91Ó°ÊÓ

A random sample of 81 female American college students were each issued a stopwatch and asked to time themselves as they prepared to attend class on the following Thursday morning. The instructions were to start the watch as soon as their feet touched the floor as they got up and to turn it off as they passed through the door of their residence on the way to class. \(x=\) "floor- to-door" time rounded to the nearest minute. $$\begin{array}{ccccccccccccccc}\hline 3 & 4 & 12 & 9 & 12 & 23 & 25 & 25 & 26 & 14 & 17 & 14 & 13 & 17 & 18 \\\30 & 28 & 37 & 19 & 18 & 20 & 22 & 38 & 38 & 42 & 38 & 41 & 26 & 23 & 29 \\\32 & 23 & 25 & 31 & 29 & 35 & 33 & 37 & 33 & 41 & 42 & 42 & 40 & 46 & 46 \\\46 & 46 & 45 & 43 & 44 & 46 & 50 & 48 & 51 & 54 & 55 & 53 & 56 & 53 & 62 \\\60 & 59 & 62 & 62 & 60 & 58 & 58 & 16 & 63 & 73 & 71 & 70 & 73 & 78 & 91 \\\89 & 98 & 83 & 79 & 75 & 76 & & & & & & & \\\\\hline\end{array}$$ a. What is the population of interest? b. Draw a histogram of the "floor-to-door" variable using multiples of 10 for class midpoints. Describe the distribution. Does it appear to be approximately normal? Explain. c. Redraw the histogram using multiples of 5 for class midpoints. Describe the visible patterns displayed by this second histogram that were not visible in the first one. Explain what causes this strange pattern. d. Would you say the histogram suggests that the variable, amount of time, is approximately normally distributed? What evidence can you find to support your answer?

Step by step solution

01

Identify the Population of Interest

In this exercise, the population of interest refers to the group being studied. So, the population of interest would be female American college students.

02

Draw the First Histogram

First, organize the data from smallest to largest. The data ranges from 3 to 98. Create a frequency distribution table to draw a histogram. Because the exercise asks to use multiples of 10 for class midpoints, the data is organized into these classes w.r.t corresponding frequencies. Then, construct the histogram using this data.

03

Description of the First Histogram

Describe the distribution of data. A histogram might have a normal or non-normal shape. A normal shape is usually symmetrical with a peak in the middle. A non-normal shape can take on several forms such as right-skewed, left-skewed etc.

04

Draw the Second Histogram

Redraw the histogram using multiples of 5 for class midpoints. Here, change the class size to fit multiples of 5, creating a new frequency table. Use this table to draw the second histogram.

05

Description of the Second Histogram

Describe the distribution in the second histogram. Also, identify the differences observed in this histogram compared to the first one, and explain why these patterns occur.

06

Assess Normality of the Data

Look at the shape of the histogram and determine whether or not it represents a normal distribution. Discuss the evidence that supports this assessment. A normal distribution forms a bell curve, while a non-normal distribution can be skewed in either direction or have other irregularities.

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Key Concepts

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

Histogram

A histogram is a type of bar chart that visually displays the distribution of numerical data. It consists of bars representing different intervals, or "bins," of the dataset. Each bar's height corresponds to the frequency - how many data points fall within that particular interval.
To create a histogram, follow these steps:

• Sort your data from smallest to largest to understand the range of values.
• Decide on the number of bins and their sizes; this can affect the interpretation of the data.
• Tally the data points into these bins, and graph the frequency of each bin as a bar.

Histograms are helpful because they reveal patterns in the data, such as symmetry, skewness, and outliers, which can indicate how data might be distributed.

Normal Distribution

Normal distribution, often called a "bell curve," is a fundamental concept in statistics, describing how data points are distributed in many natural phenomena.
This distribution is symmetrical, with most data points clustering around the mean, tapering off on both sides. The features of normal distribution include:

• Symmetry: The data is spread evenly around the central mean.
• Bell-shaped curve: It peaks around the mean and decreases as you move away.
• 68-95-99.7 Rule: Approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

Determining whether your data is normally distributed can be difficult, but examining the histogram's shape and other statistical tests can help. In our exercise, students used histograms to visually compare data distributions against a normal pattern.

Frequency Distribution

Frequency distribution is a summary of how often different values occur in a dataset. It is vital in organizing raw data to show how data points are distributed. Here's how you establish a frequency distribution:

• List the unique values or range of values in your dataset.
• Count how often each value, or interval of values, appears.
• Present this information in a table or use it to construct a histogram.

Frequency distributions are fundamental for creating histograms, as they provide the data necessary to visualize the density and spread of the dataset. They help identify how data concentrates around certain values, aiding in further statistical analysis.

Data Visualization

Data visualization involves displaying data graphically to make it easier to understand and interpret. It's crucial in identifying patterns, trends, and outliers that might not be apparent by just looking at raw data.
For instance, visual tools like histograms can convey statistical concepts like distribution, concentration, and skewness, providing quick insights at a glance. Using different visual methods allows one to communicate findings effectively and supports making informed decisions. In our exercise, creating two histograms with different bin sizes demonstrated how visual representation changes our perception of data patterns, like clustering or skewness. Understanding these concepts enhances comprehension, delivering a more profound awareness of the dataset's intrinsic qualities.