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the data represent the number of ounces of water that 26 students drank the day... test at school
8, 8, 8, 16, 16, 16, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 80, 80, 80, 80, 88, 88, 88, 88
a. create a dot plot for the data.
b. create a box plot for the data.
c. what information about the data is provided by the box plot that is not provided by the dot plot?
d. what information about the data is provided by the dot plot that is not provided by the box plot?
e. it was recommended that students drink 48 or more ounces of water. how could you use a histogram to easily display the number of students who drank the recommended amount?

Question

Accepted Answer

### Part (a): Create a Dot Plot
# Explanation:
## Step 1: Identify Data Values and Frequencies
First, we list out the data values and count how many times each value appears:
- 8: Let's count the number of 8s. From the data: 8, 8, 8 – so frequency = 3.
- 16: 16, 16, 16 – frequency = 3.
- 32: Let's count the 32s. The data has 32, 32, 32, 32, 32, 32 – wait, let's check the original data: "8 8 8 16 16 16 32 32 32 32 32 32 64 64 64 64 64 64 80 80 80 80 88 88 88" Wait, maybe I misread earlier. Let's re - extract the data properly. The data is: 8, 8, 8, 16, 16, 16, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 80, 80, 80, 80, 88, 88, 88. So:
    - 8: 3 times
    - 16: 3 times
    - 32: 6 times
    - 64: 6 times
    - 80: 4 times
    - 88: 3 times

## Step 2: Draw the Dot Plot
On a number line, mark the values 8, 16, 32, 64, 80, 88. For each value, place a dot above it corresponding to its frequency. For example, above 8, we place 3 dots; above 16, 3 dots; above 32, 6 dots; above 64, 6 dots; above 80, 4 dots; above 88, 3 dots.

### Part (b): Create a Box Plot
# Explanation:
## Step 1: Order the Data (already ordered)
The data is already in ascending order: 8, 8, 8, 16, 16, 16, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 80, 80, 80, 80, 88, 88, 88. The number of data points $ n = 25 $.

## Step 2: Find the Median (Q2)
The median is the middle value. For $ n = 25 $, the median is the $ \frac{n + 1}{2}=\frac{25+1}{2}=13^{th} $ value. Looking at the data, the 13th value is 64. So $ Q2=64 $.

## Step 3: Find the Lower Quartile (Q1)
The lower half of the data is the first 12 values (since $ n = 25 $, the lower half is from the 1st to the 12th value: 8, 8, 8, 16, 16, 16, 32, 32, 32, 32, 32, 32). The median of the lower half (Q1) is the average of the 6th and 7th values. The 6th value is 16 and the 7th value is 32. So $ Q1=\frac{16 + 32}{2}=24 $.

## Step 4: Find the Upper Quartile (Q3)
The upper half of the data is from the 14th to the 25th value: 64, 64, 64, 64, 64, 64, 80, 80, 80, 80, 88, 88, 88. The median of the upper half (Q3) is the average of the 19th and 20th values. The 19th value is 80 and the 20th value is 80. So $ Q3 = 80 $.

## Step 5: Find the Minimum and Maximum
The minimum value is 8 and the maximum value is 88.

## Step 6: Draw the Box Plot
- Draw a number line that covers the range from 8 to 88.
- Draw a box from Q1 = 24 to Q3 = 80. Inside the box, draw a line at Q2 = 64 (the median).
- Draw whiskers from the box to the minimum (8) and maximum (88) values.

### Part (c): Information from Box Plot not in Dot Plot
# Brief Explanations:
A box plot provides a summary of the data's spread and central tendency in terms of quartiles. It shows the median (Q2), the lower quartile (Q1), the upper quartile (Q3), the minimum, and the maximum. It gives a quick overview of the inter - quartile range (IQR = Q3 - Q1) which measures the spread of the middle 50% of the data. It also helps in identifying potential outliers (using the 1.5*IQR rule) more easily than a dot plot. A dot plot shows the individual data points and their frequencies, but it does not summarize the data into quartiles and the median in a concise way like a box plot. For example, we can quickly see the range of the middle 50% of the data (from Q1 to Q3) from a box plot, which is not as straightforward from a dot plot.
# Answer:
A box plot provides information about the quartiles (Q1, Q2, Q3), the inter - quartile range (IQR), and a concise summary of the spread (range of the middle 50% of data) and central tendency (median) that is not as easily discernible from a dot plot. It also helps in quickly identifying the range of the middle half of the data and potential outliers.

### Part (d): Information from Dot Plot not in Box Plot
# Brief Explanations:
A dot plot shows the frequency of each individual data value. It allows us to see the exact number of times each data point (e.g., how many students drank 8 ounces, 16 ounces, etc.) occurs. A box plot summarizes the data into quartiles, median, minimum, and maximum, but it does not show the frequency of individual data points. For example, from a dot plot, we can see that 3 students drank 8 ounces, 3 students drank 16 ounces, etc., which is not shown in a box plot.
# Answer:
A dot plot provides the frequency of each individual data value (e.g., how many times each specific number of ounces of water was drunk by students) which is not provided by a box plot. It shows the distribution of individual data points, including the count of each distinct data value.

### Part (e): Using Histogram to Show Recommended Amount
# Brief Explanations:
To use a histogram to display the number of students who drank 48 or more ounces of water, we can set up the histogram with appropriate bins. We can create a bin that includes all values from 48 to (let's say) 96 (to cover the maximum value of 88). Then, we count the number of data points (students) whose number of ounces of water is 48 or more. Looking at our data, the values 32, 64, 80, 88. Wait, no: the recommended amount is 48 or more. So we exclude 8, 16, 32. The values 64, 80, 88 are 48 or more. Let's count their frequencies: 64 occurs 6 times, 80 occurs 4 times, 88 occurs 3 times. Total $ 6 + 4+3=13 $ (wait, no, let's re - check the data. The data is 8, 8, 8, 16, 16, 16, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 80, 80, 80, 80, 88, 88, 88. So the number of data points with 48 or more: 6 (for 64) + 4 (for 80)+3 (for 88) = 13? Wait, no, the number of data points: let's count the total number of data points. 3 (8s)+3 (16s)+6 (32s)+6 (64s)+4 (80s)+3 (88s)=3 + 3+6 + 6+4 + 3=25. The number of data points with less than 48: 3 (8s)+3 (16s)+6 (32s)=12. So the number with 48 or more is $ 25-12 = 13 $. To create a histogram, we can define bins such that one of the bins is [48, 96] (or we can have more detailed bins like [48, 64), [64, 80), [80, 96]). Then, we can plot the frequency of data points in the bin(s) that correspond to 48 or more ounces. For example, if we use a bin of [48, 96], the frequency in this bin will be the number of students who drank 48 or more ounces of water.
# Answer:
To use a histogram, we can define a bin (or bins) that includes all values of 48 or more ounces (e.g., a bin from 48 to 96). Then, we count the number of data points (students) whose number of ounces of water falls within this bin. This bin's frequency will represent the number of students who drank the recommended amount (48 or more ounces). We can also use multiple bins within the 48+ range (e.g., [48, 64), [64, 80), [80, 96]) to show the distribution of the recommended - amount drinkers.

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QUESTION IMAGE

Question

Explanation:

Part (a): Create a Dot Plot

Step 1: Identify Data Values and Frequencies

Step 2: Draw the Dot Plot

Part (b): Create a Box Plot

Step 1: Order the Data (already ordered)

Step 2: Find the Median (Q2)

Step 3: Find the Lower Quartile (Q1)

Step 4: Find the Upper Quartile (Q3)

Step 5: Find the Minimum and Maximum

Step 6: Draw the Box Plot

Part (c): Information from Box Plot not in Dot Plot

Answer:

Part (d): Information from Dot Plot not in Box Plot