Question

1. model with math the scatter plot represents the numbers of hours since the city parking lots opened and the number of cars in the parking lots.

a. identify any outliers in the scatter plot.
b. identify any clusters in the scatter plot.
c. identify any gaps in the scatter plot. what situation might have caused the gap?
cars in parking lots
(chart with x-axis number of hours from 0 to 12, y-axis number of cars from 0 to 36, with data points and circled groups)

Explanation:

Part a

Step1: Recall Outlier Definition

An outlier is a data point that is far from the main cluster(s) of data.

Step2: Analyze Scatter Plot Points

• The point at (10, ~0 - 3) (around x = 10, y near 0) and the point at (1, ~36 - 39) (x=1, y high) are far from the main clusters (the lower - left cluster, the middle small cluster, and the large right - middle cluster). Wait, looking at the plot: the point at (10, low y) and the point at (1, high y) and the small cluster at (4, ~27) and the large cluster at (8 - 10, ~27 - 36) and the lower cluster at (2 - 6, ~0 - 9). Wait, the point at (10, y≈0 - 3) is far from others, and the point at (1, y≈36 - 39) is far, also the small cluster at (4, ~27) is a single - point cluster? No, the outlier: the point at (10, y near 0) (x = 10, y low) and the point at (1, y high) and the point at (4, ~27) is a small cluster? Wait, no, let's re - examine. The scatter plot has:
• A cluster at (2 - 6, 0 - 9) (lower left)
• A small cluster at (4, ~27)
• A large cluster at (8 - 10, 27 - 36)
• A point at (10, y≈0 - 3) (x = 10, y low)
• A point at (1, y≈36 - 39) (x = 1, y high)

The point at (10, y low) (x = 10, y≈0 - 3) is far from the large cluster at (8 - 10, 27 - 36), and the point at (1, y high) (x = 1, y≈36 - 39) is far from the lower - left cluster (2 - 6, 0 - 9) and the small cluster (4, 27) and the large cluster (8 - 10, 27 - 36). Also, the point at (4, ~27) is a small cluster, but the single point at (10, low y) and (1, high y) and maybe the point at (4, 27) is a small cluster, but outliers are points far from main clusters. So the outlier is the point at (10, y near 0) (x = 10, number of hours = 10, number of cars≈0 - 3) and the point at (1, y near 36 - 39) (x = 1, number of hours = 1, number of cars≈36 - 39), and the point at (4, ~27) is a small cluster? Wait, no, the outlier is the data point that is separated from the main groups. So the point at (10, y≈0) (x = 10, number of hours = 10, number of cars≈0 - 3) is an outlier, the point at (1, y≈36) (x = 1, number of hours = 1, number of cars≈36 - 39) is an outlier, and the point at (4, ~27) is a small cluster, not an outlier. Wait, maybe the outlier is the point at (10, y near 0) (x = 10, number of hours = 10, number of cars≈0 - 3) and the point at (1, y near 36) (x = 1, number of hours = 1, number of cars≈36 - 39).

Step3: Conclusion on Outliers

The outliers are the data point at (1, y - value around 36 - 39) (1 hour since opening, ~36 - 39 cars) and the data point at (10, y - value around 0 - 3) (10 hours since opening, ~0 - 3 cars), and also the single point at (4, ~27) is a small cluster, but maybe the main outliers are (1, ~36 - 39) and (10, ~0 - 3).

Part b

Step1: Recall Cluster Definition

A cluster is a group of data points that are close together, separate from other groups.

Step2: Analyze Scatter Plot Groups

• Group 1: (2 - 6, 0 - 9) (number of hours 2 - 6, number of cars 0 - 9) – this is a cluster.
• Group 2: (4, ~27) – a small cluster (1 - 2 points, close together).
• Group 3: (8 - 10, 27 - 36) – a large cluster (multiple points close together).

Part c

Step1: Recall Gap Definition

A gap is a region in the scatter plot with no data points between two clusters.

Step2: Identify Gaps

• Between the cluster (2 - 6, 0 - 9) (lower left) and the cluster (4, ~27) (small cluster at x = 4, y = 27): there is a gap between y = 9 - 27 (for x = 2 - 6) – no data points here.
• Between the cluster (4, ~27) (small cluster) and the cluster (8 - 10, 27 - 36) (large cluster): there is a gap between x = 6 - 8 (for y = 27 - 36) – no data points here.
• Between the cluster (2 - 6, 0 - 9) (lower left) and the point at (10, ~0 - 3) (x = 10, y low): gap between x = 6 - 10 (for y = 0 - 9) – no data points here.

Step3: Explain Gap Cause

For the gap between (2 - 6, 0 - 9) and (4, ~27): Maybe during the hours 2 - 6, there was a time when the parking lot had a different usage pattern, like a special event that caused a sudden jump in cars (from 0 - 9 to 27) or a lull where no cars were in that middle range. For example, maybe between 2 - 6 hours, there was a period when the parking lot was either mostly empty (0 - 9 cars) or had a sudden influx (27 cars), with no intermediate number of cars, perhaps due to a special event starting at 4 hours that brought a lot of cars, skipping the intermediate counts.

Part a Answer:

The outliers are the data point at (1, number of hours = 1, number of cars≈36 - 39) and the data point at (10, number of hours = 10, number of cars≈0 - 3) (and the point at (4, number of cars≈27) can be considered an outlier - like small cluster, but the main outliers are the points at (1, high y) and (10, low y)).

Part b Answer:

The clusters are:

• A cluster of points at (2 - 6, 0 - 9) (number of hours 2 - 6, number of cars 0 - 9)
• A small cluster of points at (4, ~27) (number of hours≈4, number of cars≈27)
• A large cluster of points at (8 - 10, 27 - 36) (number of hours 8 - 10, number of cars 27 - 36)

Part c Answer:

Gaps:

• Between the cluster (2 - 6, 0 - 9) and the cluster (4, ~27): no data points between y = 9 - 27 for x = 2 - 6.
• Between the cluster (4, ~27) and the cluster (8 - 10, 27 - 36): no data points between x = 6 - 8 for y = 27 - 36.
• Between the cluster (2 - 6, 0 - 9) and the point at (10, ~0 - 3): no data points between x = 6 - 10 for y = 0 - 9.

Cause: For the gap between low - car (0 - 9) and medium - car (27) clusters, maybe a special event (like a concert, sports game) started at 4 hours, bringing a large number of cars at once, skipping the intermediate number of cars (10 - 26 cars) during 2 - 6 hours. For the gap between medium - car (27) and high - car (27 - 36) clusters (x = 6 - 8), maybe there was a period (6 - 8 hours) when no cars entered the parking lot, or a different usage pattern (e.g., a break in events) that caused no data points here. For the gap between low - car (0 - 9) and the low - car point at (10, ~0 - 3) (x = 6 - 10, y = 0 - 9), maybe during 6 - 10 hours, the parking lot was either full (27 - 36 cars) or had very few cars (the point at 10, ~0 - 3), with no intermediate low - car counts, perhaps due to a shift in parking demand (e.g., evening events starting, so most cars are in the high - car cluster, and only a few late - arriving/leaving cars at 10 hours).

Answer:

Step1: Recall Gap Definition

A gap is a region in the scatter plot with no data points between two clusters.

Step2: Identify Gaps

• Between the cluster (2 - 6, 0 - 9) (lower left) and the cluster (4, ~27) (small cluster at x = 4, y = 27): there is a gap between y = 9 - 27 (for x = 2 - 6) – no data points here.
• Between the cluster (4, ~27) (small cluster) and the cluster (8 - 10, 27 - 36) (large cluster): there is a gap between x = 6 - 8 (for y = 27 - 36) – no data points here.
• Between the cluster (2 - 6, 0 - 9) (lower left) and the point at (10, ~0 - 3) (x = 10, y low): gap between x = 6 - 10 (for y = 0 - 9) – no data points here.

Step3: Explain Gap Cause

For the gap between (2 - 6, 0 - 9) and (4, ~27): Maybe during the hours 2 - 6, there was a time when the parking lot had a different usage pattern, like a special event that caused a sudden jump in cars (from 0 - 9 to 27) or a lull where no cars were in that middle range. For example, maybe between 2 - 6 hours, there was a period when the parking lot was either mostly empty (0 - 9 cars) or had a sudden influx (27 cars), with no intermediate number of cars, perhaps due to a special event starting at 4 hours that brought a lot of cars, skipping the intermediate counts.

Part a Answer:

The outliers are the data point at (1, number of hours = 1, number of cars≈36 - 39) and the data point at (10, number of hours = 10, number of cars≈0 - 3) (and the point at (4, number of cars≈27) can be considered an outlier - like small cluster, but the main outliers are the points at (1, high y) and (10, low y)).

Part b Answer:

The clusters are:

• A cluster of points at (2 - 6, 0 - 9) (number of hours 2 - 6, number of cars 0 - 9)
• A small cluster of points at (4, ~27) (number of hours≈4, number of cars≈27)
• A large cluster of points at (8 - 10, 27 - 36) (number of hours 8 - 10, number of cars 27 - 36)

Part c Answer:

Gaps:

• Between the cluster (2 - 6, 0 - 9) and the cluster (4, ~27): no data points between y = 9 - 27 for x = 2 - 6.
• Between the cluster (4, ~27) and the cluster (8 - 10, 27 - 36): no data points between x = 6 - 8 for y = 27 - 36.
• Between the cluster (2 - 6, 0 - 9) and the point at (10, ~0 - 3): no data points between x = 6 - 10 for y = 0 - 9.

Cause: For the gap between low - car (0 - 9) and medium - car (27) clusters, maybe a special event (like a concert, sports game) started at 4 hours, bringing a large number of cars at once, skipping the intermediate number of cars (10 - 26 cars) during 2 - 6 hours. For the gap between medium - car (27) and high - car (27 - 36) clusters (x = 6 - 8), maybe there was a period (6 - 8 hours) when no cars entered the parking lot, or a different usage pattern (e.g., a break in events) that caused no data points here. For the gap between low - car (0 - 9) and the low - car point at (10, ~0 - 3) (x = 6 - 10, y = 0 - 9), maybe during 6 - 10 hours, the parking lot was either full (27 - 36 cars) or had very few cars (the point at 10, ~0 - 3), with no intermediate low - car counts, perhaps due to a shift in parking demand (e.g., evening events starting, so most cars are in the high - car cluster, and only a few late - arriving/leaving cars at 10 hours).