Business Statistics Ch 1 (Week 1)
The stemplot below displays midterm exam scores for 34 students taking a calculus course. The highest possible test score was 100. The teacher declared that an exam grade of 65 or higher was good enough for a grade of C or better. The percent of students earning a grade of C or higher (as declared by the teacher) is closest to:
65%
What is the difference between a histogram and a bar chart?
A bar chart displays a categorical variable on the horizontal axis, whereas a histogram does not.
Categorical Variable
A categorical variable places an individual into one of several groups or categories.
Symmetric and Skewed Distributions
A distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other. A distribution is skewed to the right if the right side of the histogram (containing the half of the observations with the larger values) extends much farther out than the left side. It is skewed to the left if the left side of the histogram extends much farther out than the right side.
Histogram
A graph of vertical bars representing the frequency distribution of a set of data. Histograms show the distribution of a quantitative variable by using bars whose height represents the number of individuals who take on a value within a particular class. -Appropriate for quantitative variables that take many values and/or large datasets. -Divide the possible values into classes (equal widths). -Count how many observations fall into each interval (may change to percents). -Draw a picture representing the distribution―bar heights are equivalent to the number (percent) of observations in each interval. ¨EXAMPLE: Freshman Graduation Rate, or FGR, Data for 2013 - 2014
Quantitative Variable
A quantitative variable takes numerical values for which arithmetic operations such as adding and averaging make sense. The values of a quantitative variable are usually recorded with a unit of measurement such as seconds or kilograms.
Time Plot
A time plot of a variable plots each observation against the time at which it was measured. Always put time on the horizontal scale of your plot and the variable you are measuring on the vertical scale. Connecting the data points by lines helps emphasize any change over time. - A time plot shows behavior over time. - Time is always on the horizontal axis, and the variable being measured is on the vertical axis. - Look for an overall pattern (trend), and deviations from this trend. Connecting the data points by lines may emphasize this trend. - Look for patterns that repeat at known regular intervals (seasonal variations).
Trend
A type of overall pattern in a time plot - a long-term upward or downward movement overtime.
Variables
A variable is any characteristic of an individual. A variable can take different values for different individuals. Categorical Variable - A categorical variable places an individual into one of several groups or categories. Quantitative Variable - A quantitative variable takes numerical values for which arithmetic operations such as adding and averaging make sense. The values of a quantitative variable are usually recorded with a unit of measurement such as seconds or kilograms.
As part of a database of new births at a hospital, some variables recorded are the age of the mother, the marital status of the mother (such as single, married, or divorced), the weight of the baby, and the sex of the baby. Of these variables: A. the individuals described are mothers and babies involved in births at a hospital. B. the age of the mother and the weight of the baby are quantitative variables. C. the sex of the baby and the marital status of the mother are categorical variables. D. All of the answer options are correct.
All of the answer options are correct.
Some of the variables from a survey conducted by the U.S. Census Bureau are the number of people living in a household, the total household gross income, and the ages of household residents. Which of the variables is quantitative? a. the number of people living in a household b. the total household gross income c. the ages of household residents d. All of the answer options are correct.
All of the answer options are correct.
Bar Graphs
Bar graphs are used to represent two elements of a single subject. Bar Graphs only: EXAMPLE: What sources do Americans aged 12-24 years use to keep up-to-date and learn about music?
A poll was conducted of more than 50,000 buyers of new cars, 90 days after the cars were purchased. The data on problems per 100 vehicles for cars made by Toyota and General Motors (GM) are given in the time plot below for the years 1998-2004. The solid line is for GM and the dashed line is for Toyota. Which of the following is a true statement?
The number of problems was higher for GM than for Toyota in each year.
Distributions
Describing Distributions - A distribution is symmetric if the right and left sides of the graph are approximately mirror images of each other. - A distribution is skewed to the right (right-skewed) if the right side of the graph (containing the half of the observations with larger values) is much longer than the left side. - A distribution is skewed to the left (left-skewed) if the left side of the graph is much longer than the right side.
Stemplots
EXAMPLE: Weight Data—Introductory Statistics Class
Pie Charts or Bar Graphs?
EXAMPLE: What do 1.5 million full-time, first-year students plan to study? Here are data on the percents of post-secondary, first-year students who plan to major in several discipline areas:
Examining a Histogram
In any graph of data, look for the overall pattern and for striking deviations from that pattern. You can describe the overall pattern of a histogram by its shape, center, and variability. You will sometimes see variability referred to as spread. An important kind of deviation is an outlier, and individual value that falls outside the overall pattern.
Individuals
Individuals are the objects described by a set of data. Individuals may be people, but they may also be animals or things.
Cycles
Looking at the overall patterns and strong deviations in patterns in a time plot.
Consumers' Union measured the gas mileage per gallon of thirty-eight 1998-99 model automobiles on a special test track. The following pie chart provides information about the country of manufacture of the cars that Consumers' Union used. Based on this pie chart, we may conclude that:
More than half of the cars in the study were from the United States.
Outliers
Outliers are observations that lie outside of the overall pattern of a distribution.
Pie Chart
Pie charts show the distribution of a categorical variable as a "pie" whose slices are sized by the counts or percents for the categories. EXAMPLE: What do the 1.5 million full-time, first-year students plan to study? Here are data on the percents of post-secondary, first-year students who plan to major in several discipline areas:
Statistics
Statistics is the science of data. The first step in dealing with data is to organize your thinking about the data: Individual: an object described by a set of data Variable: a characteristic of the individual
Stemplot
Stemplots separate each observation into a stem and a leaf that are then plotted to display the distribution while maintaining the original values of the variable. To make a stemplot: 1. Separate each observation into a stem, consisting of all but the final (rightmost) digit, and a leaf, the final digit. Stems may have as many digits as needed, but each leaf contains only a single digit. 2. Write the stems in a vertical column with the smallest at the top, and draw a vertical line at the right of this column. Be sure to include all the stems needed to span the data, even when some stems will have no leaves. 3. Write each leaf in the row to the right of its stem, in increasing order out from the stem. - If there are very few stems (when the data cover only a very small range of values), then we may want to create more stems by splitting the original stems. EXAMPLE: If all of the data values were between 150 and 179, then we may choose to use the following stems:
Roundoff error
The difference between the calculated approximation of a number and its exact mathematical value.
Categorical Data
The distribution of a categorical variable lists the categories, and gives the count or percent of individuals who fall into each category. -Pie Charts show the distribution of a categorical variable as a "pie" whose slices are sized by the counts or percents for the categories. -Bar Graphs represent each category as a bar whose height shows the category counts or percents.
Quantitative Data
The distribution of a quantitative variable tells us what values the variable takes on and how often it takes those values. -Histograms show the distribution of a quantitative variable by using bars whose height represents the number of individuals who take on a value within a particular class. -Stemplots separate each observation into a stem and a leaf that are then plotted to display the distribution while maintaining the original values of the variable.
Distribution of a Variable
The distribution of a variable tells us what values it takes and how often it takes these values. The values of categorical variable are labels for the categories. The distribution of a categorical variable lists the categories and gives either the count or the percent of individuals who fall into each category.
Exploratory Data Analysis
The use of statistical tools and ideas to help us examine the data in order to describe their main features. Exploring Data 1. Begin by examining each variable by itself. Then move on to study the relationships among the variables. 2. Begin with a graph or graphs. Then add numerical summaries of specific aspects of the data.
Veterinary researchers wanted to know if housing might be related to whether or not a horse develops enteroliths. Attached are side-by-side pie charts of housing for horses with enteroliths (cases) and horses without (controls). Based on these charts, it is reasonable to conclude that:
cases are less likely to be housed in a pasture.
Employees at a large company are surveyed about their health insurance status. Employees are coded as "1" if health insurance is obtained through the company's benefit program, "2" if health insurance is obtained from another source (such as through a spouse's employment benefit program), or "0" if the employee does not have health insurance. This variable is: A. numerical. B. categorical. C. quantitatively categorical. D. All of the answer options are correct.
categorical.
Cross-Sectional Data
data collected at the same or approximately the same point in time (but are at different locations - for example)
Time Series Data
data collected over several time periods
Enteroliths are calcifications that form in the gut of horses. The stones can cause considerable morbidity and mortality. A study was conducted to investigate factors (such as age, diet, and environment) that may be related to the formation of enteroliths. The histogram of age for the horses in the enteroliths study is:
slightly left-skewed.
A description of different houses for sale includes the square footage of the house, whether or not the house has a finished basement, and the monthly electric bill. Which of the variables is categorical? A. the square footage of the house B. whether or not the house has a finished basement C. the monthly electric bill D. All of the answer options are correct.
whether or not the house has a finished basement