Chapter 1

Question 1

Population

Answer 1

The entire collection of individuals or objects about which information is desired

Question 2

Census

Answer 2

When all of the desired information is avaiable for all objects or individuals in the population

Question 3

Sample

Answer 3

A subset of the population because of limited time, resources, money, etc.

Question 4

Types of Variables

Answer 4

1) Categorical 2) Quantitative or Numerical

Question 5

Categorical Variable

Answer 5

A categorical variable places an individual or object into one of several groups or categories Ex) Gender, race, type-of-job, hair color

Question 6

Quantitative or Numerical Variable

Answer 6

A Quantitative or Numerical Variable takes numerical values for which arithmetic operations such as adding and calculating an average value makes sense Ex) Age, salary

Question 7

Discrete numerical data

Answer 7

Numerical data is discrete if it’s set of possible values are finite Ex) Your year in college (1, 2, 3 or 4)

Question 8

Continous numerical data

Answer 8

Numerical data is continuous if it’s set of possible values form an entire interval on the number line Ex) Weight/height of an individual

Question 9

Univariate Data

Answer 9

Observations made on a single variable for each object in the dataset Ex) The unemployment rate of each state (state = object and unemployment rate = single variable)

Question 10

Multivariate Data

Answer 10

Observations made on multiple variables for each object in the dataset Ex) Each person -> age, gender, race, salary, job-type

Question 11

Bivariate Data

Answer 11

Bivariate data is a special case of multivariate data, where observations are made on two variables for each object in the dataset

Question 12

Branches of Statistics

Answer 12

Descriptive Statistics; Inferential Statistics; Probability

Question 13

Descriptive Statistics

Answer 13

• Objective is to merely summarize and describe important features of the data that is collected - Graphical approach: stem-and-leaf plots, histograms, box-plots, pie-charts, scatter-plots etc. - Numerical approach: calculation of numerical summary measures such as arithmetic mean, median, mode, standard deviation, correlation coefficient etc.

Question 14

Inferential Statistics

Answer 14

• Objective is to use information in the sample to make some sort of a conclusion (or inference) about the population from which the sample was selected - Includes Point-estimation, Hypothesis Testing, Confidence Interval Estimation, ANOVA, Linear Regression, etc.

Question 15

Probability

Answer 15

• Forms a bridge between descriptive and inferential statistics - Probability makes assumptions about the structure of the population, and then asks questions about what might result from selecting a sample from the population (deductive reasoning)

Question 16

Stem-Leaf Plot Construction

Answer 16

Separate each observation into a ‘stem” consisting of all but the final (rightmost) digit and a “leaf,” the final digit

Question 17

Stem-Leaf Plot: Pros and Cons

Answer 17

Pros: - a quick way of describing and ordering data - generally easy to construct - displays the actual data values - easy way to obtain a general idea of the distribution of the data (e.g. symmetric, skewed, bimodal) - to be able to describe the data from a stem-and-leaf plot, look for: a typical or a representative value (e.g. median); extent of spread about the typical value; presence of any gaps in the data; number and location of peaks; outliers Cons: - not always easy to construct an appropriate stem

Question 18

Relative Frequency

Answer 18

The proportion of times the value occurs in the sample For continuous data -> we have to create class-intervals and find the relative frequency for each class-interval

Question 19

Construction of Histograms for Discrete Data

Answer 19

Area of each rectangle is proportional to the relative frequency of the value

Question 20

Positive Skew of Histogram

Answer 20

A unimodal histogram with the right or upper tail stretched out more compared to the left or lower tail

Question 21

Symmetric Histogram

Answer 21

Left half of the histogram is mirror image of right half

Question 22

Unimodal Histogram

Answer 22

Histogram rises to a single peak and then declines

Question 23

Bimodal Histogram

Answer 23

Histogram has two different peaks

Question 24

Multimodal Histogram

Answer 24

A histogram with more than two peaks

Question 25

Number of Peaks in a Histogram

Answer 25

The larger the number of class intervals, the more likely it is that bimodality or unimodality will manifest itself

Question 26

Negative Skew of Histogram

Answer 26

A unimodal histogram with the left or lower tail stretched out more compared to the right or upper tail

Question 27

Density Estimate

Answer 27

A smoothed histogram

Question 28

Sample Size

Answer 28

* The number of observations in a sample * Notation = *n* * If two samples are simultaneously under consideration, either m and n; or n₁ and n₂

Question 29

The Mean

Answer 29

* The most familiar and useful measure of the center is the mean, or arithmetic average of the set - The only point at which a fulcrum can be placed to balance the system of weights is the point corresponding to the value of x-bar. * Notation: Sample Mean = x-bar * Notation: Population Mean = µ​ * µ = (sum of the N population values)/N

Question 30

Deficiencies of The Mean

Answer 30

- Outlier can greatly affect the mean and make the mean an inappropriate measure of the center

Question 31

Sample Median

Answer 31

* Obtained by first ordering the *n* observations from smallest to largest (with any repeated values included so that every sample observation appears in the ordered list)

Question 32

Median Distribution

Answer 32

Question 33

Quartiles

Answer 33

* Quartiles divide the data set into 4 equal parts, with the observations above the **3rd quartile** constituting the upper quarter of the data set, the **2nd quartile** being identical to the median, and the **1st quartile** separating the lower quartile from the upper 3 quartiles

Question 34

Percentiles

Answer 34

* Similarly, a data set (sample or population) can be even more finely divided using percentiles; the 99th percentile separates the highest 1% from the bottom 99%

Question 35

Trimmed Mean

Answer 35

* a compromise between the sample mean and the sample median * a 10% **trimmed mean**, for example, would be computed by eliminating the smallest 10% and the largest 10% of the sample and then averaging what remains

Question 36

Range

Answer 36

* A measure of variability * Computed by finding the difference between the largest and smallest sample values * Defect = **Range** only depends on the two most extreme observations and disregards the positions of the remaining n-2 values

Question 37

Deviations from The Mean

Answer 37

* A deviation will be positive if the observation is larger than the mean (to the right of the mean on the measurement axis) and negative if the observation is smaller than the mean * If all the deviations are small in magnitude, then all xis are close to the mean and there is little variability. * Alternatively, if some of the deviations are large in magnitude, then some xis lie far from the mean suggesting a greater amount of variability.

Question 38

Sum of Deviations

Answer 38

* The average deviation is always zero

Question 39

Sample Variance

Answer 39

* denoted by *s*²

Question 40

Sample Standard Deviation

Answer 40

* denoted by *s* * Note that *s*² and *s* are both nonnegative. The unit for *s* is the same as the unit for each of the x_is.

Question 41

Population Variance

Answer 41

* Denoted by σ² * For the population, the divisor is *N* and not *N*-1 * Note that σ² involves squared deviations about the population mean µ. * If we actually knew the value of µ, then we could define the sample variance as the average squared deviation of the sample x_is about µ. * However, the value of µ is almost never known, so the sum of squared deviations about x-bar must be used. * But the x_is tend to be closer to their average sample median than to the population average µ, so to compensate for this the divisor n – 1 is used rather than n

Question 42

Population Standard Deviation

Answer 42

* Denoted by σ

Question 43

Five Number Summary

Answer 43

* The **five number summary** consists of the smallest observation, the first quartile, the median, the third quartile, and the largest observation, written in order from smallest to largest 1. Minimum 2. Q₁ 3. M 4. Q₃ 5. Maximum

Question 44

To Calculate The Quartiles

Answer 44

1. Arrange the observations in increasing order and locate the 50th percentile, or the median M in the ordered list of obsercations 2. The 25th percentile, or the first quartile Q₁ is the median of the observations whose position in the ordered list is to the left of the overall median 3. The 75th percentile, or the third quartile Q₃ is the median of the observations whose position in the ordered list is to the right of the overall median

Question 45

Box-Plot

Answer 45

* A box-plot is a graph of the five number summary * Box-plots are most useful for side-by-side comparison of several distributions * Describes **location** and **spread** of a sample * Procedure: 1. Draw a rectangle with lower and upper edges at the 25th percentile, or 1st quartile, and the 75th percentile, or 3rd quartile 2. Draw a horizontal line across the rectangle at the median 3. Extend verticle lines, or whiskers, from the middle of the upper and lower edges of the rectangle to the minimum and maximum values