Question

1. a survey of 36 local trampoline parks was conducted on their admission prices. the results are show in the box plot below. a. what percentage of trampoline parks charge over $19.75 for admission? b. what percentage of trampoline parks charge under $20.50 for admission? c. how many trampoline parks charge between $15.25 and $18.25? d. how many trampoline parks charge under $20.00? e. what is the 5-number summary of the box plot above?

Explanation:

Part a

Step1: Recall box plot percentiles

In a box plot, the data is divided into quartiles. The median (Q2) is at 50%, Q1 at 25%, Q3 at 75%. The whiskers represent the minimum and maximum, and the box spans Q1 to Q3. Values above Q3 (75th percentile) to max, and within the box, each quartile is 25%. But first, we need to identify the position of $19.75$. From the box plot, let's assume the upper whisker or the data: typically, in a box plot, the right end of the box is Q3, and then the whisker to max. Wait, actually, the key is that in a box plot, the data is split into four parts, each 25%. So if $19.75$ is the upper quartile (Q3) or beyond? Wait, no, let's think again. Wait, the question is about percentage over $19.75$. Let's assume that $19.75$ is the 75th percentile? No, wait, maybe the box plot has minimum, Q1, median, Q3, maximum. Let's assume the minimum is $15.25$, Q1 is $18.25$? Wait, no, the x-axis is 15,16,17,18,19,20,21. Wait, the left whisker starts at 15.5 (maybe 15.25), then the box starts at 18.25? Wait, no, the first vertical line on the left whisker is at 15.5 (let's say 15.25), then the box starts at 18.25? Wait, maybe the 5 - number summary: minimum, Q1, median, Q3, maximum. Let's assume from the plot: minimum is $15.25$, Q1 is $18.25$, median is $19.00$, Q3 is $19.75$, maximum is $20.50$. Wait, that makes sense. So:

• Minimum: $15.25$ (start of left whisker)
• Q1: $18.25$ (start of box)
• Median (Q2): $19.00$ (line in box)
• Q3: $19.75$ (end of box)
• Maximum: $20.50$ (end of right whisker)

So, in a box plot, the data is divided into quartiles:

• 25% below Q1
• 25% between Q1 and median
• 25% between median and Q3
• 25% above Q3

So, if Q3 is $19.75$, then the percentage of data above Q3 (i.e., over $19.75$) is 25%? Wait, no: wait, Q3 is the 75th percentile, so data above Q3 is 100 - 75 = 25%. Wait, yes. Because Q3 is the value where 75% of data is below it, so 25% is above. So if $19.75$ is Q3, then percentage over $19.75$ is 25%? Wait, but maybe the maximum is $20.50$, so between Q3 and max is 25%? Wait, no, the quartiles: Q1 (25th), Q2 (50th), Q3 (75th). So the data is split into four groups:

1. Min to Q1: 25%
2. Q1 to Q2: 25%
3. Q2 to Q3: 25%
4. Q3 to Max: 25%

So, if $19.75$ is Q3, then data above Q3 (i.e., >19.75) is the 4th group, which is 25% of the data.

Step1: Identify Q3

Assume from the box plot, Q3 (third quartile) is at $19.75$.

Step2: Calculate percentage above Q3

Since Q3 is the 75th percentile, the percentage of data above Q3 is $100\% - 75\% = 25\%$.

Step1: Identify maximum

From the box plot, the maximum (end of right whisker) is $20.50$.

Step2: Calculate percentage below maximum

Since the maximum is the highest value, all data (100%) is below the maximum (or equal, but since we are asked "under $20.50$", and the maximum is $20.50$, but in box plots, the whisker ends at maximum, so all data is ≤ maximum. But if we consider "under" as <, but since the maximum is the highest, the percentage of data under $20.50$ is 100% (because all data is ≤20.50, and if the maximum is 20.50, then the number of data points with value <20.50 would be total minus the number at maximum. But wait, maybe the maximum is 20.50, and the data is continuous? Wait, no, the survey is of 36 parks. Wait, maybe the right whisker ends at 20.50, so the maximum is 20.50. So all data is ≤20.50, so the percentage under $20.50$ is 100% (since no data is above 20.50). Wait, but maybe the maximum is 20.50, so data under 20.50 is all except the ones at 20.50. But in box plots, the whisker goes to the minimum and maximum, so the maximum is the highest value in the data. So if the maximum is 20.50, then the percentage of data under 20.50 is (36 - number of parks with 20.50)/36 100. But since we don't have the exact count, but from the box plot structure, the maximum is the end of the whisker, so typically, the data is such that the whisker includes all data except outliers, but here it's a survey of 36, so no outliers (since whiskers are present). So the maximum is the highest value, so all data is ≤20.50, so the percentage under 20.50 is 100% (because "under" could be <, but if the maximum is 20.50, then the data points at 20.50 are included in "under"? Wait, no, "under" is <, so we need to see how much is <20.50. But since the maximum is 20.50, the data above Q3 (19.75) is 25% (from part a), which is from Q3 (19.75) to max (20.50). So that 25% is between 19.75 and 20.50, so all of that is under 20.50? Wait, no, 20.50 is the maximum, so the data from Q3 to max is 25% (9 parks, since 360.25=9). So the percentage under 20.50 is 100%, because even the maximum is 20.50, but if "under" is <, then it's 100% - (number of parks with 20.50)/36 *100. But since we don't have that, but from the box plot, the whisker ends at 20.50, so the maximum is 20.50, so all data is ≤20.50, so the percentage under 20.50 is 100% (assuming "under" includes ≤, but the question says "under", which is <. Wait, maybe the maximum is 20.50, and the data points at 20.50 are part of the 25% above Q3. So the percentage under 20.50 is 100% - 0% (if no data is exactly 20.50) or 100% - (number of 20.50)/36. But since we can infer from the box plot structure, the maximum is the end of the whisker, so the data from Q3 (19.75) to max (20.50) is 25% of the data, so all of that is ≤20.50, so the percentage under 20.50 is 100% (because even the 25% is ≤20.50, and the rest is below Q3). Wait, no, the 25% above Q3 is between Q3 and max, so they are ≤max (20.50). So all data is ≤20.50, so the percentage under 20.50 is 100% (if "under" is <, then it's 100% - (number of max values)/36, but since we don't have that, and in box plots, the whisker represents the range, so we can assume that the maximum is the highest value, and all data is below or equal, so the percentage under 20.50 is 100%.

Wait, maybe a better approach: in a box plot, the five - number summary is Min, Q1, Median, Q3, Max. So if Max is 20.50, then all data is ≤20.50. So the percentage of data under 20.50 (i.e., <20.50) is 100% (because the maximum is the highest value, so no data is above 20.50)…

Step1: Identify Min and Q1

From the box plot, assume the minimum (Min) is $15.25$ and the first quartile (Q1) is $18.25$.

Step2: Calculate percentage between Min and Q1

In a box plot, the data between the minimum and Q1 represents $25\%$ of the total data (since Q1 is the 25th percentile).

Step3: Calculate number of parks

The total number of parks is 36. To find the number of parks between $15.25$ and $18.25$, we calculate $25\%$ of 36.
The formula for finding a percentage of a number is $\text{Number}=\text{Total}\times\text{Percentage}$.
So, $\text{Number}=36\times0.25 = 9$.

Step1: Find percentile range

Data between Min ($15.25$) and Q1 ($18.25$) is 25% of total.

Step2: Calculate number of parks

Number = $36\times0.25 = 9$.

Answer:

25%

Part b