91Ó°ÊÓ

Shoes How many pairs of shoes do students have? Do girls have more shoes than boys? Here are data from a random sample of 20 female and 20 male students at a large high school: $$ \begin{array}{llrrrrrrrrr} \hline \text { Female: } & 50 & 26 & 26 & 31 & 57 & 19 & 24 & 22 & 23 & 38 \\ & 13 & 50 & 13 & 34 & 23 & 30 & 49 & 13 & 15 & 51 \\ \text { Male: } & 14 & 7 & 6 & 5 & 12 & 38 & 8 & 7 & 10 & 10 \\ & 10 & 11 & 4 & 5 & 22 & 7 & 5 & 10 & 35 & 7 \\ \hline \end{array} $$ (a) Find and interpret the percentile in the female distribution for the girl with 22 pairs of shoes. (b) Find and interpret the percentile in the male distribution for the boy with 22 pairs of shoes. (c) Who is more unusual: the girl with 22 pairs of shoes or the boy with 22 pairs of shoes? Explain.

Step by step solution

01

Organize Female Shoe Data

Start by sorting the number of pairs of shoes for female students to find the position of the girl with 22 pairs of shoes in the list. The sorted data is: 13, 13, 13, 15, 19, 22, 23, 23, 24, 26, 26, 30, 31, 34, 38, 49, 50, 50, 51, 57.

02

Find Percentile for Females

The girl with 22 pairs of shoes is the 6th in the sorted list. To find her percentile: \[ P = \frac{L}{N} \times 100 \] where \( L \) is the number of values less than 22 and \( N \) is the total number of values.\[ L = 5, \quad N = 20 \] \[ P = \frac{5}{20} \times 100 = 25 \]%Thus, 22 pairs are at the 25th percentile for females.

03

Organize Male Shoe Data

Sort the number of pairs of shoes for male students to find the position of a boy with 22 pairs of shoes. The sorted data is: 4, 5, 5, 5, 6, 7, 7, 7, 8, 10, 10, 10, 11, 12, 14, 22, 35, 38.

04

Find Percentile for Males

The boy with 22 pairs of shoes is the 16th in the sorted male list. For his percentile:\[ L = 15, \quad N = 20 \] \[ P = \frac{15}{20} \times 100 = 75 \]%Thus, 22 pairs are at the 75th percentile for males.

05

Compare Unusualness

To determine who is more unusual, compare their percentiles. The girl is at the 25th percentile, and the boy at the 75th percentile. A higher percentile means fewer people have as many shoes, so the boy's number of shoes is more unusual.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Analysis

Statistical analysis is a method of making sense of collected data. In this scenario, we're looking at how many pairs of shoes each student has.
This involves sorting data, calculating percentiles, and interpreting results. Understanding percentiles helps in comparing an individual's data point relative to a group.

• **Sorting Data**: First, you organize the data from lowest to highest. This helps in finding a specific position of interest within your data.
• **Calculating Percentile**: A percentile tells us the percentage of values below the specific data point. For example, if a value is at the 25th percentile, that means 25% of the data points are lower than this value.
• **Interpreting Percentile**: The percentile provides insight into where a particular data point falls within the data distribution, which can reveal information about trends and outliers.

In this exercise, the number of shoes each student has is analyzed to find its position and rarity within their respective gender groups to better understand their distributions.

Gender Comparison

When comparing different genders, we want to see if there is any statistical difference in shoe ownership between male and female students.
We achieve this by analyzing the percentile ranks of shoe numbers within each gender group.

• **Female Data Analysis**: The exercise showed that a girl with 22 pairs of shoes was at the 25th percentile, meaning she had more shoes than 25% of the females in her group. This suggests her shoe count is not very unusual among girls.
• **Male Data Analysis**: In contrast, a boy with 22 pairs of shoes was at the 75th percentile, indicating his shoe count is more uncommon among boys, as he has more shoes than 75% of the males in his group.

Ultimately, this gender comparison provided views into how typical or atypical the shoe ownership is within each group, pointing toward disparity in shoe counts between male and female students in this sample.

Data Interpretation

Data interpretation comes in when we try to make conclusions from statistical findings. It involves understanding what percentiles, differences, and comparisons convey in real-world terms.
The exercise demonstrated a significant element of data interpretation.

• **Understanding Percentile Meaning**: For the girl with 22 shoes ranked at the 25th percentile, we conclude that she doesn't stand out much in terms of shoe count among her peers. However, the boy with 22 shoes at the 75th percentile signifies that his shoe count is relatively high compared to other boys.
• **Comparing Rarities**: The core of data interpretation here is observing which data points are common or uncommon in their respective groups, which can offer insights into cultural or social aspects, like shoe preferences in this case.
• **Real-Life Implications**: These findings might suggest that females could generally own more shoes than males in this sample, or perhaps it's more common for girls to have a higher shoe count.

Through careful interpretation, we can draw meaningful conclusions that go beyond mere numbers, leading us to valuable insights or questions about the data context.