Data Analysis: Broadway Shows The annual gross ticket sales...

Data Analysis: Broadway Shows The annual gross ticket sales $S$ (in millions of dollars) for Broadway shows in New York City from 1995 through 2011 are given by the following ordered pairs. (Spreadsheet at LarsonPrecalculus.com) (Source: The Broadway League) (a) Use a graphing utility to create a scatter plot of the data. Let $t=5$ represent $1995 .$ (b) Use the regression feature of the graphing utility to find the equation of the least squares regression line that fits the data. (c) Use the graphing utility to graph the scatter plot you created in part (a) and the model you found in part (b) in the same viewing window. How closely does the model represent the data? (d) Use the model to predict the annual gross ticket sales in $2017 .$ model in the context of the problem.

Data Analysis: Broadway Shows The annual gross ticket sales $S$ (in millions of dollars) for Broadway shows in New York City from 1995 through 2011 are given by the following ordered pairs. (Spreadsheet at LarsonPrecalculus.com) (Source: The Broadway League)
(a) Use a graphing utility to create a scatter plot of the data. Let $t=5$ represent $1995 .$
(b) Use the regression feature of the graphing utility to find the equation of the least squares regression line that fits the data.
(c) Use the graphing utility to graph the scatter plot you created in part (a) and the model you found in part (b) in the same viewing window. How closely does
the model represent the data?
(d) Use the model to predict the annual gross ticket sales in $2017 .$ model in the context of the problem.

Key Concepts

Data Visualization (Scatter Plot)

A scatter plot is a graph used to visually display the relationship between two quantitative variables by plotting data points for each observation. This method provides an immediate visual impression of trends, clustering, and potential outliers, which is essential in exploratory data analysis before applying any formal modeling techniques.

Regression Analysis (Least Squares Regression)

Regression analysis involves modeling the relationship between a dependent variable and one or more independent variables. The least squares regression method specifically finds the line that minimizes the sum of the squared differences between the observed values and the values predicted by the model, serving as a primary tool for linear data modeling.

Model Fitting and Evaluation

Model fitting is the process of choosing a mathematical function that best represents the underlying patterns in the data. Evaluating the model involves checking how closely this fitted model represents the actual data points, often using visual assessments, residual analysis, or statistical measures of goodness of fit.

Extrapolation and Predictions

Extrapolation refers to using a fitted model to predict values outside the range of the observed data. While useful, predictions made by extrapolating should be treated with caution because the relationship observed within the data range might not hold when extended beyond that range.

Transcript

00:01 Given our gross ticket sales for these years, we're asked to first create a scatter plot with all the data points we have.

00:09 So i already have my scatter plot up here.

00:12 This is what i get when we plot the points.

00:15 And they say t equals 5 represents the year 1995.

00:20 So that's why my time only uses numbers that start at 5.

00:27 Now, once you've plotted this using a graphing utility, the next part is.

00:32 To use the regression feature.

00:34 So the regression feature on, for instance, a graphing calculator, it's going to give you, it's going to fill a line to this data, and it's going to give you an equation.

00:48 So when you use it, it can give you many different forms.

00:52 You can have the line in the form of y equals mx plus b, which is the standard, that's what i used.

00:57 And i get an equation of y is equal to 4 .0 .59.

01:04 I rounded, because those are long decimals times t plus two zero four point two seven so that there is our model using the regression tool the next thing we need to do is using the same graphing utility you need to graph the scatter plot and the line from this equation here on the same plot so i have that here right so this line represents my regression, that equation that the graphing tool gave us, and of course our scatterplot.

01:53 So it's actually a really good fit.

01:55 And well, that's the point of the regression tool is to fit it as best possible...

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