Problem 18 - Algebra 1

Question

Use the table which shows the number of movie theater screens (in thousands) from 1975 to 1995. $$ \begin{array}{|l|c|c|c|c|c|}\hline \text { Year } & 1975 & 1980 & 1985 & 1990 & 1995 \\\\\hline \text { Indoor screens (in thousands) } & 11 & 14 & 18 & 23 & 27 \\\\\hline \text { Drive-in screens (in thousands) } & 4 & 4 & 3 & 1 & 1 \\\\\hline\end{array} $$ Make a scatter plot of the number of drive-in movie screens in terms of the year \(t .\) Let \(t\) represent the number of years since 1975.

Step-by-Step Solution

Answer

The scatter plot should represent the decline in the number of drive-in screens (in thousands) over the specified years.

Step 1: Represent years since 1975

To start with, let's represent the number of years since 1975 as \(t\), so our \(t\)-values will start from 0 to 20 in increments of 5. This represents 1975, 1980, 1985, 1990 and 1995 respectively.

Step 2: Determine the number of drive-in screens

From the table given, the number of drive-in screens (in thousands) corresponding to these years are 4, 4, 3, 1, 1 respectively. These will be our y-values for the scatter plot.

Step 3: Plot the data

Using the \(t\)-values and the corresponding number of drive-in screens, plot each point on a graph. Each \(t\)-value corresponds to a 'x'-value (years since 1975), and each corresponding screen count corresponds to a 'y'-value.

Step 4: Connect the dots

Once the points are on the plot, connect them with lines to make the pattern easy to understand. This will be our scatter plot.

Key Concepts

Data Representation

Data representation is a crucial concept in visualizing information, especially when dealing with statistical data. When you represent data, you transform it into a form that's suitable for analysis and interpretation. In the context of the exercise, we used a table to display the number of drive-in movie screens over a series of years.

Key aspects of data representation include:

• Accuracy: Ensure the data is correct and matches the source information.
• Clarity: Present data in a way that's easy to understand. Tables, graphs, and charts can help make complex data more interpretable.
• Consistency: Use the same scales, units, and formats throughout data presentation for uniformity.

In this exercise, data was presented in a tabular form where each year (1975 to 1995) was associated with the number of drive-in screens. This approach helps in identifying patterns and making logical comparisons between different time periods.

Coordinate Graphing

Coordinate graphing involves plotting points on a graph to visually represent relationships between variables. In this exercise, we used a two-dimensional coordinate graph, where the x-axis represented the years since 1975, denoted by variable \( t \), and the y-axis represented the number of drive-in screens.

Here's how to approach coordinate graphing:

• Axes Labeling: Clearly label the x-axis and y-axis to indicate what each represents. This is vital for understanding what each data point signifies.
• Scale Selection: Choose an appropriate scale that fits your data and makes differences in values clear and visible.
• Plotting Points: For each data pair (\( t \), number of screens), find the corresponding position on the graph and mark it with a dot or marker.

Coordinate graphing provides a visual overview, aiding in the quick assessment of trends and patterns that might not be immediately obvious in a numerical table. It is a fundamental tool in many fields of studies, from mathematics to social sciences.

Time Series Analysis

Time series analysis is a statistical method focused on analyzing data points collected or recorded at specific time intervals. In this task, we performed a time series analysis by observing how the number of drive-in screens changed over the years from 1975 to 1995.

The following steps help in conducting time series analysis:

• Identify Time Intervals: The first step is to identify uniform time intervals, which, in this case, was every 5 years.
• Trend Analysis: Use the graphical representation to identify any upward or downward trends over time.
• Pattern Recognition: Look for recurring patterns or unusual data points to understand the behavior of the variable over time.
• Prediction: Based on observed trends, time series analysis can help in predicting future values.

From our scatter plot, it was observed that the number of drive-in screens decreased over time, highlighting a downward trend. Time series analysis, therefore, is essential for both historical analysis and future forecasting.