Midterm 1 Flashcards
1
Q
What is descriptive stats?
A
consists of methods for organizing and summarizing information
2
Q
What is inferential statistics?
A
consists of methods for drawing and measuring the reliability of conclusions about a population based on information obtained from a sample of the population
3
Q
What is a population?
A
The collection of all individuals or items under consideration in a statistical study
4
Q
What is a sample?
A
That part of the population from which information is actually obtained
5
Q
What is an observational study?
A
- researchers simply observe characteristics and take measurements, as in a sample survey - Observation studies can reveal association, but not causation
6
Q
What is a designed experiment?
A
- researchers impose treatments and controls and then observe characteristics and take measurements - Designed experiments (done properly) reveals both association and causation.
7
Q
What is a census?
A
A survey that includes every member of the population
8
Q
What are the issues with censuses?
A
- If the population is large, it can be very costly and difficult (perhaps impossible) to collect information from every member of the population. -Since a census is usually too costly or takes too long, most statistical information is gathered by sampling or experimentation
9
Q
What is sampling?
A
collecting the information from a sample rather than the entire population. - Since the of sampling is to make decisions about the corresponding population, it is important that the results obtained from sampling closely match the results that we would obtain by conducting a census. - This means sampling must be done very carefully so as to obtain a representative sample. - One method of sampling is to try to choose elements of the population so each element has an equal chance of being included in the sample.
10
Q
What is simple random sampling?
A
A sampling procedure for which each possible sample of a given size is equally likely to the one obtained.
11
Q
What is a simple random sample?
A
A sample obtained by simple random sampling. - Using an SRS (simple random sample) is a common way of obtaining a representative sample. - Samples may be selected with or without replacement. - In sampling with replacement, each time an element is chosen from the population, it is put back in the population - thus any element may be chosen more than once for a sample. - In sampling without replacement , an element of the population is removed from the population once it has been chosen - thus any element can only appear only once in the sample
12
Q
What is a study?
A
The process of sampling a population, and collecting the information of interest
13
Q
What is raw data?
A
Data recorded in the sequence in which they are collected and before they are processed or ranked
14
Q
What is a variable?
A
A characteristic that varies from one individual to another
15
Q
What is qualitative or categorical variable?
A
A non-numerically valued variable Examples of qualitative (i.e. categorical) variables include eye colour, first letter in a persons last name, type of automobile a person drives.
16
Q
What is a quantitative variable?
A
A numerically valued variable. - Examples of quantitative variables include height, weight, age, speed of traffic (the various vehicles) at a certain location and time and number of stars that can be observed in a particular part of sky. - There are two types discrete and continuous
17
Q
What is a discrete variable?
A
A quantitative variable whose possible values can be listed.
18
Q
What is a continuous variable?
A
A quantitative variable whose possible values form some interval of numbers.
19
Q
What is data?
A
Values of a variable
20
Q
What is Qualitative or Categorical data
A
Values of a qualitative or categorical variable
21
Q
What is quantitative data?
A
Values of a quantitative variable
22
Q
What is discrete data?
A
Values of a discrete variable
23
Q
What is continuous data?
A
Values of a continuous variable
24
Q
What is a data set?
A
The collection of all observations for a particular variable
25
What is an observation?
Each piece of individual data
26
What is frequency distribution for qualitative (categorical) data?
lists all categories and the number of elements that belong to each of the categories
27
What is relative frequency?
a category is obtained by dividing the frequency of that category by the sum of all frequencies - the relative frequency shows what fractional part or proportion of the total frequency belongs to the corresponding category
28
What is the relative frequency distribution?
lists the relative frequencies for all categories. - Relative frequency must always add up to 1.00
29
How is a percentage of a category obtained?
by multiplying the relative frequency of that category by 100 -
30
What is a percentage distribution list?
lists the percentages for all categories - percentage must always add up to 100
31
What is a bar graph or bar chart?
A graph made of bars whose heights represent the frequencies or relative frequencies or percent frequencies of respective categories
32
What is a pie chart?
A circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories - we multiply 360 by the relative frequency of each category to obtain the degree measure or size of the angle for the corresponding category.
33
What is a class, category or bin?
For quantitative data, an that includes all the values that fall within two numbers, the lower and upper limits - Each class (category, bin) is defined by an interval - these are chosen so every measurement in the data set falls into exactly one interval.
34
What is a frequency?
are the number of values that belong to different classes and are denoted by f
35
What is frequency distribution for quantitative data?
lists all the classes and the number of values that belong to each class. Data presented in the form of a frequency distribution are called .group data
36
What is single value grouping?
- In single-value grouping, each distinct number in the set of measurements forms a class. - Single-value grouping is appropriate for discrete data in where there are only a small number of distinct values. - Make a table of frequencies of each distinct value, in order from smallest to largest.
37
What is limit grouping?
- This method is appropriate for data. - In this method, we define the classes by providing numbers called class limits - For each class, there is a lower limit and upper limit, selected by the person doing the analysis. - The lower limit of a class is the smallest number that could be in the class. - The upper limit of a class is the largest number that could be in the class.
38
What is lower class limit?
The smallest value that could go into a class
39
What is upper class limit?
The largest value that could go into a class
40
What is class width?
The difference between the lower limit of a class and the lower limit of the next-higher class
41
What is class mark?
The average of the two class limits of a class
42
What are the guidelines for grouping?
1. The number of classes should be small enough to provide an effective summary but large enough to display the relevant characteristics of the data. Generally 5 to 20 classes are used. 2. Each observation must belong to one, and only one, class. 3. Whenever feasible, all classes should have the same width
43
What is cutpoint grouping?
- This method is similar to Limit Grouping - define classes by choosing two values. - there is an important difference in how one of these values is defined. - This modification makes this method appropriate for continuous data although it may also be used for discrete data.
44
What is lower class cutpoint?
The smallest value that could go in a class
45
What is upper class cutpoint?
The smallest value that could go in the next-higher class (equivalent to the lower cutpoint of the next-higher class)
46
What is class width in cutpoint grouping?
The difference between the cutpoints of a class
47
What is class midpoint in cutpoint grouping?
The average of two cutpoints of a class
48
What is the relative frequency of a class?
is the frequency of that class divided the sum of all frequencies (i.e. the number of measurements).
49
What is a histogram?
displays the classes of the quantitative data on a horizontal axis and frequencies or relative frequencies or percent frequencies of those classes on a vertical axis. The frequency (relative, percent) of each class is represented by a vertical bar whose height is equal to the frequency (relative, percent) of that class. The bars should be drawn so that consecutive classes share a common side (bars are touching)
50
What is single-value grouping in a histogram?
For single-value grouping, we use the distinct values of the observations to label the bars, with each such value centered under its bar.
51
What are other groupings in a histogram?
For limit grouping or cutpoint grouping, we use the lower class limits (or equivalently, lower class cutpoints) to label the bars.
52
What are the different types of shapes of a histogram?
- symmetric (identical on both sides of its central point) - skewed (is asymmetric. For a skewed histogram, the tail on one side is longer than the tail on the other side can be skewed to the left or the right) - uniform or rectangular (has the same frequency for each uniform rectangular histogram class)
53
What can make graphs misleading?
1. Changing the scale 2. Truncating the frequency axis 3.
54
What is a stem and leaf display?
For quantitative data, each value is divided into two portions-a stem and a leaf. The leaves for each stem are shown separately in a display. - A common method for stem diagrams is to use the last digit of each data value as the leaves and all the preceding digits as stems - If one or more of the stems has too many leaves, the stem-and-leaf display doesn't look good and may be difficult to fit on a page horizontally. - The remedy for this is to use two stems of the same value - the first stem having leaves with digits 0 through 4 and the second stem having leaves with digits 5 to 9.
55
What is a dotplot?
- All the data values from smallest to largest are listed on a horizontal scale. - A dot is placed above each data value for each occurrence of the data value -Dotplots can help us detect outliers (also called extreme values) in a data set. - Stem and Leaf and dotplots are easy ways to graph sets of data in a comparative way. - If you graph two or more sets of data with stem and leaf or dotplot, make sure the scales line up and scaling is identical so a comparison of the resulting graphs can be made
56
What is an outlier or extreme value?
Values that are very small or very large relative to the majority of the values in a data set are called outliers or extreme values.
57
What is index notation?
Use of a letter usually x with a subscript to represent the data values x without a subscript refers to the entire data set and x subscript i would be an unspecified member of the data set
58
What greek letter is the summation symbol?
sigma Σ
59
What does a measure of central tendency provide?
The center of a histogram or a frequency distribution curve
60
What is the mean of a sample?
sum of all the values divided by the number of values in the data set
61
What is the median of a sample?
Is the midpoint of the distributed values - represented by M half the numbers are smaller than M and half the numbers are larger than M
62
What is mode of a sample?
Find the frequency of each value in the data set. If no value occurs more than once, the data set has no mode. Otherwise, any value that occurs with the greatest frequency is the mode value of the dataset.
63
Which is most sensitive to outliers - mean, mode or median?
mean
64
What is deviation?
Measures the variation of the data
65
What is the range of a sample?
The difference of the largest value to the smallest value: Range = Largest value - Smallest value
66
What is standard deviation?
is the square root of the variance. Is used when you use mean
67
What is variance?
68
What is Chebyshev's Theorem?
For any number greater than 1,
1-1/k2
at least of the data values lie with standard deviations of the mean. is the number of standard deviations.
69
What is the empirical rule?
For a bell-shaped distribution, approximately
1. 68% of the observations lie within one standard deviation of the mean,
2. 95% of the observations lie within two standard deviations of the mean,
3. 99.7% of the observations lie within three standard deviations of the mean.
70
What is the quartiles of a sample?
are three numbers Q1 Q2 Q,3 and which show the locations where the data set is split into four equal parts.
71
What is the five number summary?
The five number summary of a distribution consists of the smallest observation, the first quartile, the median, the third quartile and the largest observation, written in order from smallest to largest. In symbols the five number summary is
minimum Q1 M Q3 maximum.
72
What is the interquartile range?
The interquartile range of a set of data is the difference of the third quartile to the first quartile.
IQR = Q3 - Q1
73
How is the lower limit and upper limit of a data set calculated?
L = Q1 - 1.5xIQR
U=Q3 + 1.5xIQR
74
What is a potential outlier?
An observation is suspected to be a potential outlier if it falls outside the lower and upper limits.
75
What is an adjacent value?
The smallest value from the dataset greater than the lower limit and the largest value from the dataset less than the upper limit are called adjacent values.
76
How is the mean of a population calculated?
Suppose *N* is the number of measurements taken of some quantitative variable for every individual in the population of interest. Let be the values of those measurements. Then the population mean value of these measurements is denoted by and is calculated by
77
How is standard deviation of a population calculated?
Suppose *N* is the number of measurements taken of some quantitative variable for every individual in the population of interest. Let x1, x 2, x3 ..... xn be the values of those measurements and let be the population mean (calculated as above). Then the population standard deviation of these measurements is denoted by and is defined by
78
What is a parameter?
A descriptive measure for a population (egs µ include and ð ).
79
What is a statistic?
A descriptive measure for a sample (egs x include and s).
80
What is the standardized version and standardized variable?
For a variable taken *x* from a population with mean µ and standard deviation σ the variable is called the standardized version of *x* or the standardized variable corresponding to the variable *x.*
81
What is the z-score?
For an observed value of a variable *x*, the corresponding value of the variable *z* is called the of the observation. It also called the standard score.
The z-score of an observation is the number of standard deviations the observation is away from the mean.
A negative z-score indicates an observation is less than the mean.
A positive z-score indicates an observation is greater than the mean.
Based on Chebyshev's theorem, at least 75% of the -scores of a dataset must be between -2 and 2 (within two standard deviations of the mean) and at least 89% must be between -3 and 3 (within three standard deviations of the mean).
82
Explain experiment, outcomes and sample space
An experiment is a process that, when performed, results in one and only one of many observations. These observations are called the outcomes of the experiment. The set of all outcomes for an experiment is called a sample space.
83
What graphical representation are used for sample space and the possible outcomes in that space?
(a) Venn diagram is a picture, in which all the possible outcomes of the experiment can be placed as dots within the rectangle.
(b) tree diagram in which each outcome is at the end of a branch of the tree
84
What is an event?
is a collection of outcomes for the experiment, that is, any subset of the sample space of an experiment.
An event is said to occur if and only if the outcome of the experiment is a member of the event.
85
Describe the intersection of events
Suppose *S* is a sample space with events *A* and *B* taken from *S*. If we form the set (i.e. the event) that has all the outcomes common to both the events *A* and *B*, this new event is called the intersection of the events *A* and *B*.
86
Define mutually exclusive events
Suppose *S* is a sample space with events *A* and *B* taken from The events *A* and *B* are said to be mutually exclusive if the two events can never occur together. That is, if *A* occurs, then *B* does not occur and if the *B* event occurs, then the event *A* does not occur.
87
Explain union of events
Suppose S is a sample space with events A and B taken from S. If all the outcomes of events A and B are combined into a new set, this new set is also an event called the union of events A and B.
88
Explain compliment of events
Suppose S is a sample space with an event C, Then the compliment of the event C is the event of all the outcomes from sample space. that are not in event C. The compliment of C is written (not C),
89
What is probability?
is a numerical measure of the likelihood that a specific event will occur.
90
What is probability mode?
is the mathematical description of a random phenomenon consisting of two parts: sample space S and a way of assigning probabilities to the event
91
What is the formula for the special addition rule?
Suppose we have a sample space S with events A and B which are mutually exclusive. Then
P(A or B)= P(A) + P(B).
92
What is the complementation rule?
Suppose E is an event in sample space S then
P(E) = 1 - P (not E)
which can also be written as
P(E) + P(not E) = 1
93
What is the general addition rule?
Suppose we have a sample space S with events A and B then
P (A or B) = P(A) + P(B) - P(A&B)
which can also be written as
P(A or B) + P(A&B) = P(A) + P(B)
94
How do you calculate the probability of events?
95
Define equiprobable
Suppose S is a sample space. If all the outcomes that make up sample space S have the same probability of occurrence, we say the sample is equiprobable
96
How do you calculate the probability of events in an equiprobable sample space
P(A) = number of outcomes in A/number of outcomes in S
97
What is classical probability?
In the classical probability model, it is assumed that the probabilities of outcomes for an experiment are all the same.
Earlier we had called this type of model equiprobable.
The classical model is used in many situations: flipping a coin, rolling a die, rolling dice, drawing one of several identical items from a container, bingo balls, etc.
98
Explain relative frequency in relation to probability
There are many situations where the probabilities of outcomes are not equal.
Suppose we have an experiment with some set of outcomes and we can repeat the experiment as often as we want. By repeated observations, we note the frequency of each elementary outcome.
After many such observations, we define the probability of an outcome to be the ratio of the frequency of that outcome to the total number of experiments.
That is, the probability is the relative frequency of the occurrences of outcomes to total outcomes after the experiment has been done many, many times.
P(A) = f/n
99
What is the law of large numbers?
If an experiment is repeated again and again, the probability of an event obtained from the relative frequency approaches the actual or theoretical probability
100
What is subjective probability?
is the probability assigned to an event based on the subjective judgment, experience, information, and belief
101
What are the basic properties of probability?
1: The probability of an event always lies in the range between
0 and 1. That is for any event A
0 \<=P(A) \<=1
2. The probability of an event that cannot occur is zero.
3. The probability of an event that must occur is one.
102
What is marginal probability?
is the probability of a single event without consideration of any other event. Marginal probability is also called
simple probability.
103
What is joint probability?
The probability of the intersection of two events
104
How is joint probability calculated?
Using, set analysis if we know all the outcomes from both events, we can
nd the intersection set.
105
What is conditional probability?
is the probability that an event will occur given that another event has already occurred in the same experiment. If A and B are two events, then the conditional probability of B
given A is written as
P (B|A)
and read as the probability of B given that A occurred
106
What are independent events?
Two events are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other. In other words, A and B are independent events if either
P(A|B) = P(A) or P(B|A) = P(B)
107
What is the general rule of multiplication as it relates to probabilities?
* The General Multiplication Rule is just a rearrangement of the formula for conditional probability given earlier.
* Notice that the General Multiplication Rule relates the joint probability, marginal probability and conditional probability in one formula
* Suppose we have a sample space S with events
A and B. The general multiplication rule is
P(A&B) = P(A) P(B|A)
* NOTE: Do not use the General Multiplication Rule as a way to find joint probabilities unless the conditional probabilities are already known or if the experiment involves multiple stages.
* In the case of multiple stages, we can figure out the conditional probability by set analysis and then use the General Multiplication Rule to find the joint probability.
108
What are two-way tables?
* In some probability problems, a set of individuals is considered and two different pieces of information are extracted from each individual.
* This creates two distinct sample spaces on the same set of individuals.
* We wish to consider how to analyze probability problems with two sample spaces.
109
What is the basic counting rule (BCR)?
* If there is a situation in which we have several sets of items from which we choose one item each, the basic counting rule calculates all the possible ways these items can be selected.
* Suppose there are actions *r* to be performed in a definite order. Further suppose that there are *m1* ways to perform
the first action, *m2 *ways to perform the second action and so on, up to *mr *ways to perform the last action. *m1* , *m2, m3* . .*mr* , are all positive integers (they may have different values and they can possibly equal 1). Then the number of possible ways the *r* actions can be performed is the product *m1* x *m2, x m3* x . . . x *mr*
110
What are factorials?
The factorial of a positive integer *n* is written *n!* (read it as "n
factorial") and is given by
n! = n x (n-1) x (n-2) x ... x3 x 2 x 1
if n \>= 1
Notice 1! = 1, The factorial of 0 is also defined given by 0! = 1
* The factorial obeys this rule if *n* is an integer greater than 1 (this is called a recursive formula - it is often
used to define the factorial).
111
What is the permutation rule?
* The number of possible permutations *r* of objects from a collection of *m* objects is given by the formula in the image
* Used where there is a set from which we wish to select a subset of a certain size and for which the order in the subset matters.
* Each choice for the ordered subset is called a permutation and we wish to calculate the number of possible permutations.
112
What is the combination rule?
* The third counting rule is also concerned with the number of ways of selecting items from a group of *m* distinct objects*,* but this time without regard to the order the items are listed.
* This is similar to the permutations except we do not have to count the number of ways of sorting the items: thus number of combinations is less
than the number of permutations
* The number of possible combinations of *r* objects
from a collection of *m* objects is given by the formula in the image
113
What is another way to write mCr in the combination rule formula?
114
115
What is a random variable?
is a quantitative variable whose value
* in some cases a random variable may be the outcomes from a sample space
* in other cases the random variable involves an additional calculation using the outcomes from the sample space
116
What is a discrete random variable?
A random variable that assumes countable values
* in this class countable values mean a finite number of values
* the upper case letter represent the value of an outcome of a random variable
117
What is discrete probability distribution of a discrete random variable?
is a table which lists all the possible values that the
random variable can assume and their corresponding probabilities
118
What is a discrete probability histogram?
is a graph of the probability distribution that displays the possible values of the discrete random variable on the horizontal axis and the probabilities of those values on the vertical axis. The probability of each value is represented by a vertical bar
whose height equals the probability.
119
What is the interpretation of a probability distribution?
In a large number of independent observations of a random variable X, the proportion of times each possible value occurs will approximate the probability distribution of X; or, equivalently,
the proportion histogram will approximate the probability histogram for X
120
Using mathematical notation for discrete probability how do you express less than 8?
P(X\<=7)
121
Using mathematical notation for discrete probability how do you express no more than 7?
P(X\<=7)
122
Using mathematical notation for discrete probability how do you express at most 7?
P(X\<=7)
123
Using mathematical notation for discrete probability how do you express 7 at least?
P(X\<=7)
124
Using mathematical notation for discrete probability how do you express more than 6 or at least 7?
P(X=\>7) =1 - P(X\<=6)
125
Using mathematical notation for discrete probability how do you express no less than 7 or 7 or more?
P(X\>=7) = 1 - P(X\<=6)
126
Using mathematical notation for discrete probability how do you express between 5 and 9 inclusively?
P(5\<=X\<=9) = P(X\<=9) - P(X\<=4)
127
Using mathematical notation for discrete probability how do you express more than 5 but less than 9, between 5 and 9 not inclusively?
P(6\<=X\<=8) = P(X\<=8) - P(X\<=5)
128
What are Bernoulli trials?
Repeated trials of an experiment if the following three conditions are satisfied:
* the experiment (each trial), has two possible outcomes, denoted generically *s* for success and *f* for failure
* the trials are independent
* the probability of success, called the success probability, and denoted by *p*, remains the same from trial to trial
129
What is binomial distribution?
is the probability distribution for the number of successes in a sequence of Bernoulli trials
Every binomial distribution has three numbers
* *n* which is the number of trials and must be a positive integer
* *p* which is the probability of a success on a single trial P(*s*)
* 1 - *p* which is the probability of a failure of a single trial P(*f*)
* The random variable for the binomial distribution is the number of successes *x* out of the possible outcomes given *n* and *p*
* *x* must be a nonnegative integer up to and including *n*
130
What is a continuous variable?
one in which there are an infinite number of possible values
e.g. height, time to take a test, weight etc.
131
What is a polygon plot?
Lines connecting the centre of each rectangle on a histogram plot
132
What is a density curve?
is a curve that:
* is always on or above the horizontal axis and has exactly area equal to one between the graph and the horizontal axis.
* A density curve is also called a probability distribution curve or a probability density function, since is used to find probabilities in the distribution of a continuous variable.
Two Characteristics of a Density Curve
1. The probability that assumes a value in any interval lies in the range 0 to 1.
2. The total probability of all the (mutually exclusive) intervals within *x* which can assume a value is 1.0.
The probability is measured as the area under the density curve
133
What is normal distribution?
a normal distribution is described by a normal density curve which when plotted gives a bell shaped curve:
1. The total area under the curve is 1.0.
2. The curve is symmetric about the mean.
3. The standard deviation of a Normal Distribution is the distance from the center to the change-of-curvature points on either side.
4. The two tails of the curve extend indefinitely.
The mean, µ, and the standard deviation, ð, are called the parameters of the normal distribution. Given the values of these two parameters, we can find the area under
a normal distribution curve for any interval
134
What is the 68-95-99.7 rule?
There are many possibilities for µ and ð, but all normal density curves satisfy some common properties
In a normal distribution with mean µ and standard
deviation ð
* Approximately 68% of the observations fall within 1ð of the mean µ
* Approximately 95% of the observations fall within 2ð of the mean µ
* Approximately 99.7% of the observations fall within 3ð of the mean µ
135
What is a z-score?
a standardized value
* tells us how many standard deviations the original observation falls away from the mean, and in which direction
* Observations larger than the mean are positive when standardized; observations smaller than the mean are negative; the mean becomes zero as a z-score
* Since all normal distributions have the same properties, it definitely most convenient to work with just one fixed normal distribution.
* The result of standardizing creates that one xed normal distribution.
136
137
What is standard normal distribution?
*N* (0,1) is called the standard normal distribution if the variable *x* has any normal distribution *N*(µ, ð) with mean µ and standard deviation ð then the standard variable z = x - µ/ð has the standard normal distribution
To find the area in the standard normal distribution, a Standard
Normal Table is available
138
What is a parameter?
is a number that describes the population. In statistical practice, the values of a parameter is not known because we cannot examine the entire population.
139
What is a statistic?
is a number that can be computed from the sample data without making any use of any unknown parameter. In practice, we often use a statistic to estimate value of a parameter
* Statistics come from samples.
* Parameter come from populations.
To keep this straight we use Greek letters to represent parameters of the population and Latin letters to represent statistics of samples.
e.g. For the standard deviation, the population is represented by ð while the standard deviation for the sample is s
140
What is sampling distribution?
is the distribution of values taken by the statistic in all possible samples of the same size taken from the same population.
* The sampling distribution is the ideal pattern that would emerge if we looked at all possible samples of size *n* from out population.
141
What is a sampling error?
is the error that results from using
a sample statistic to estimate a population parameter
142
What is the shape of population distribution?
* The shape depends on the population distribution
* if the distribution of the population is normal, the distribution of the mean will also be normal
