Question

5 from unit 1, lesson 2 the data set represents the responses, in degrees fahrenheit, collected to answer the question “how hot is the sidewalk during the school day?”. 92 95 95 95 98 100 100 100 103 105 105 111 112 115 115 116 117 117 118 119 119 119 119 119 119 a. create a dot plot to represent the distribution of the data. 92 b. create a histogram to represent the distribution of the data. c. which display gives you a better overall understanding of the data? explain your reasoning.

Explanation:

Part (a): Dot Plot Creation

Step 1: Identify Data Values and Frequencies

First, we list out each unique data value and count how many times it appears (frequency):

• \( 92 \): 1 time
• \( 95 \): 3 times
• \( 98 \): 1 time
• \( 100 \): 3 times
• \( 103 \): 1 time
• \( 105 \): 2 times
• \( 111 \): 1 time
• \( 112 \): 1 time
• \( 115 \): 2 times
• \( 116 \): 1 time
• \( 117 \): 2 times
• \( 118 \): 1 time
• \( 119 \): 6 times

Step 2: Draw the Dot Plot

• On the horizontal axis, mark the data values in order: \( 92, 95, 98, 100, 103, 105, 111, 112, 115, 116, 117, 118, 119 \).
• For each value, place a dot above it corresponding to its frequency. For example:
• Above \( 92 \): 1 dot
• Above \( 95 \): 3 dots
• Above \( 98 \): 1 dot
• Above \( 100 \): 3 dots
• Above \( 103 \): 1 dot
• Above \( 105 \): 2 dots
• Above \( 111 \): 1 dot
• Above \( 112 \): 1 dot
• Above \( 115 \): 2 dots
• Above \( 116 \): 1 dot
• Above \( 117 \): 2 dots
• Above \( 118 \): 1 dot
• Above \( 119 \): 6 dots

Part (b): Histogram Creation

Step 1: Determine Bin Width (Class Interval)

We can choose a bin width (class interval) to group the data. Let's use a bin width of \( 10 \) (ranges like \( 90 - 99 \), \( 100 - 109 \), \( 110 - 119 \)) or a smaller width for more detail. Let's use \( 5 \) as the bin width:

• \( 90 - 94 \): Contains \( 92 \) → Frequency = 1
• \( 95 - 99 \): Contains \( 95, 95, 95, 98 \) → Frequency = 4
• \( 100 - 104 \): Contains \( 100, 100, 100, 103 \) → Frequency = 4
• \( 105 - 109 \): Contains \( 105, 105 \) → Frequency = 2
• \( 110 - 114 \): Contains \( 111, 112 \) → Frequency = 2
• \( 115 - 119 \): Contains \( 115, 115, 116, 117, 117, 118, 119, 119, 119, 119, 119, 119 \) → Frequency = 12

Step 2: Draw the Histogram

• On the horizontal axis, mark the bin intervals: \( 90 - 94, 95 - 99, 100 - 104, 105 - 109, 110 - 114, 115 - 119 \).
• On the vertical axis, mark the frequency.
• For each bin, draw a bar whose height corresponds to the frequency of that bin. For example:
• Bar for \( 90 - 94 \): Height = 1
• Bar for \( 95 - 99 \): Height = 4
• Bar for \( 100 - 104 \): Height = 4
• Bar for \( 105 - 109 \): Height = 2
• Bar for \( 110 - 114 \): Height = 2
• Bar for \( 115 - 119 \): Height = 12

Part (c): Comparing Dot Plot and Histogram

• A dot plot shows the exact frequency of each individual data point, which is useful for seeing the precise distribution and identifying outliers or individual values.
• A histogram groups data into intervals, which is useful for seeing the overall shape of the distribution (e.g., skewness, peaks) and for larger data sets where individual points might be less meaningful.
• If we want to see the exact count of each temperature (e.g., how many times \( 119^\circ \) occurred), the dot plot is better. If we want to see the general trend (e.g., most temperatures are in the \( 115 - 119 \) range), the histogram is better. However, for a better "overall understanding" of the distribution's shape and spread, the histogram might be better because it simplifies the data into groups, making it easier to see patterns like the large cluster at \( 115 - 119 \) and the smaller clusters at lower temperatures. Alternatively, the dot plot is better for precise frequency of each value. The choice can depend on the goal, but typically, the histogram gives a clearer picture of the overall distribution's shape.

Final Answers (Summarized)

a. Dot plot with dots above each value (as per frequencies).
b. Histogram with bars for each bin (as per frequencies).
c. Example: "The histogram gives a better overall understanding because it groups the data into intervals, making it easier to see the overall shape of the distribution (e.g., the large cluster of higher temperatures) compared to the dot plot, which shows individual values but can be more cluttered for identifying broad patterns." (Or vice versa, depending on reasoning.)

Answer:

• A dot plot shows the exact frequency of each individual data point, which is useful for seeing the precise distribution and identifying outliers or individual values.
• A histogram groups data into intervals, which is useful for seeing the overall shape of the distribution (e.g., skewness, peaks) and for larger data sets where individual points might be less meaningful.
• If we want to see the exact count of each temperature (e.g., how many times \( 119^\circ \) occurred), the dot plot is better. If we want to see the general trend (e.g., most temperatures are in the \( 115 - 119 \) range), the histogram is better. However, for a better "overall understanding" of the distribution's shape and spread, the histogram might be better because it simplifies the data into groups, making it easier to see patterns like the large cluster at \( 115 - 119 \) and the smaller clusters at lower temperatures. Alternatively, the dot plot is better for precise frequency of each value. The choice can depend on the goal, but typically, the histogram gives a clearer picture of the overall distribution's shape.

Final Answers (Summarized)

a. Dot plot with dots above each value (as per frequencies).
b. Histogram with bars for each bin (as per frequencies).
c. Example: "The histogram gives a better overall understanding because it groups the data into intervals, making it easier to see the overall shape of the distribution (e.g., the large cluster of higher temperatures) compared to the dot plot, which shows individual values but can be more cluttered for identifying broad patterns." (Or vice versa, depending on reasoning.)