91Ó°ÊÓ

Statisticians often need to know the shape of a population to make inferences. Suppose that you are asked to specify the shape of the population of weights of all college students. a. Sketch a graph of what you think the weights of all college students would look like. b. The following data give the weights (in pounds) of a random sample of 44 college students (F and M indicate female and male, respectively). \(\begin{array}{llllllll}123 \mathrm{~F} & 195 \mathrm{M} & 138 \mathrm{M} & 115 \mathrm{~F} & 179 \mathrm{M} & 119 \mathrm{~F} & 148 \mathrm{~F} & 147 \mathrm{~F} \\ 180 \mathrm{M} & 146 \mathrm{~F} & 179 \mathrm{M} & 189 \mathrm{M} & 175 \mathrm{M} & 108 \mathrm{~F} & 193 \mathrm{M} & 114 \mathrm{~F} \\ 179 \mathrm{M} & 147 \mathrm{M} & 108 \mathrm{~F} & 128 \mathrm{~F} & 164 \mathrm{~F} & 174 \mathrm{M} & 128 \mathrm{~F} & 159 \mathrm{M} \\ 193 \mathrm{M} & 204 \mathrm{M} & 125 \mathrm{~F} & 133 \mathrm{~F} & 115 \mathrm{~F} & 168 \mathrm{M} & 123 \mathrm{~F} & 183 \mathrm{M} \\ 116 \mathrm{~F} & 182 \mathrm{M} & 174 \mathrm{M} & 102 \mathrm{~F} & 123 \mathrm{~F} & 99 \mathrm{~F} & 161 \mathrm{M} & 162 \mathrm{M} \\ 155 \mathrm{~F} & 202 \mathrm{M} & 110 \mathrm{~F} & 132 \mathrm{M} & & & & \end{array}\) i. Construct a stem-and-leaf display for these data. ii. Can you explain why these data appear the way they do? c. Construct a back-to-back stem-and-leaf display for the data on weights, placing the weights of the female students to the left of the stems and those of the male students to the right of the stems. (See Exercise \(2.89\) for an example of a back-to-back stem-and-leaf plot.) Does one gender tend to have higher weights than the other? Explain how you know this from the display.

Step by step solution

01

Sketching the Graph

Since we do not have any given data, a reasonable assumption to be made is that the weights of all college students follow a normal distribution. Hence, the graph should look like a bell curve which is symmetric about the mean.

02

Constructing a stem-and-leaf display

Arrange the weights in increasing order and separating the last digit (leaf) from the rest (stem). For example, for the weights 102, 108, and 116, the stem would be 10 and the leafs would be 2, 8, and 6. Do this for all weights.

03

Interpretation of the data

The column of stems gives the distribution of tens digits while the individual leaves give you a sense of how many times this tens digit turns up in the data and where most of the observations lie.

04

Constructing a back-to-back stem-and-leaf display

First separate the weights of female students from male students. Then use these separate data to construct a stem-and-leaf display with weights of female students to the left of the stems and weights of male students to the right of the stems.

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Key Concepts

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

Population Shape

When statisticians talk about the shape of a population, they're referring to how the data is distributed across different values. In this case, we're interested in the distribution of weights for college students. The population shape can help us understand the commonness of certain weight ranges and if there are any unusual patterns, such as clusters or outliers. Often, when one thinks of data like weights, a normal distribution is assumed. This means the data would usually resemble a bell-shaped curve.
In a bell curve, most students would have weights around the average, with fewer students having extremely low or high weights. This is helpful because it allows statisticians to make predictions about the population based on a sample. If the population shape is not normal, it might indicate skewness or the presence of more than one peak.
Understanding the shape of the population is crucial for making informed inferences about the data.

Stem-and-Leaf Display

A stem-and-leaf display is a tool used by statisticians to visualize data where each value is split into a 'stem' and a 'leaf'. This method helps in maintaining the original data as much as possible while allowing us to quickly see the shape and distribution. In the context of the exercise, each student's weight is split with the tens digit as a 'stem' and the units digit as a 'leaf'.
This provides a simple way to glance at the data to notice patterns. For example, multiple leaves on a single stem suggest many data points within that range. Constructing this plot involves organizing the weight data in ascending order and positioning them around their respective stems.
The stem-and-leaf plot is particularly useful for small data sets where it is important to observe the distribution visually while retaining the actual data values.

Normal Distribution

The normal distribution, sometimes referred to as a Gaussian distribution, is a key concept in statistics. It's what the population shape would look like if most students' weights are grouped near the average, with fewer students having weights lower or higher than this average. This distribution is symmetric, meaning it looks the same on the left and right of the center point.
Why do we care if the data is normally distributed? If the dataset approximates this distribution, it makes analyzing and interpreting the data simpler, and applying statistical methods less error-prone. For the weights of college students, visualizing them as normally distributed allows us to use many standard statistical tools and models.
In many cases, datasets in social sciences and natural sciences align to some degree with normal distribution, though real-world data can have varying degrees of skewness.

Data Interpretation

Interpreting data involves making sense of rows of numbers to answer specific questions or predict outcomes. In the example of college students' weights, the aim is to decide how these weights are distributed across genders and what inferences can be drawn. The stem-and-leaf plots offer a foundation for understanding the cluster and spread of the data.
Using the back-to-back stem-and-leaf display, differences in weight distributions between genders can be observed. Noticing which side (male or female) tends to have more leaves in the higher stems is a way to determine which gender tends to weigh more on average.
Thoughtfully analyzing the spread and shape of the data in terms of distribution allows us to ask pertinent questions: "Are males generally heavier?", "Does the data support or contradict preconceived notions?" Interpretation is the core process where statisticians extract meaningful conclusions and actionable insights from data.