit by choosing Linear under the Regression Model drop down menu.

The equation of the line of best fit is y=1.0159x+645.7778.

When analyzing the quadratic curve of best fit, we can see that the curve does not model the data better than the linear model. In fact, the section of the parabola shown does not have much curve to it. This means that prediction made with either model would be similar.

Ideally, Ronaldo would like an apartment that is 1100\text{ ft}^2. Predict the monthly rental price of an apartment of this size.

In the previous part, we found the equation of the line of best fit to be y=1.01586x + 645.7778, where x represents the size of an apartment in square feet and y represents the monthly rental price in dollars. We can substitute x=1100 into the equation to find the monthly rental price.

An 1100\text{ ft}^2 apartment will cost about \$1763 per month.

This prediction was made with interpolation because it falls within the range of the known data values. However, the prediction is not very strong because the points are not tightly clustered around the line.

Ronaldo's budget is \$1650. Predict the size of the apartment he can afford.

The monthly rental price is the dependent variable \left(y\right), and the size of the apartment is the independent variable \left(x\right). We must subsitute y=1650 into the equation of the line of best fit, and solve for the x-value.

\$1650 a month can get Ronaldo an apartment with about 988.5 square feet of space.

Draw a conclusion by answering the statistical question from part (b) and summarize the results of the investigation.

The statistical question from part (b) was, "What is the price range of two-bedroom apartments with 1000–1200 square feet?"

If Ronaldo wants a two-bedroom apartment that is 1100\text{ ft}^2, he should expect to pay about \$1763 per month. This is outside of his budget, so he should look for apartments that are around 988\text{ ft}^2 to stay within his desired price range.

However, according to the raw data, the montly rental price of an apartment with 1000–1200 square feet ranges from \$1300–\$2300. This shows that it is possible to find an 1100\text{ ft}^2 apartment within the \$1650 price range.

There are most likely other factors, such as the neighborhood or distance from downtown Norfolk, that affect the price of the property that Ronaldo should take into consideration when making his final decision.

These results could help Ronaldo make a decision about the apartment he would like to rent, or it could lead him to ask another question. For example, Ronaldo might ask the question, "How does the size of an apartment impact the monthly rental price of a one-bedroom or two-bedroom apartment?" He could use the slope of the line of best fit to conclude that for each 1 square foot increase in apartment size he can expect to pay around \$1.02 more per month.

This might lead Ronolado to explore one-bedroom apartments instead. He could repeat the data cycle, collecting data on one-bedroom apartment sizes and prices. Then, he can plot the data on the same scatterplot in part (d), but use a different color for the points representing one-bedroom apartments.