Advanced Statistics.pptx

ADVANCED
STATISTICS
RENATO N. PACPAKIN

ADVANCED STATISTICS
What is the definition of advanced statistics?
A) Advanced statistics refers to the use of complex mathematical techniques
to analyze and interpret data. B) Advanced statistics involves collecting and
organizing data for research purposes. C) Advanced statistics focuses on
basic statistical concepts such as mean, median, and mode. D) Advanced
statistics is a term used to describe data visualization techniques.
Answer: A) Advanced statistics refers to the use of complex mathematical
techniques to analyze and interpret data.
Explanation: Advanced statistics goes beyond basic statistical concepts
and involves the application of sophisticated mathematical techniques to
analyze data. This can include techniques such as regression analysis,
multivariate analysis, time series analysis, and more.

ADVANCED STATISTICS
Which of the following is an example of advanced statistics?
A) Calculating the average of a dataset. B) Conducting a t-test to
compare the means of two groups. C) Counting the frequency of
different categories in a dataset. D) Plotting a bar graph to
visualize the distribution of a variable.
Answer: B) Conducting a t-test to compare the means of two
groups.
Explanation: A t-test is an example of advanced statistics as it
involves comparing the means of two groups to determine if they
are significantly different. It requires a deeper understanding of

ADVANCED STATISTICS
What is the purpose of advanced statistics?
A) To simplify data analysis for non-experts. B) To provide basic
statistical summaries of a dataset. C) To identify complex
patterns and relationships in data. D) To collect and organize
data for research purposes.
Answer: C) To identify complex patterns and relationships in
data.
Explanation: Advanced statistics aims to uncover intricate
patterns and relationships that may not be easily detectable
using basic statistical techniques. It helps researchers and
analysts gain a deeper understanding of the data and make more

ADVANCED STATISTICS
Which of the following is not a characteristic of advanced
statistics?
A) Involves the use of specialized software or programming
languages. B) Requires a solid foundation in basic statistical
concepts. C) Relies solely on descriptive statistics. D) Utilizes
advanced mathematical techniques.
Answer: C) Relies solely on descriptive statistics.
Explanation: Advanced statistics goes beyond descriptive statistics, which
involve summarizing and describing data using measures such as mean,
median, and mode. Advanced statistics employs more complex techniques,
including inferential statistics, modeling, and predictive analytics.

ADVANCED STATISTICS
What is one example of a statistical technique used in advanced
statistics?
A) Chi-square test. B) Calculating the range of a dataset. C)
Creating a frequency distribution table. D) Computing the
standard deviation of a variable.
Answer: A) Chi-square test.
Explanation: The chi-square test is a statistical technique used in advanced
statistics. It is used to determine if there is a significant association between
two categorical variables in a dataset. The test helps analyze whether the
observed frequencies differ significantly from the expected frequencies,
indicating a relationship between the variables.

Unit 1: Introduction to
Statistical Analysis

Distinguish between quantitative and
categorical variables
1. What type of variable is "Age"?
A) Quantitative B) Categorical
2. What type of variable is "Gender"?
3. What type of variable is "Height in centimeters"?
4. What type of variable is "Favorite color"?
5. What type of variable is "Income in dollars"?

Answers:
1.A) Quantitative
2.B) Categorical
3.A) Quantitative
4.B) Categorical
5.A) Quantitative

Explanation:
1."Age" represents a numerical value, making it a
quantitative variable.
2."Gender" represents categories (e.g., male, female),
making it a categorical variable.
3."Height in centimeters" represents a numerical
measurement, making it a quantitative variable.
4."Favorite color" represents categories (e.g., red, blue,
green), making it a categorical variable.
5."Income in dollars" represents a numerical value, making it
a quantitative variable.

Describe the difference between a population and a sample
and be able to distinguish between a parameter and a
statistic.
1. A group of all individuals, objects, or events of interest is
known as:
A) Population B) Sample
2. A subset of individuals, objects, or events selected from a
population is known as: A) Population B) Sample
3. A numerical characteristic of a population is called a:
A) Parameter B) Statistic
4. A numerical characteristic of a sample is called a:
5. Which one represents the entire population?

statistic.
Answers:
1.A) Population
2.B) Sample
3.A) Parameter
4.B) Statistic
5.A) Parameter

statistic.
Explanation:
1. A population refers to the entire group of individuals, objects, or
events of interest.
2. A sample is a subset of the population, selected to represent the
larger group.
3. A parameter is a numerical characteristic or measurement that
describes a population.
4. A statistic is a numerical characteristic or measurement that
describes a sample.
5. A parameter represents the entire population, whereas a statistic
represents the sample and is used to estimate parameters or draw
inferences about the population.

Given a type of measurement, identify the correct level of
measurement: nominal, ordinal, interval, or ratio.
1. Categories of favorite music genres (e.g., rock, pop, jazz): A)
Nominal B) Ordinal C) Interval D) Ratio
2. Ratings of movie reviews on a scale of 1 to 5 stars: A) Nominal
B) Ordinal C) Interval D) Ratio
3. Temperatures recorded in Celsius or Fahrenheit: A) Nominal
4. Rank order of students based on their GPA: A) Nominal B)
Ordinal C) Interval D) Ratio
5. Measurement of weight in kilograms or pounds: A) Nominal

Answers:
1. A) Nominal
2. B) Ordinal
3. C) Interval
4. B) Ordinal
5. D) Ratio

Explanation:
1.Categories of favorite music genres represent qualitative data
without any inherent order, so it is measured at the nominal level.
2.Ratings of movie reviews have an inherent order (1 star, 2 stars,
etc.), making it an ordinal measurement.
3.Temperatures recorded in Celsius or Fahrenheit have equal intervals
between values but do not have a meaningful zero point, making it
an interval measurement.
4.Rank order of students based on GPA has an order but the
differences between ranks may not be equal, making it an ordinal
measurement.
5.Measurement of weight in kilograms or pounds has equal intervals
and a meaningful zero point (absence of weight), making it a ratio
measurement.

Calculate the mean, median, and mode for a set of data,
and compare and contrast these measures of center.
1.Find the mean, median, and mode for the following dataset: 2, 4, 6, 6, 8, 10.
A) Mean: 6, Median: 6, Mode: 6 B) Mean: 6, Median: 6, Mode: None C) Mean:
6.33, Median: 6, Mode: 6 D) Mean: 6.33, Median: 6.5, Mode: 6
2.Find the mean, median, and mode for the following dataset: 3, 4, 5, 6, 7, 8.
A) Mean: 5.5, Median: 5.5, Mode: None B) Mean: 5.5, Median: 6, Mode: None C)
Mean: 5.5, Median: 6, Mode: 5 D) Mean: 5.5, Median: 5.5, Mode: 5
3.Find the mean, median, and mode for the following dataset: 1, 2, 3, 4, 5, 6, 7.
A) Mean: 4, Median: 4, Mode: None B) Mean: 4, Median: 4, Mode: 1 C) Mean: 4,
Median: 4, Mode: 4 D) Mean: 4, Median: 4.5, Mode: 4
4.Find the mean, median, and mode for the following dataset: 2, 4, 4, 6, 6, 8, 10.
A) Mean: 6, Median: 4, Mode: 4 B) Mean: 5.71, Median: 4, Mode: 4 C) Mean: 6,
Median: 6, Mode: 4 D) Mean: 5.71, Median: 6, Mode: 6
5.Find the mean, median, and mode for the following dataset: 2, 4, 6, 8, 10.
A) Mean: 6, Median: 6, Mode: None B) Mean: 6, Median: 6, Mode: 2 C) Mean: 6,
Median: 6, Mode: 6 D) Mean: 6, Median: 7, Mode: None

Answers:
1. A) Mean: 6, Median: 6, Mode: 6
2. D) Mean: 5.5, Median: 5.5, Mode: 5
3. C) Mean: 4, Median: 4, Mode: 4
4. D) Mean: 5.71, Median: 6, Mode: 6
5. B) Mean: 6, Median: 6, Mode: 2

Explanation:
1.The mean is the sum of all values divided by the number of values, which
in this case is (2+4+6+6+8+10)/6 = 6. The median is the middle value,
which is 6. The mode is the most frequently occurring value, which is 6.
2.The mean is (3+4+5+6+7+8)/6 = 5.5. The median is the middle value,
which is 5.5. There is no mode since no value occurs more than once.
3.The mean is (1+2+3+4+5+6+7)/7 = 4. The median is the middle value,
4.The mean is (2+4+4+6+6+8+10)/7 ≈ 5.71. The median is the middle value,
5.The mean is (2+4+6+8+10)/5 = 6. The median is the middle value, which
is 6. There is no mode since no value occurs more than once.

Comparison:
• Mean: It is the average value calculated by summing all values and
dividing by the total number of values. It is sensitive to extreme values.
• Median: It is the middle value when the data is arranged in ascending or
descending order. It is less affected by extreme values.
• Mode: It is the value that occurs most frequently in the dataset. There can
be one mode, more than one mode (multimodal), or no mode (no value
repeats).
• These measures of center provide different insights into the dataset and
are useful in different scenarios based on the nature of the data and the
specific question being addressed.

Identify the symbols and know the formulas for
sample and population means.
1.What is the symbol used to represent the population mean?
A) x
̄ (x-bar) B) μ (mu) C) σ (sigma) D) N (population size)
2.What is the symbol used to represent the sample mean?
A) x
̄ (x-bar) B) μ (mu) C) σ (sigma) D) N (sample size)
3.What is the formula for calculating the population mean?
A) μ = (Σx) / N B) μ = (Σx) / n C) μ = (Σx) / (N - 1) D) μ = (Σx) /
n^2
4.What is the formula for calculating the sample mean?
A) x
̄ = (Σx) / N B) x
̄ = (Σx) / n C) x
̄ = (Σx) / (N - 1) D) x
̄ = (Σx) /
n^2
5.Which symbol represents the sum of all individual values in a
dataset? A) x
̄ (x-bar) B) Σ (sigma) C) μ (mu) D) N (population
size)

Answers:
1.B) μ (mu)
2.A) x
̄ (x-bar)
3.A) μ = (Σx) / N
4.B) x
̄ = (Σx) / n
5.B) Σ (sigma)

Explanation:
1.The symbol μ (mu) represents the population mean.
2.The symbol x
̄ (x-bar) represents the sample mean.
3.The formula for calculating the population mean is μ = (Σx) / N,
where Σx represents the sum of all individual values in the
population and N represents the population size.
4.The formula for calculating the sample mean is x
̄ = (Σx) / n, where
Σx represents the sum of all individual values in the sample and n
represents the sample size.
5.The symbol Σ (sigma) represents the sum of all individual values in a
dataset. It is used to calculate the sum of values in both the
population and the sample.

Calculate the midrange, weighted mean,
percentiles, and quartiles for a data set.
1.What is the formula for calculating the midrange of a data set? A) Midrange =
(Maximum Value + Minimum Value) / 2 B) Midrange = (Sum of all Values) /
Number of Values C) Midrange = Median of the Data Set D) Midrange = Mode of
the Data Set
2.How do you calculate the weighted mean of a data set? A) Summing all the
values and dividing by the number of values B) Summing the product of each
value and its corresponding weight, and dividing by the sum of the weights C)
Taking the middle value when the data set is arranged in ascending order D)
Calculating the average of the smallest and largest value in the data set
3.What are percentiles used to measure in a data set? A) Measure of center B)
Measure of spread or dispersion C) Measure of relative standing or position D)
Measure of the shape of the distribution
4.What is the range of percentiles commonly used to calculate quartiles? A) 0 to 25
B) 0 to 50 C) 0 to 75 D) 25 to 75
5.How many quartiles are there in a data set? A) 1 B) 2 C) 3 D) 4

Answers:
1. A) Midrange = (Maximum Value + Minimum Value) / 2
2. B) Summing the product of each value and its corresponding
weight, and dividing by the sum of the weights
3. C) Measure of relative standing or position
4. D) 25 to 75
5. C) 3

Explanation:
1.The midrange is calculated by taking the average of the maximum
value and the minimum value in the data set.
2.The weighted mean is calculated by summing the product of each
value and its corresponding weight, and then dividing by the sum of
the weights.
3.Percentiles are used to measure the relative standing or position of
a particular value in a data set.
4.Quartiles divide a data set into four equal parts, and they are
commonly calculated within the range of 25 to 75 percentiles.
5.There are three quartiles in a data set: the lower quartile (Q1), the
median (Q2), and the upper quartile (Q3).

Calculate the range, the interquartile range, the standard deviation,
and the variance for a population and a sample, and know the
symbols, formulas, and uses of these measures of spread
1.What is the formula for calculating the range of a dataset? A) Range =
Maximum Value - Minimum Value B) Range = Sum of all Values / Number
of Values C) Range = Upper Quartile - Lower Quartile D) Range = Median
- Mean
2.What is the symbol used to represent the population standard deviation?
A) σ (sigma) B) s (lowercase s) C) μ (mu) D) R (uppercase R)
3.What is the formula for calculating the sample standard deviation? A) σ =
√(Σ(x - μ)² / N) B) σ = √(Σ(x - μ)² / (N - 1)) C) s = √(Σ(x - μ)² / N) D) s =
√(Σ(x - μ)² / (N - 1))
4.What is the formula for calculating the population variance? A) σ² = Σ(x -
μ)² / N B) σ² = Σ(x - μ)² / (N - 1) C) s² = Σ(x - μ)² / N D) s² = Σ(x - μ)² / (N -
1)
5.What is the main use of the interquartile range? A) Measure the spread of
data around the mean B) Identify outliers in a dataset C) Determine the
central tendency of a dataset D) Measure the relative position of a value in

Answers:
1. A) Range = Maximum Value - Minimum Value
2. A) σ (sigma)
3. D) s = √(Σ(x - μ)² / (N - 1))
4. A) σ² = Σ(x - μ)² / N
5. B) Identify outliers in a dataset

Explanation:
1.The range is calculated by subtracting the minimum value from the
maximum value in the dataset.
2.The symbol σ (sigma) is used to represent the population standard
deviation.
3.The formula for calculating the sample standard deviation uses the
symbol s (lowercase s) and includes dividing by (N - 1) to provide an
unbiased estimate.
4.The population variance is calculated by squaring the standard
deviation formula and dividing by the population size (N).
5.The interquartile range (IQR) is used to identify outliers in a dataset
and measures the spread of the middle 50% of the data, specifically
the difference between the third quartile (Q3) and the first quartile
(Q1).

Unit 2: Visualizations
of Data

Read and make frequency tables for a data set.
1. What is a frequency table? A) A table that shows the frequency of each unique value in a data set B) A
table that lists the data values in ascending order C) A table that displays the cumulative frequencies of a
data set D) A table that shows the range of values in a data set
2. Which of the following is a correct frequency table for the data set: 2, 2, 3, 5, 5, 5, 7, 8, 8, 8?
A) Value Frequency 2 2 3 1 5 3 7 1 8 3
B) Value Frequency 2 3 3 1 5 3 7 1 8 3
C) Value Frequency 2 2 3 2 5 3 7 1 8 4
D) Value Frequency 2 2 3 1 5 3 7 2 8 3
3. How is the frequency calculated in a frequency table? A) By counting the total number of values in the
data set B) By dividing the sum of all values by the number of values C) By finding the difference
between the minimum and maximum values D) By counting the occurrences of each unique value in the
data set
4. What is a cumulative frequency table? A) A table that displays the cumulative sum of the frequencies up
to a certain point in the data set B) A table that shows the range of values in a data set C) A table that
lists the data values in descending order D) A table that shows the relative frequencies of each unique
value in a data set
5. How can a frequency table be useful in analyzing data? A) It helps to calculate the mean and median of a
data set B) It provides a visual representation of the distribution of values in a data set C) It identifies
outliers in a data set D) It determines the standard deviation of a data set

Answers:
1. A) A table that shows the frequency of each unique value in a
data set
2. A) Value Frequency 2 2 3 1 5 3 7 1 8 3
3. D) By counting the occurrences of each unique value in the
data set
4. A) A table that displays the cumulative sum of the frequencies
up to a certain point in the data set
5. B) It provides a visual representation of the distribution of
values in a data set

Explanation:
1. A frequency table is a table that displays the frequency or count of each
unique value in a data set.
2. The correct frequency table for the given data set is option A. It correctly
lists each unique value with its corresponding frequency.
3. The frequency in a frequency table is calculated by counting the
occurrences or repetitions of each unique value in the data set.
4. A cumulative frequency table shows the running or cumulative sum of the
frequencies up to a particular point in the data set.
5. A frequency table is useful in analyzing data as it provides a visual
summary of the distribution of values, allowing us to identify patterns,
trends, and the most common or frequent values in the data set. It does
not directly calculate measures like mean, median, standard deviation, or
identify outliers.

Identify and translate data sets to and from a histogram, a relative frequency
histogram, a frequency polygon, an ogive, a bar chart, a pie graph, a dot plot, a stem-
and-leaf plot, a scatterplot, and a box-and-whisker plot.
1.Which graphical representation is best suited for displaying the distribution of
continuous numerical data? A) Histogram B) Bar chart C) Pie graph D) Dot plot
2.A histogram displays data using: A) Bars of equal width with heights representing
the frequency or relative frequency of the data B) Segments of a circle
representing proportions of a whole C) Dots placed along a number line D) Points
plotted in a coordinate system
3.What does a relative frequency histogram represent? A) The count or frequency
of each data value B) The proportion or percentage of each data value relative to
the total number of data points C) The pattern of data points plotted in a
coordinate system D) The distribution of data across different categories
4.Which graphical representation is best suited for displaying cumulative
frequencies or cumulative relative frequencies? A) Ogive B) Frequency polygon
C) Scatterplot D) Stem-and-leaf plot
5.A bar chart uses: A) Rectangular bars of equal width with heights representing
the frequency or relative frequency of the data B) Segments of a circle
representing proportions of a whole C) Dots placed along a number line D) Points
plotted in a coordinate system

histogram, a frequency polygon, an ogive, a bar chart, a pie graph, a dot plot, a stem-
and-leaf plot, a scatterplot, and a box-and-whisker plot.
6. A pie graph represents: A) The count or frequency of each data value B) The proportion or
percentage of each data value relative to the total number of data points C) The pattern of data points
plotted in a coordinate system D) The distribution of data across different categories
7. A dot plot uses: A) Dots placed along a number line to represent individual data points B) Bars of
equal width with heights representing the frequency or relative frequency of the data C) Points plotted
in a coordinate system to show relationships between two variables D) Segments of a circle
representing proportions of a whole
8. Which graphical representation is best suited for displaying both the distribution and individual data
points of a numerical dataset? A) Scatterplot B) Box-and-whisker plot C) Histogram D) Frequency
polygon
9. A stem-and-leaf plot organizes data by: A) Separating the data into categories and displaying the
counts or frequencies for each category B) Showing the distribution of data using bars of equal width
C) Plotting points in a coordinate system to display relationships between two variables D) Grouping
data by stems and leaves to show the individual values
10. A box-and-whisker plot represents: A) The count or frequency of each data value B) The
proportion or percentage of each data value relative to the total number of data points C) The
distribution and key summary statistics, such as quartiles and outliers, of a numerical dataset D) The
relationship between two variables by plotting points in a coordinate system

histogram, a frequency polygon, an ogive, a bar chart, a pie graph, a dot plot,
a stem-and-leaf plot, a scatterplot, and a box-and-whisker plot.
Answers:
1.A) Histogram
2.A) Bars of equal width with heights representing the frequency or relative frequency of the
data
3.B) The proportion or percentage of each data value relative to the total number of data
points
4.A) Ogive
5.A) Rectangular bars of equal width with heights representing the frequency or relative
frequency of the data
6.B) The proportion or percentage of each data value relative to the total number of data
points
7.A) Dots placed along a number line to represent individual data points
8.A) Scatterplot
9.D) Grouping data by stems and leaves to show the individual values
10.C) The distribution and key summary statistics, such as quartiles and outliers, of a
numerical dataset

histogram, a frequency polygon, an ogive, a bar chart, a pie graph, a dot plot,
a stem-and-leaf plot, a scatterplot, and a box-and-whisker plot.
Explanation:
1. Histograms are best suited for displaying the distribution of continuous numerical data using bars of equal
width.
2. A histogram uses bars of equal width, with heights representing the frequency or relative frequency of the
data in each interval.
3. A relative frequency histogram represents the proportion or percentage of each data value relative to the
total number of data points.
4. Ogives, also known as cumulative frequency graphs, display cumulative frequencies or cumulative
relative frequencies.
5. A bar chart uses rectangular bars of equal width, with heights representing the frequency or relative
frequency of the data.
6. A pie graph represents proportions or percentages of each data value relative to the total number of data
points.
7. A dot plot uses dots placed along a number line to represent individual data points.
8. Scatterplots are used to display the relationship between two numerical variables, showing individual
data points.
9. A stem-and-leaf plot organizes data by grouping values into stems and leaves to show individual values.
10.A box-and-whisker plot represents the distribution and key summary statistics, such as quartiles and
outliers, of a numerical dataset.

Identify graph distribution shapes as skewed or symmetric,
and describe the basic implication of these shapes.
1.A histogram has a longer tail on the right side and the majority of the data is
concentrated on the left side. How would you classify this distribution shape? A)
Skewed right (positively skewed) B) Skewed left (negatively skewed) C)
Symmetric D) Uniform
2.A frequency polygon has a bell-shaped curve that is symmetrical. How would you
classify this distribution shape? A) Skewed right (positively skewed) B) Skewed
left (negatively skewed) C) Symmetric D) Uniform
3.A scatterplot shows data points forming a diagonal line that starts from the bottom
left and goes towards the top right. How would you classify this distribution
shape? A) Skewed right (positively skewed) B) Skewed left (negatively skewed)
C) Symmetric D) Uniform
4.A box-and-whisker plot has a box that is evenly positioned between the lower and
upper quartiles, with whiskers extending equally in both directions. How would
you classify this distribution shape? A) Skewed right (positively skewed) B)
Skewed left (negatively skewed) C) Symmetric D) Uniform
5.A bar chart displays bars of equal width where the heights represent the
frequencies of different categories, and the bars are roughly symmetrical. How
would you classify this distribution shape? A) Skewed right (positively skewed) B)
Skewed left (negatively skewed) C) Symmetric D) Uniform

Answers:
1. A) Skewed right (positively skewed)
2. C) Symmetric
3. C) Symmetric
4. C) Symmetric
5. C) Symmetric

Explanation:
1.When a histogram has a longer tail on the right side and the majority of the data is
concentrated on the left side, it is classified as skewed right (positively skewed). This
implies that the distribution is asymmetrical, with a tail extending towards the higher
values.
2.When a frequency polygon has a bell-shaped curve that is symmetrical, it is classified as
symmetric. This implies that the data is evenly distributed around the center, with no
skewness towards one side or the other.
3.When a scatterplot shows data points forming a diagonal line from the bottom left to the
top right, it is classified as symmetric. This implies a positive linear relationship between
the variables, without any skewness.
4.When a box-and-whisker plot has a box that is evenly positioned between the lower and
upper quartiles and whiskers extending equally in both directions, it is classified as
symmetric. This implies a balanced distribution without any skewness.
5.When a bar chart displays bars of equal width, where the heights represent the
frequencies of different categories, and the bars are roughly symmetrical, it is classified
as symmetric. This implies an even distribution across the categories without any
skewness.

Compare distributions of univariate data
(shape, center, spread, and outliers).
1.Which statistical measure is used to describe the center of a
distribution? A) Mean B) Standard deviation C) Variance D)
Skewness
2.A distribution with a longer tail on the right side and the majority of
data concentrated on the left side is known as: A) Positively skewed
B) Negatively skewed C) Symmetric D) Uniform
3.Which measure of spread is affected by extreme outliers in a
dataset? A) Range B) Standard deviation C) Interquartile range D)
Variance
4.What is the best measure of center to use when a dataset has
extreme outliers? A) Median B) Mode C) Mean D) Standard
deviation
5.How does the presence of outliers affect the shape of a distribution?
A) Outliers do not affect the shape of the distribution B) Outliers
cause the distribution to become positively skewed C) Outliers cause
the distribution to become negatively skewed D) Outliers may distort

Answers:
1. A) Mean
2. A) Positively skewed
3. A) Range
4. A) Median
5. D) Outliers may distort the shape of the distribution

Explanation:
1.The measure used to describe the center of a distribution is the
mean.
2.A distribution with a longer tail on the right side and the majority of
data concentrated on the left side is positively skewed.
3.The measure of spread affected by extreme outliers in a dataset is
the range, as it is influenced by the extreme values.
4.The best measure of center to use when a dataset has extreme
outliers is the median, as it is less affected by extreme values than
the mean.
5.The presence of outliers can distort the shape of a distribution.
Outliers can pull the tail of the distribution in the direction of the
outlier, causing the shape to appear skewed or asymmetrical.

Calculate the values of the five-number
summary.
1.Given the dataset: 7, 8, 10, 12, 15, 18, 20, 24, 28, 35, what is
the value of the minimum (smallest value)? A) 7 B) 8 C) 10 D)
35
the value of the lower quartile (Q1)? A) 7 B) 10 C) 15 D) 18
the value of the median (Q2)? A) 10 B) 15 C) 18 D) 20
the value of the upper quartile (Q3)? A) 15 B) 18 C) 24 D) 28
the value of the maximum (largest value)? A) 15 B) 18 C) 24 D)
35

summary.
Answers:
1.A) 7
2.D) 18
3.D) 20
4.C) 24
5.D) 35

summary.
Explanation:
1.The minimum value is the smallest value in the dataset, which is 7.
2.The lower quartile (Q1) is the median of the lower half of the dataset,
which is 18.
3.The median (Q2) is the middle value of the dataset when it is arranged in
ascending order, which is 20.
4.The upper quartile (Q3) is the median of the upper half of the dataset,
which is 24.
5.The maximum value is the largest value in the dataset, which is 35.
• The five-number summary includes the minimum, lower quartile (Q1),
median (Q2), upper quartile (Q3), and maximum. These values provide
information about the central tendency, spread, and distribution of the
dataset.

Describe the effects of changing units on
summary measures.
1.If you convert a dataset from pounds to kilograms, how will it affect the mean? A)
The mean will increase. B) The mean will decrease. C) The mean will remain the
same. D) The mean will become zero.
2.If you convert a dataset from inches to centimeters, how will it affect the range?
A) The range will increase. B) The range will decrease. C) The range will remain
the same. D) The range will become zero.
3.If you convert a dataset from Fahrenheit to Celsius, how will it affect the standard
deviation? A) The standard deviation will increase. B) The standard deviation will
decrease. C) The standard deviation will remain the same. D) The standard
deviation will become zero.
4.If you multiply a dataset by a positive constant, how will it affect the variance? A)
The variance will increase. B) The variance will decrease. C) The variance will
remain the same. D) The variance will become zero.
5.If you add a constant to each value in a dataset, how will it affect the quartiles? A)
The quartiles will increase by the constant. B) The quartiles will decrease by the
constant. C) The quartiles will remain the same. D) The quartiles will become
zero.

summary measures.
Answers:
1.C) The mean will remain the same.
2.C) The range will remain the same.
3.C) The standard deviation will remain the same.
4.A) The variance will increase.
5.A) The quartiles will increase by the constant.

summary measures.
Explanation:
1.Changing units of measurement does not affect the mean because it is a measure of
central tendency and is not influenced by changes in units.
2.Changing units of measurement does not affect the range because it represents the
difference between the maximum and minimum values and is independent of units.
3.Changing units of measurement does not affect the standard deviation because it is a
measure of dispersion that considers the differences between data points and is not
affected by changes in units.
4.Multiplying a dataset by a positive constant will result in the variance being multiplied by
the square of that constant. Therefore, the variance will increase.
5.Adding a constant to each value in a dataset will shift the entire distribution by that
constant. As a result, the quartiles will increase by the constant value.
• Changing units of measurement only affects summary measures that involve scaling or
multiplying/dividing the data, such as variance and quartiles. Measures like mean, range,
and standard deviation are not affected by changes in units as they are relative measures
that rely on the differences between data points.

List simple events and sample spaces.
1.Consider the roll of a fair six-sided die. List one simple event from the sample
space. A) Rolling an odd number B) Rolling a multiple of 3 C) Rolling a
number greater than 6 D) Rolling a negative number
2.A bag contains red and blue marbles. List one simple event from the sample
space of selecting a marble from the bag. A) Selecting a red marble B) Selecting
a green marble C) Selecting a square-shaped marble D) Selecting two marbles
at once
3.A deck of playing cards is shuffled. List one simple event from the sample space
of drawing a card from the deck. A) Drawing a heart B) Drawing a diamond and a
club simultaneously C) Drawing a card with a face value of 10 D) Drawing two
cards at once
4.A coin is flipped three times. List one simple event from the sample space. A)
Getting heads on all three flips B) Getting tails on exactly two flips C) Getting both
heads and tails on the first flip D) Getting four heads in three flips
5.A box contains three red balls and two blue balls. List one simple event from the
sample space of selecting two balls from the box without replacement. A)

Answers:
1.A) Rolling an odd number
2.A) Selecting a red marble
3.A) Drawing a heart
4.B) Getting tails on exactly two flips
5.A) Selecting two red balls

Explanation:
1.A simple event in the sample space of rolling a fair six-sided die is
rolling an odd number (1, 3, or 5).
2.In the sample space of selecting a marble from a bag containing red
and blue marbles, a simple event is selecting a red marble.
3.A simple event in the sample space of drawing a card from a
shuffled deck is drawing a heart.
4.A simple event in the sample space of flipping a coin three times is
getting tails on exactly two flips.
5.In the sample space of selecting two balls from a box containing
three red and two blue balls without replacement, a simple event is
selecting two red balls.

Know the symbols and operations of unions
and intersections of sets.
1.What symbol is used to represent the union of two sets? A) ∪
(union) B) ∩ (intersection) C) ⊂ (subset) D) ∈ (element of)
2.What symbol is used to represent the intersection of two sets? A) ∪
(union) B) ∩ (intersection) C) ⊂ (subset) D) ∈ (element of)
3.Which operation includes all elements that are present in either of
the two sets being combined? A) Union B) Intersection C) Subset D)
Complement
4.Which operation includes only the elements that are common to
both sets being combined? A) Union B) Intersection C) Subset D)
Complement
5.Given two sets A = {1, 2, 3} and B = {3, 4, 5}, what is the result of A
∪ B (union)? A) {1, 2, 3} B) {1, 2, 3, 4, 5} C) {1, 4, 5} D) {2, 3, 4}

Answers:
1.A) ∪ (union)
2.B) ∩ (intersection)
3.A) Union
4.B) Intersection
5.B) {1, 2, 3, 4, 5}

Explanation:
1.The symbol ∪ (union) is used to represent the union of two sets,
which includes all elements that are present in either of the sets.
2.The symbol ∩ (intersection) is used to represent the intersection of
two sets, which includes only the elements that are common to both
sets.
3.The operation of union (∪) combines all elements from both sets,
including duplicates, resulting in a set that contains all unique
elements.
4.The operation of intersection (∩) selects only the elements that are
common to both sets, resulting in a set that includes only shared
elements.
5.The union of sets A and B, denoted as A ∪ B, results in the set {1, 2,

Know and use the Complement Rule to
calculate the probability of an event.
1.What is the Complement Rule? A) The rule that states the probability of an
event occurring is equal to 1 minus the probability of the event not
occurring. B) The rule that states the probability of an event occurring is
equal to the sum of the probabilities of all possible outcomes. C) The rule
that states the probability of an event occurring is equal to the product of
the probabilities of all independent events. D) The rule that states the
probability of an event occurring is equal to the ratio of favorable outcomes
to total outcomes.
2.If the probability of event A occurring is 0.7, what is the probability of event
A not occurring? A) 0.7 B) 0.3 C) 1.0 D) 0.5
3.If the probability of getting a head in a coin toss is 0.6, what is the
probability of getting a tail? A) 0.6 B) 0.4 C) 1.0 D) 0.5
4.If the probability of drawing a red card from a standard deck of 52 playing
cards is 0.25, what is the probability of not drawing a red card? A) 0.75 B)
0.25 C) 1.0 D) 0.5
5.The probability of event B occurring is 0.8. What is the probability of event
B not occurring? A) 0.8 B) 0.2 C) 1.0 D) 0.5

Answers:
1.A) The rule that states the probability of an event occurring is
equal to 1 minus the probability of the event not occurring.
2.B) 0.3
3.B) 0.4
4.A) 0.75
5.B) 0.2

Explanation:
1.The Complement Rule states that the probability of an event
occurring is equal to 1 minus the probability of the event not
occurring.
2.If the probability of event A occurring is 0.7, the probability of event A
not occurring is 1 - 0.7 = 0.3.
3.If the probability of getting a head in a coin toss is 0.6, the probability
of getting a tail is 1 - 0.6 = 0.4.
4.If the probability of drawing a red card is 0.25, the probability of not
drawing a red card is 1 - 0.25 = 0.75.
5.If the probability of event B occurring is 0.8, the probability of event
B not occurring is 1 - 0.8 = 0.2.

Calculate probabilities using the Addition Rule for
mutually exclusive and non-mutually exclusive
events.
1. What is the Addition Rule for mutually exclusive events? A) The probability of two mutually exclusive
events occurring is equal to the sum of their individual probabilities. B) The probability of two
mutually exclusive events occurring is equal to the product of their individual probabilities. C) The
probability of two mutually exclusive events occurring is always zero. D) The probability of two
mutually exclusive events occurring is always one.
2. If the probability of event A occurring is 0.4 and the probability of event B occurring is 0.6, what is
the probability of either event A or event B occurring for mutually exclusive events? A) 0.4 B) 0.6 C)
1.0 D) 0.2
3. What is the Addition Rule for non-mutually exclusive events? A) The probability of two non-mutually
exclusive events occurring is equal to the sum of their individual probabilities minus the probability
of their intersection. B) The probability of two non-mutually exclusive events occurring is equal to
the product of their individual probabilities. C) The probability of two non-mutually exclusive events
occurring is always zero. D) The probability of two non-mutually exclusive events occurring is
always one.
4. If the probability of event A occurring is 0.3 and the probability of event B occurring is 0.5, and the
probability of their intersection is 0.1, what is the probability of either event A or event B occurring
for non-mutually exclusive events? A) 0.3 B) 0.5 C) 0.6 D) 0.8
5. If event C is mutually exclusive with both event A and event B, and the probability of event A
occurring is 0.2 and the probability of event B occurring is 0.3, what is the probability of either event
A or event B or event C occurring? A) 0.2 B) 0.3 C) 0.5 D) 0.6

events.
Answers:
1.A) The probability of two mutually exclusive events occurring is
equal to the sum of their individual probabilities.
2.C) 1.0
3.A) The probability of two non-mutually exclusive events
occurring is equal to the sum of their individual probabilities
minus the probability of their intersection.
4.D) 0.8
5.C) 0.5

events.
Explanation:
1.The Addition Rule for mutually exclusive events states that the probability of two
mutually exclusive events occurring is equal to the sum of their individual
probabilities since they cannot happen simultaneously.
2.For mutually exclusive events, the probability of either event A or event B
occurring is equal to 0.4 + 0.6 = 1.0.
3.The Addition Rule for non-mutually exclusive events states that the probability of
two non-mutually exclusive events occurring is equal to the sum of their individual
probabilities minus the probability of their intersection to avoid double counting.
4.For non-mutually exclusive events, the probability of either event A or event B
occurring is equal to 0.3 + 0.5 - 0.1 = 0.8.
5.Since event C is mutually exclusive with both event A and event B, the probability
of either event A or event B or event C occurring is equal to 0.2 + 0.3 + 0 = 0.5,
as the events cannot occur simultaneously.

Calculate probabilities using the Multiplication Rule
for independent and non-independent events.
1.What is the Multiplication Rule for independent events? A) The probability of two independent
events occurring is equal to the sum of their individual probabilities. B) The probability of two
independent events occurring is equal to the product of their individual probabilities. C) The
probability of two independent events occurring is always zero. D) The probability of two
independent events occurring is always one.
2.If the probability of event A occurring is 0.4 and the probability of event B occurring is 0.6, what
is the probability of both event A and event B occurring for independent events? A) 0.4 B) 0.6
C) 1.0 D) 0.24
3.What is the Multiplication Rule for non-independent events? A) The probability of two non-
independent events occurring is equal to the sum of their individual probabilities. B) The
probability of two non-independent events occurring is equal to the product of their individual
probabilities. C) The probability of two non-independent events occurring is always zero. D)
The probability of two non-independent events occurring is always one.
4.If the probability of event A occurring is 0.6 and the probability of event B occurring, given that
event A has occurred, is 0.3, what is the probability of both event A and event B occurring for
non-independent events? A) 0.6 B) 0.3 C) 0.18 D) 0.9
5.If event C is independent of both event A and event B, and the probability of event A occurring
is 0.4 and the probability of event B occurring is 0.3, what is the probability of both event A and
event B and event C occurring? A) 0.4 B) 0.3 C) 0.12 D) 0.72

Answers:
1.B) The probability of two independent events occurring is equal
to the product of their individual probabilities.
2.D) 0.24
3.B) The probability of two non-independent events occurring is
equal to the product of their individual probabilities.
4.C) 0.18
5.C) 0.12

Explanation:
1.The Multiplication Rule for independent events states that the probability
of two independent events occurring is equal to the product of their
individual probabilities since they are unrelated and do not affect each
other's outcomes.
2.For independent events, the probability of both event A and event B
occurring is equal to 0.4 × 0.6 = 0.24.
3.The Multiplication Rule for non-independent events states that the
probability of two non-independent events occurring is equal to the product
of their individual probabilities since their outcomes are dependent on
each other.
4.For non-independent events, the probability of both event A and event B
occurring is equal to 0.6 × 0.3 = 0.18, given that event A has occurred.
5.Since event C is independent of both event A and event B, the probability
of all three events A, B, and C occurring is equal to 0.4 × 0.3 × 1 = 0.12, as
the events are unrelated and occur independently.

Calculate combinations and permutations.
1.Question: In a group of 8 people, how many different ways can you
select a committee of 3 people? A) 56 B) 24 C) 336 D) 84
2.Question: How many different ways can you arrange the letters of
the word "APPLE"? A) 60 B) 120 C) 24 D) 720
3.Question: A lock has 4 dials, each with 10 digits (0-9). How many
possible combinations are there? A) 100 B) 1,000 C) 10,000 D) 40
4.Question: How many different ways can you choose a president,
vice-president, and secretary from a group of 10 candidates? A) 720
B) 120 C) 2,520 D) 1,200
5.Question: In a deck of playing cards, how many different 5-card
hands can be dealt? A) 52 B) 10 C) 2,598,960 D) 120

Calculate combinations and permutations.
1.Answer: A) 56 Explanation: To calculate the number of ways to select a committee of 3
people from a group of 8, we use the combination formula. The formula for combinations
is nCr = n! / ((n - r)! * r!), where n is the total number of items and r is the number of items
to be selected. Plugging in the values, we get 8C3 = 8! / ((8 - 3)! * 3!) = 56.
2.Answer: B) 120 Explanation: To find the number of ways to arrange the letters in the word
"APPLE," we use the permutation formula. The formula for permutations is nPr = n!,
where n is the total number of items. In this case, "APPLE" has 5 letters. Therefore, 5! = 5
x 4 x 3 x 2 x 1 = 120.
3.Answer: C) 10,000 Explanation: The lock has 4 dials, and each dial can be set to one of
the 10 digits (0-9). To calculate the number of possible combinations, we multiply the
number of choices for each dial together. In this case, it's 10 x 10 x 10 x 10 = 10,000.
4.Answer: A) 720 Explanation: To select a president, vice-president, and secretary from a
group of 10 candidates, we need to find the number of permutations. Using the
permutation formula, we have 10P3 = 10! / (10 - 3)! = 10! / 7! = 10 x 9 x 8 = 720.
5.Answer: C) 2,598,960 Explanation: In a standard deck of playing cards, there are 52
cards. To calculate the number of different 5-card hands that can be dealt, we use the
combination formula. The number of combinations is 52C5 = 52! / ((52 - 5)! * 5!) =
2,598,960.

Use two-way tables as sample spaces for calculating
joint, marginal, and conditional probabilities.
1. Question: Consider the following two-way table representing the results of a survey on
the preferred mode of transportation for men and women. Calculate the joint probability of a
randomly selected person being a man and preferring cars. A) 0.4 B) 0.25 C) 0.3 D) 0.2
Answer: A) 0.4 Explanation: The joint probability is calculated by dividing the number of favorable
outcomes (men who prefer cars) by the total number of outcomes. In this case, there are 40 men who
prefer cars out of a total of 100 people surveyed, so the joint probability is 40/100 = 0.4.
2. Question: Using the same two-way table as in question 1, calculate the marginal probability of a
randomly selected person preferring bicycles. A) 0.35 B) 0.15 C) 0.2 D) 0.3
Answer: C) 0.2 Explanation: The marginal probability is the probability of an event occurring
irrespective of the other variable. To calculate the marginal probability of preferring bicycles, we sum
the probabilities across the row for bicycles. In this case, there are 20 men and 15 women who prefer
bicycles out of a total of 100 people surveyed, so the marginal probability is (20 + 15)/100 = 0.2.
Cars Bicycles Walk
Men 40 20 10
Women 30 15 25

Use two-way tables as sample spaces for calculating
joint, marginal, and conditional probabilities.
3. Question: Consider the following two-way table representing
the outcomes of rolling two dice. Calculate the conditional
probability of getting a sum of 7 given that the first die rolled is a 4.
A) 1/6 B) 1/12 C) 1/3 D) 1/36
Answer: A) 1/6 Explanation: The conditional probability is calculated
by dividing the number of favorable outcomes (cases where the sum is 7 and the first die is a 4)
by the total number of outcomes where the first die is a 4. From the table, we see that there is
only one outcome with a sum of 7 when the first die is a 4. Since the first die has 6 possible
outcomes (1, 2, 3, 4, 5, 6), the conditional probability is 1/6.
4. Question: Using the same two-way table as in question 3, calculate the marginal probability of
getting a sum of 9.
A) 1/9 B) 1/12 C) 1/6 D) 1/36
Answer: C) 1/6 Explanation: The marginal probability is the probability of an event occurring
irrespective of the other variable. To calculate the marginal probability of getting a sum of 9, we
sum the probabilities across the row and column for outcomes with a sum of 9. From the table,
we see that there are 4 outcomes out of a total of 36 outcomes where the sum is 9. Therefore,
the marginal probability is 4/36 = 1/9 = 1/6.
1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12

Use probabilities to analyze real-world
problems and make decisions.
1. A company manufacturing light bulbs finds that, on average, 5% of its products are
defective. If a customer purchases 10 bulbs, what is the probability that at least one bulb is
defective?
A) 0.05 B) 0.5 C) 0.41 D) 0.95
2. In a local election, a candidate has a 70% chance of winning. What is the probability that
the candidate will lose?
A) 0.7 B) 0.3 C) 0.2 D) 0.5
3. A basketball player has a 75% free-throw shooting average. If the player takes 4 free
throws, what is the probability of making exactly 3 of them?
A) 0.421875 B) 0.703125 C) 0.328125 D) 0.109375
4. A bag contains 5 red balls and 3 green balls. If two balls are drawn without replacement,
what is the probability of getting two red balls?
A) 0.2 B) 0.625 C) 0.4 D) 0.833
5. A weather forecast predicts a 30% chance of rain and a 20% chance of wind. What is the
probability of having both rain and wind on a given day?
A) 0.06 B) 0.5 C) 0.2 D) 0.1

Use probabilities to analyze real-world
problems and make decisions.
1. Answer: C) 0.41 Explanation: To find the probability that at least one bulb is defective, we can use
the complement rule. The complement of "at least one defective bulb" is "no defective bulbs." The
probability of no defective bulbs can be calculated as (1 - 0.05)^10 since each bulb is independent.
Therefore, the probability of at least one defective bulb is 1 - (1 - 0.05)^10 ≈ 0.41.
2. Answer: B) 0.3 Explanation: The probability of losing can be calculated as 1 - 0.7 since the sum of
the probabilities of winning and losing is 1. Therefore, the probability of the candidate losing is 0.3.
3. Answer: C) 0.328125 Explanation: To calculate the probability of making exactly 3 free throws out of
4, we need to use the binomial probability formula. The probability of making a free throw is 0.75,
and the probability of missing a free throw is 1 - 0.75 = 0.25. Using the formula, the probability is
given by 4C3 * (0.75)^3 * (0.25)^1 = 4 * 0.75^3 * 0.25 = 0.328125.
4. Answer: B) 0.625 Explanation: When two balls are drawn without replacement, the probability of
getting two red balls can be calculated by multiplying the probability of drawing a red ball on the first
draw (5/8) by the probability of drawing a red ball on the second draw (4/7) since there are 5 red
balls out of 8 in the first draw and 4 red balls out of 7 remaining in the second draw. Therefore, the
probability is (5/8) * (4/7) = 0.625.
5. Answer: A) 0.06 Explanation: To find the probability of both rain and wind occurring, we multiply the
probabilities of rain and wind together. Therefore, the probability is 0.30 * 0.20 = 0.06.

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