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MTTC Professional Readiness Examination
Statistics and Probability Review

Populations versus samples: A population is all members of a group or set. A sample is a subset
or portion of the population.
o Population parameters are measures of the entire population. Every member of the
population must be measured for the value to be considered a population parameter.
Examples of population parameters are the population mean (𝜇) and population
standard deviation (𝜎).
o Sample statistics are measures from a sample of the population. In others words not all
members of the population are included in the sample. The sample mean (𝑥̅ ) and
sample standard deviation (𝑠) are two such statistics.
o Why use samples? We are always interested in population parameters; however,
typically every member of a population cannot be measured or is impractical to
measure. So, we measure a representative sample of the population and then apply
statistical ideas to those measures to make inferences about the population parameters.
For example, we measure the sample mean and then use statistical concepts to draw
conclusions about the population mean.
Examples: Which of the following are samples and which are population measures?
a) the average age of all citizens of the United States
b) the proportion of voters in a poll who voted for the President in the most recent election
c) average GPA in a subset of all math classes at SVSU
d) the proportion of SVSU students who average more than seven hours of sleep per night during the fall
semester.
Answers: a) population, because the group of people is a census of all citizens of the US
b) sample, because the voters are from a poll which is sample of all voters
c) sample, because a subset of all math classes is a sample
d) population, because it is a census of all SVSU students (editorial note, this proportion is probably
rather small.)

Bias is a difference between the population parameter and sample statistic that is not caused by
random error. Sources of bias arise from how the sample is selected and how the data is
collected. In others words bias can come from sampling and measurement errors.
o Sampling errors produce samples that do not represent the population and thereby
produce false measures and conclusions about the population. For this reason, sample
selection is critically important to collecting useful data. For example, sample selection
is often responsible for discrepancies between different political polls.
o Data collection errors produce nonrandom, inaccurate measurements of individuals in
the sample. Calibration errors of physical measurement tools are one source of such
errors. Another source of such errors is biased or leading questions in surveys.
Examples: Determine the type of error and explain how the error could be prevented:
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Statistics and Probability p. 2
a) A voluntary survey of local news viewers found that the majority saw that station as having the best
local news.
b) The blood cholesterol levels of a random sample of people from a city were measured using an
improperly calibrated instrument.
c) A random sample of the US population was asked the following question: do you think road funding
should be increased given the horrible state of our roads?
Answers: a) The survey is self-selected and not random, so it does not represent the views of the
population. The bias could be eliminated with a well-constructed sample of randomly selected viewers
of all local news from the entire local population. Voluntary and convenience sampling is easily
conducted, but is dangerous because it can easily lead to false inferences.
b) An inaccurate calibration produces a non-random, systematic error. As a result the mean value will be
lower or greater than the actual value. This problem can be corrected by properly calibrating the
instrument and verifying the calibration.
c) The emotionally charged phrase “given the horrible state of our roads” will likely influence some
respondent’s answers. Questions need to be written in as neutral a manner as possible. For example the
question, “Should road funding be decreased, unchanged or increased?” would be less biased.

Observational and experimental studies are two broad types of statistical studies
o Observational studies simply obtain data without manipulating or controlling the
situation being studied. An example of an observational study is recording the number
of drivers who come to a complete stop at a stop sign from a concealed location.
Another example is a survey of a group, such as likely voters. Observational studies
cannot establish or reject cause-and-effect relationships. They are used instead to
understand the state of a population, like the driving habits at a specific stop sign
o Experimental studies manipulate the independent variables. Such studies are used to
establish or reject cause-and-effect relationships. For instance a double blind study to
test the effectiveness of a pharmaceutical drug would be an example of an experimental
study.
Examples: Identify the study type and conclusion that can be drawn from the study.
a) From a survey of college students, the plot of GPA versus hours spent studying per week is made and
strong positive linear correlation is shown.
b) Students were randomly assigned to sections of a math course. Half the sections included a once a
week recitation session and half did not. The mean course percentage was significantly greater for those
students in the recitation sections than the other sections.
Answers: a) Since the investigation involved a simple survey, the study was observational. The
conclusion that can be drawn is that hours spent studying and GPA are correlated. No conclusion about
causation can be made, because the study was not an experimental study.
b) This is an experimental study, since the independent variable of recitation session was manipulated. It
can be concluded that the recitation sessions increased course percentage since that data was obtained
from an experimental study and recitation sessions was the manipulated independent variable.
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
Statistics and Probability p. 3
Measures of center are numerical values that represent the location of the middle portion of
the distribution. The common measures of center are…
o Mean: Also known as the average. It is calculated by adding all the values and dividing
the sum by the number of values. An example of mean is an average exam score.
o Median: The middle value. Found by arranging all the values in order from smallest to
largest, and then crossing off pairs of values at the extremes until only the one or two
middle values remain. If only one value remains, the median is the middle value. If two
values remain, the median is the average of these two middle values. A good example is
median family income. Fifty percent of values are greater than the median and 50% are
less.
o Mode: The most frequent value in the distribution or the peak value. A distribution has
no mode if no values are repeated. A distribution can have multiple modes if multiple
values are repeated the same number of most frequent times. Distributions with
multiple modes have multiple peaks.
o Weighted Average: An average where some values count more than others. Each value
has a weight associated with it. The product of the weights and values are summed and
divided by the sum of the weights. A good example of a weighted average is grade point
average. The values are the grade points per credit hour. The weights are the credit
hours.
o Relationship between measures of center: If the distribution is symmetric, the mean,
median, and mode will have the same value. If the distribution is skewed the mean will
be moved towards the direction of skew, but the median will remain near the peak,
since very large or small values affect the mean but have little impact on the median.
This is why median income values are reported, so the income of the ultra-rich, like Bill
Gates, doesn’t distort the measure of center towards high income.
Examples:
A) Find the mean, median, and mode of the following set of values: 4, 6, 6, 7, 9, 12.
Answer:
4 + 6 + 6 + 7 + 9 + 12
̅̅̅
Mean =
= 7. ̅33
6
Median: 4, 6, 6, 7, 9, 12
medain =
6+7
2
= 6.5
Mode = 6 (Because 6 is the most frequent occurring value)
B) For the distribution of household income in the U.S., which is a better measure of center, mean or
median?
Answer: Median household income is a better measure of the center of the distribution than mean
family income. This is because the household income distribution is skewed to the right by the ultra-rich,
which will significantly impact the mean, but have little impact on the median.
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Statistics and Probability p. 4
C) The weighted average (or GPA) on a four-point scale is computed by:
GPA =
𝑐𝑟𝑒𝑑𝑖𝑡𝑠1 ∙𝑔𝑟𝑎𝑑𝑒1 +𝑐𝑟𝑒𝑑𝑖𝑡𝑠2 ∙𝑔𝑟𝑎𝑑𝑒2 +𝑐𝑟𝑒𝑑𝑖𝑡𝑠3 ∙𝑔𝑟𝑎𝑑𝑒3 +...
𝑡𝑜𝑡𝑎𝑙 𝑐𝑟𝑒𝑑𝑖𝑡𝑠
Find the GPA for someone who got an A in a 3 credits class, a B in a 2 credit class, and a C in a 4 credit
class.
Answer: GPA =

3(4)+2(3)+4(2)
3+2+4
=
26
9
≈ 2.89
Measures of Variation are numerical values that measure the spread of a distribution. Note that
distributions can have identical measures of center and still have radically different measures of
variation. A measure of center without a measure of variation gives an inadequate description
of the distribution. Common measures of variation are…
o Percentiles and Quartiles: A percentile is the value which has that percentage of the
distribution less than it. For example the 90th percentile in height for women is the
height which 90% of women are shorter than and 10% of woman are taller than. The 1st,
2nd and 3rd quartiles are the 25th, 50th, and 75th percentiles respectively, so quartiles are
just special percentiles. The 50th percentile is also the median.
o Range: The difference between the maximum and minimum values.
o Interquartile Range: The difference between the 75th and 25th percentiles.
o Variance: The average square of the distance from the mean. For a population the
equation for variance (𝜎 2 ) the sum of the squares of the differences from the mean
divided by sample size. For a sample variance (𝑠) the denominator is the one less than
the sample size.
o Standard deviation is simply the square root of the variance.
∑(𝑥𝑖 −𝜇)2
𝜎=√
𝑁
, for a population standard deviation.
∑(𝑥𝑖 −𝑥̅ )2
𝑠=√
𝑛−1
, for a sample standard deviation.
where or 𝑥̅ is the mean and N or n is the size of the population or sample.
Example: Find the 25th, 50th, and 75th percentiles, range, interquartile range, variance, and standard
deviation of the following set of values: 4, 6, 6, 7, 9, 12. Assume the data set is from a sample.
Answer:
50th percentile = 6.5 (the median, see previous example)
25th percentile (median of the lower 50%): 4, 6, 6
25th percentile = 6
th
75 percentile (median of upper 50%) : 7, 9, 12
75th percentile = 9
Range: maximum – minimum = 12 – 4 = 8
Interquartile Range = 75th percentile – 25th percentile = 9 - 6 = 3
Sample Variance: Remember from the earlier example 𝑥̅ = 7.33
𝑠2 =
∑(𝑥𝑖 −𝑥̅ )2
𝑛−1
=
(4−7.33)2 +(6−7.33)2 +(6−7.33)2 +(7−7.33)2 +(9−7.33)2 +(12−7,.33)2
6−1
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= 7.87
MTTC Professional Readiness Review
Statistics and Probability p. 5
Sample Standard Deviation: 𝑠 = √7.87 = 2.81

In the graph of a probability distribution, the x-axis of the graph represents the observed or
measured value. The height represent the frequency or likelihood of that value occurring
o Area between two x values represents the probability of obtaining a result between
them.
o A Normal Distribution is a symmetrical, bell-shaped probability distribution. Normal
distributions with greater means will be centered farther to the right. Distributions with
greater standard deviations will be broader and shorter.
 The probability of being within 1 standard deviation from the mean is
approximately 68%. In others words, 68% of the observations would be
expected to be within 1 standard deviation.
 The probability of being within 2 standard deviations from the mean is
approximately 95%. In others words, 95% of the observations would be
expected to be within 2 standard deviations.
 The probability of being within 3 standard deviations from the mean is
approximately 99.7%. In others words, 99.7% of the observations would be
expected to be within 2 standard deviations.
 Generally observations beyond two standard deviations are considered to be
unusual since there is a small probability they occur by just random chance.
o The z-score is the number of standard deviations the observation is from the mean. The
farther the z-score is from zero the more unlikely the observation occurred by just
random chance. The closer the z-score is to zero the more likely the observation can be
explained by random chance. The equation for z-score is
𝑧=
𝑥−𝜇

where 𝑥 is the observed or measured value, 𝜇 is the population mean, and  is the
population standard deviation
Examples:
A) What is the probability of a measured value being within a z-score of -2 and 2?
Answer: Since a z-score of 2 is equivalent to two standard deviations and 95% of the data lies within two
standard deviations, the probability of an observation being within a z-score of -2 and 2 is approximately
95%.
B) Distribution A has a mean of 20 and standard deviation of 10. Distribution B has a mean of 30 and
standard deviation of 5. Which distribution has its center farther to the right? Which is tall and
narrower? Which is shorter and wider?
Answer: The distribution with the largest mean will have its center located farther to the right. So the
center of distribution B is farther to the right, since it has a larger mean. Distributions with smaller
standard deviations are narrower and taller. Those with larger standard deviations are wider and
shorter. So distribution A, because it has a larger standard deviation is wider and shorter than
distribution B, and distribution B is narrower and taller.
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Statistics and Probability p. 6
C) For a distribution with a mean of 15 and standard deviation of 5, what is the z-score of an
observation with a value of 7? Also would this observation be considered unusual based on random
chance?
Answer: 𝑧 =
𝑥−𝜇
𝑠
=
7−15
5
= −1.6 This observation would not be considered unusual, because it is
within 2 standard deviations of the mean.

Graphical representations of data are used to enhance visualization of both correlations in
bivariate data and the distribution of uni-variant data. Often the first step in statistical analysis is
to tabulate and graph the data to visually look for relationships within the data. Much can be
initially gleaned by examining the data visually in tabular or graphical form.
o Bar charts are used when one variable is categorical.
o Scatterplots are more useful when both variables are quantitative.
o Histograms are often used to examine the distribution of data.
o Pie charts are helpful to examine the distribution of categorical data.
Examples: What type of graph would be most useful for each data set?
a) The relationship between gas mileage and car weight
b) temperature trends during the year
c) distribution of eye color in the US population
d) hours spent studying as a function of day of the week
e) family income distribution in the US.
Answers: a) Since both gas mileage and car weight are quantitative a scatterplot would be a good
choice. Any correlation between mpg and weight would likely be visible in the scatterplot.
b) A line graph or time chart would be a good representation of temperature versus time data and help
identify yearly trends.
c) Since eye color is categorical data, a pie chart would show the distribution of the population. A bar
chart would also be suitable.
d) Likewise since days of the week are categorical, either a bar chart or pie chart would work well.
e) Since income is quantitative, a histogram with a suitable class width to create about 10-20 bars would
be a good choice to represent the distribution of family income in the US.

Bivariate Data is data that involves two variables. Examples of such data are height versus age,
income versus education level, life expectancy versus income, GPA versus hours spend studying
per week.
o Scatter Plots on an x-y coordinate system are often used to represent and help analyze
bivariate data. Such plots help visually observe the presence or absence correlation
between the two variables plotted on the x- and y-axes. If the trend rises moving left to
right, the slope is positive. If it declines moving from left to right, the slope is negative.
o Pearson’s Correlation Coefficient tells how strongly two variables are associated.
Bivariate data with a correlation coefficient of 1 is perfectly correlated in an upward
trend, meaning all points lie on the same line with a positive slope. Likewise a
correlation coefficient of -1 is a perfectly correlated downward trend. A correlation
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Statistics and Probability p. 7
coefficient of 0 has no association between the variables. When the correlation
coefficient is 0 the points on the x-y plot are typically scattered, although they could all
lie on a horizontal straight line. Data with correlations coefficients between -1 and 1
have points scattered around a trend line. The the larger the correlational coefficient (in
absolute value) the more tightly the points are grouped around the trend line. Examples
are shown below (source: http://ordination.okstate.edu/STAT
o
o
Correlation is not the same as causation! Correlation only means there is a
mathematical trend between the two variables. Correlation does not mean a change in
one variable causes a response in the other variable. For example, a correlation
between ice cream sales at a beach and the number of lifeguard rescues does not mean
that ice cream causes people to swim more dangerously. It just means the two variables
follow the same trend. A more likely explanation is that the two variables are correlated
because on days that there are more sales of ice cream more people are at the beach
and the number of rescues tends to increase because there are more people at the
beach. So ice cream sales and rescues are correlated simply because both are related to
the number of people at the beach. The important point is that correlation does not
imply causation! Observational studies do not establish causation. Experimental studies
where the independent variables are manipulated in a controlled matter are necessary
to make inferences about causation.
Linear regression mathematically finds the best fit line for data that has a statistically
significant correlation. Such lines can be used to predict the expected value of the
second variable from the value of the first variable given the trend exists. Remember
that such predictions do not imply causation, for the two variables may be linked to
another variable.
Example: Pizza prices and the Consumer Price Index have a correlation coefficient of 0.85. Describe the
meaning of the correlation, appearance of the plot of the two variables on an x-y graph, and the
causation between the variables.
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Answer: A correlation coefficient of 0.85 is considered a strong correlation. Remember the strength of
the correlation increases as the correlation coefficient approaches 1 or -1. The data on the x-y graph
would be expected to be fairly tightly scattered around an upward trend line. The strong correlation
does not imply causation. Correlations never establish causation. In this case the Consumer Price Index
obviously doesn’t cause pizza prices to increase, since the CPI is only a mathematic index used to
measure the economy, not a driving force and causation of change in the prices.

Probability is the likelihood of an event occurring. It is simply the number of ways the desired
event can occur divided by the number of all possible events. For example, the likelihood of
obtaining a five on a single roll of a six-sided dice is 1/6. There is only one way for the dice to
show a five. There are six possible events. So the probability is 1 divided by 6 or 1/6.
o Without replacement are situations where items from the population are selected only
once. Dealing cards from a deck are one example. Once a card is dealt to a person, the
same card cannot be dealt to another person during the same round. In others words
the cards are not replaced into the deck immediately after each one is dealt. In
situations without replacements, the number of items available to choose from declines
by one after each selection since one fewer of the items remain, and therefore the
probability of selection of that item changes with each selection.
o With replacement are situations where the probability of selecting any item remains
the same, because after each selection all items are returned to the population prior to
the next selection. An example of with replacement is sequential throwing a dice. On
each throw all six possible outcome exist and have the same probability.
o Sample space is a listing of all possible outcomes. For example if you flip two different
coins simultaneously the sample spaces is simply the possible outcomes, which are
heads-heads, heads-tail, and tails-tails.
o Fundamental counting rule is used to calculate the number of possible outcomes. If the
first event can occur m ways and the second can occur n ways, the two events together
can occur 𝑚 ∙ 𝑛 ways
Example:
How many possible can a six character passwords are possible if characters can be repeated? Answer:
There are 26 letters and ten digits (0 through 9) available for each of the six characters. So, there are 36
possibilities for each character. So there are 36 ∙ 36 ∙ 36 ∙ 36 ∙ 36 ∙ 36 = 366 = 2,176,782,336 possible
passwords.
Practice Problems:
1. What is the mean, median, and mode and standard deviation of a sample with values of 10, 11, 17,
13, 29, 10?
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Statistics and Probability p. 9
2. Is 53 an usual or unusual value for a normal distribution with an mean of 60 and standard deviation
of 4?
a. Not unusual, because the z-score is less than -2.
b. Not unusual, because the z-score is not less than -2.
c. Unusual, because the z-score is less than -2.
d. Unusual, because the z-score is not less than -2.
3. Is 31 an unusual or usual age for a normal distribution with and average population age of 21 and a
standard deviation of 4.2?
a. Not unusual, because the z-score is greater than 2.
b. Not unusual, because the z-score is less than or equal to 2.
c. Unusual, because the z-score is greater than 2.
d. Unusual, because the z-score is less than or equal to 2.
4. If an observational study found that weight correlates with age with a correlation coefficient of 0.85
what can be conclude?
a. Weight increases with age, and age causes weight gain.
b. Weight decreases with age, and age cause weight loss.
c. Weight and age are related, and weight tends to increase with age.
d. Weight and age are related, and weight tends to decrease with age.
e. There is no relationship between age and weight.
5. Which kind of graph could best be used to represent the distribution of ages of citizens of the United
States?
a. A x-y scatterplot
b. A histogram
c. Pie chart
d. Bar chart
6.
a.
b.
c.
d.
Which kind of graph could not be used to represent the trend in daily high temperature over time?
A x-y scatterplot
A histogram
Pie chart
A run chart
7.
a.
b.
c.
d.
Which would be a parameter of a population?
The mean score of a sample five students from all of those passing MATH 110.
The mean of a poll for a presidential election.
The standard deviation of the age of fifty SVSU students who are non-traditional.
The standard deviation of the age of all current SVSU students.
8. Which is non-biased?
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Statistics and Probability p. 10
a.
b.
c.
d.
e.
A poll of your friend’s opinions on their favorite candy bar.
A survey of opinions regarding Medicare of visitors to the AARP Internet site
A survey that asks: “Should there be term limits to fix the do-nothing Congress?”
A Gallop poll sampling voters views on the state of public education.
The ratings of SVSU professors on RateMyProfessor.com
9.
a.
b.
c.
d.
Which is an experimental study?
A science project investigating how controlling the level of fertilizer affected plant growth.
A comparison of the heights and weights of residents of the United States.
An examination of personal income and eye color.
The distribution of number of television sets per household in the United States.
10. How many possible license plate combinations can be made for plates with six characters (letters
and numbers) if the characters cannot be repeated.
a. 6!
b. 36 ∙ 35 ∙ 34 ∙ 33 ∙ 32 ∙ 31
c. 66
d. 366
11. How many possible license plate combinations can be made for plates with six characters (letters
and numbers) if the characters can be repeated.
a. 6!
b. 36 ∙ 35 ∙ 34 ∙ 33 ∙ 32 ∙ 31
c. 66
d. 366
Answers: See the solutions document for detailed calculations and explanations
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Mean =15; median = 12; mode = 10; standard deviation = 7.35
B
C
C
B
C
D
D
A
B
D
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