Left, right or middle

LEFT, RIGHT OR MIDDLETAILS IN DISTRIBUTIONSMINITAB 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)

LEFT, RIGHT OR MIDDLE

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

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Distribution PlotBinomial, n=12, p=0.72

LEFT, RIGHT OR MIDDLETails

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

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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

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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”

LEFT, RIGHT OR MIDDLETails

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

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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

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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 above 7 because of the < sign)


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) = 0.16060.18

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Distribution PlotPoisson, 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) = 0.44570.18

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0.4457

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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

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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?



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!!!

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Distribution PlotGeometric, 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

https://www.youtube.com/watch?v=x9vsoP8wLAM

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Date post:	17-Jul-2015
Category:	Education
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