Left, right or middle

LEFT, RIGHT OR MIDDLE
TAILS IN DISTRIBUTIONS
MINITAB EXPLANATIONS
BINOMIAL, POISSON & GEOMETRIC
B Heard
(This is not to copied,
posted or shared without
my permission, students are
welcome to download a
copy for personal use)

First on all of these, we will use the “Graph” feature in Minitab
Our initial steps in Minitab will be
Graph >> Probability Distribution Plots >> THEN Click “View
Probability”

The next menu allows us to choose which distribution. We choose
the distribution based on the problem and the information entered is
based on the distribution

We then click the shaded area tab

This is where we pick our tails and provide more information

At this point, I am going to assume that you understand about using
Graph >> Probability Distribution Plots >> Clicking View Probability
For the problems noted, I will give the logic behind the tails

Let’s say we have done a recent survey and found that 72% of
statistics students use Minitab. We randomly select 12 statistics
students and ask them if they use Minitab.
Find the probability of the following numbers of the 12 using Minitab.
Exactly 9
At least 9
More than 9
At most 9
Less than 9

Let’s look at these first… and see what they mean
Exactly 9 (No doubt about this)
At least 9 (This means 9 or more, in our case, it means 9,10,11 or
12)
More than 9 (This means 10 or more, in our case, it means 10,11 or
12)
At most 9 (This means 9 or less, in our case, it means
0,1,2,3,4,5,6,7,8 or 9)
Less than 9 (This means 8 or less, in our case, it means
0,1,2,3,4,5,6,7,8 or 9)

Using probability notation, we would have
Exactly 9 is P(x=9) (No doubt about this)
At least 9 is P(x≥9) (This means 9 or more, in our case, it means
9,10,11 or 12)
More than 9 is P(x>9) (This means 10 or more, in our case, it means
10,11 or 12)
At most 9 is P(x≤9) (This means 9 or less, in our case, it means
0,1,2,3,4,5,6,7,8 or 9)
Less than 9 is P(x<9) (This means 8 or less, in our case, it means
0,1,2,3,4,5,6,7,8 or 9)

Tails
With problems want exact values like P(x=9) we use the middle
feature after clicking the shaded area tab… So P(x=9) = 0.2511
0.25
0.20
0.15
0.10
0.05
0.00
X
Probability
9
0.2511
4 12
Distribution Plot
Binomial, n=12, p=0.72

Tails
With problems with greater than or greater than or equal to signs we
use “Right Tail” So for P(x≥9) we use the right tail feature after
clicking the shaded area tab… So P(x≥9) = 0.5548
0.25
0.20
0.15
0.10
0.05
0.00
X
Probability
9
0.5548
4
Distribution Plot

Tails
P(x>9) is another right tail problem, but notice x>9 does not include
the 9, so it means 10 or more. So P(x>9) = 0.3037
0.25
0.20
0.15
0.10
0.05
0.00
X
Probability
10
0.3037
4
Distribution Plot

Remember Right Tails are used when the sign “points right”… think of
it as being an arrow. You do have to get your starting point correct!
P(x>9) P(x≥9)
They “Point Right”

Tails
With problems with less than or less than or equal to signs we use
“Left Tail” So for P(x≤9) we use the left tail feature after clicking the
shaded area tab… So P(x≤9) = 0.6963
0.25
0.20
0.15
0.10
0.05
0.00
X
Probability
9
0.6963
12
Distribution Plot

Tails
P(x<9) is another left tail problem, but notice x<9 does not include
the 9, so it means 8 or less. So P(x<9) = 0.4452
0.25
0.20
0.15
0.10
0.05
0.00
X
Probability
8
0.4452
12
Distribution Plot

Remember Left Tails are used when the sign “points left”… think of it
as being an arrow. You do have to get your starting point correct!
P(x<9) P(x≤9)
They “Point Left”

Test yourself… Answers on the following page. Which tail or middle
would you use in Minitab? And what would the x value be?
P(x=3)
P(x<7)
P(x≥12)
P(x≤2)
P(x>5)

Test yourself… Answers on the following page. Which tail or middle
would you use in Minitab? And what would the x value be?
P(x=3) Use Middle with both x values being 3
P(x<7) Use Left Tail with x value being 6
P(x≥12) Use Right Tail with x value being 12
P(x≤2) Use Left Tail with x value being 2
P(x>5) Use Right Tail with x value being 6

Same Problem, different questions:
Let’s say we have done a recent survey and found that 72% of
statistics students use Minitab. We randomly select 12 statistics
students and ask them if they use Minitab.
Find the probability
P(4<x<10)
P(5≤x ≤8)
P(7<x ≤10)

On all of these we would use “Middle,” but what would our x values
be?
P(4<x<10) (Our x values would be 5 and 9, because we are looking
for values greater than 4 and less than 10, but NOT including them)
P(5≤x ≤8) (Our x values would be 5 and 8, because we are looking
for values greater than or equal to 5 and less than or equal to 8, this
includes both endpoints)
P(7<x ≤10) (Our x values would be 8 and 10, because we are looking
for values greater than 7 and less than or equal to 10, this includes
just the right endpoint because of the ≤ sign and I must go one

By now you should know the steps, but the answers would be as
follows:
P(4<x<10) = 0.6903
P(5≤x ≤8) = 0.4392
P(7<x ≤10) = 0.6646

Again this problem had n=12, p = 0.72 and it was a Binomial
What is the mean? Simply np or (12)(0.72) = 8.64 This means I
would “expect” about 9 out of 12 of the statistics students to use
Minitab.
What is the variance? Simply npq or (12)(0.72)(1-0.72) = 2.4192
What is the standard deviation? Simply 𝑛𝑝𝑞 or (12)(0.72)(0.28) =
2.4192

What about Poisson questions?
They are just as easy…
Question: Use technology to solve the following Poisson problem.
The mean number of speeding tickets an officer gives during rush
hour is 5. Find the probability that during rush hour, (a) an officer
gives 5 tickets, (b) at most 5 tickets and (c) more than 5 tickets.

Ok, it’s a Poisson and the mean is 6
Find the probability that during rush hour,
(a) an officer gives 5 tickets - P(x=5) So this will be a Poisson using a
“Middle”
(b) at most 5 tickets – P(x≤5) So this will be a Poisson using a “Left
Tail”
(c) more than 5 tickets - P(x>5) So this will be a Poisson using a
“Right Tail”

Graph >> Probability Distribution Plot >> Click View Probability
Choose Poisson for the Distribution, Enter 6 as the mean
Then click the Shaded Area Tab

P(x=5)
After clicking Shaded Area tab,
Click Radial button next to X Value, Choose Middle, Enter 5 for both X
values
You will find
P(x=5) = 0.1606
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
X
Probability
5
0.1606
0 15
Distribution Plot
Poisson, Mean=6

P(x≤5)
Click Radial button next to X Value, Choose Left Tail, Enter 5 for X
value
You will find
P(x≤5) = 0.4457
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
X
Probability
5
0.4457
15
Distribution Plot
Poisson, Mean=6

P(x>5)
Click Radial button next to X Value, Choose Right Tail, Enter 6 for X
value
You will find
P(x>5) = 0.5543
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
X
Probability
6
0.5543
0
Distribution Plot
Poisson, Mean=6

Same goes for a geometric distribution. But let’s look at a few other things
also.
Question: Assume that a gate guard at a security check estimates than 12%
of drivers do not wear their seatbelts. They decide to do a spot check and
stop 20 cars on Friday to check for seatbelt usage.
What is the mean of the distribution?
What is the variance?
What is the standard deviation?
How many cars should they expect to stop before finding a driver whose
seatbelt is not buckled?
What is the probability that they will find someone without a seatbelt in the
first 6 cars that pass through?

Question: Assume that a gate guard at a security check estimates
than 12% of drivers do not wear their seatbelts. They decide to do a
spot check and stop 20 cars on Friday to check for seatbelt usage.
What is the mean of the distribution?
The formula for the mean of the geometric distribution is
µ =
1
𝑝
=
1
0.12
= 8.3
Where “p” is just the decimal form of the percentage.

What is the variance?
The formula for the variance of the geometric distribution is
σ2 =
𝑞
𝑝2 =
0.88
0.122 = 61.1
Remember q is just 1-p, so our q would be 1- 0.12 or 0.88

What is the Standard Deviation?
The formula for the standard deviation of the geometric distribution
is
σ =
𝑞
𝑝2 =
0.88
0.122 = 61.1 = 7.8
If you have already calculated the variance, you simply take the
square root of it.

How many cars should they expect to stop before finding a driver
whose seatbelt is not buckled?
You already answered this by finding the mean.
You expect to stop 8.3 cars on average… before finding someone
without a seatbelt

Same goes for a geometric distribution. But let’s look at a few other
things also.
What is the probability that they will find someone without a seatbelt
in the first 6 cars that pass through?

Graph >> Probability Distribution Plot >> Click View Probability
Choose Geometric, Enter decimal form of p (12% is 0.12)

Click Shaded Area Tab
Choose x value radial button and left tail
Then enter 6 for the x value
Why left tailed?
The questions was “What is the probability
that they will find someone without a seatbelt
in the first 6 cars that pass through?”
That means 6 cars or less or P(x<6)

The answer is 0.5356…. Seems odd, since the mean was 8.3, but
look at the shape of the distribution!!!
0.12
0.10
0.08
0.06
0.04
0.02
0.00
X
Probability
6
0.5356
42
Distribution Plot
Geometric, p=0.12
X = total number of trials.

I put a lot of work into this….
If you want to say thanks, help a child with math or give me a thumbs
up on one of my YouTube Storytelling videos… I don’t get paid for
doing these, but I enjoy it and the more views and THUMBS UP I get
the better chance I have of getting to tell stories in other places.
https://www.youtube.com/watch?v=x9vsoP8wLAM
https://www.youtube.com/watch?v=q4993u8ZRdY
https://www.youtube.com/watch?v=SbXeza5v4b0

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