Week 4 Three Extra Homework Examples (3x9)/WMA 221 Statistics for Decision Making

Professor Brent HeardNot to be copied or linked to without my permission 3 x 9

WS4DM

(3x9)/W•Disclaimer

▫Please note that I might do things a little differently than your instructor, your tutor, your friends, your mother, your dog, etc.

▫The reason I post these examples is to help you▫There is no requirement for you to even look at

these▫Therefore, if I help you – great. If I don’t – I’m

sorry.▫This is something I simply do as a service to

the students because I like them!

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•Number 5 Example▫On number 5 in the homework, they are

just trying to get you to use the Binomial to find the probability of certain things happening based on a given n and p.

▫It’s really easy. Let’s look at an example…

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•Forty-two percent of households say they would feel secure if they had six months of expenses in the bank. You randomly select 9 households and ask them if they would feel secure having six months worth of their expenses in the bank.▫Find the probability exactly 6 would say

yes.▫Find the probability more than 6 would say

yes.▫Find the probability that at most 6 would

say yes.•Solution next page

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•Based on our question, we know we are asking 9 households, so n=9. Also we know that 42% feel secure having that amount saved, so p=0.42 (the decimal form of 42%)

•The rest is easy using Minitab…▫I use the same approach in Minitab to solve

all three parts of this question▫I recently discovered that it works very

well

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•Once in Minitab▫Go to Graph >> Probability Distribution

Plot Once There Click the Fourth Option “View

Probability” and then Click “OK”

(3x9)/W• Now in Probability

Distribution Plot▫ Now for this problem,

change the Distribution Dropdown to “Binomial” and input the n (number of trials) and p (Event probability) for this problem (Remember 9 households, so n=9. Forty-two percent, so p = 0.42

(3x9)/W• Now click the Shaded Area Tab and pay attention

(3x9)/W• For this problem we

will be putting in “X values

• For Exact values like P(x=6), we will use “Middle,” for less than problems we will use “Left Tail” and for greater than problems, we will use “Right Tail”

• But pay attention to the wording!

(3x9)/W• Now part “a” was to find P(6) based on n=9 and p=0.42

Therefore, P(6) = 0.08996 or 0.090 rounded to three decimal places

(3x9)/W• Now part “b” was to find “more than 6” based on n=9 and p=0.42. “More

than 6 implies “7 or more.” This is a “Right Tail.”

Therefore, P(x>6) = 0.03338 or 0.033 rounded to three decimal places

(3x9)/W• Now part “c” was to find the probability of “no more than 6” based on n=9

and p=0.42. “No more than 6” implies “6 or less.” This is a “Left Tail.”

Therefore, P(x<=6) is 0.9666 or 0.967 rounded to three decimal places

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•These are easy, just follow the steps and remember on those “More than 6” type problems, you are dealing with a right tail where “more than 6” means 7 or more…

•Steps again - Graph >> Probability Distribution Plots

•Next page

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•Order

Pick Tails and values based on problem

See Answer

PracticePracticePractice

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•Number 13 Example▫On number 13 in the homework, they are

just trying to get you to use the Geometric distribution to find the probability of something happening on the 4th time or 5th or a certain “one shot” time. In other words, when the event happens – game over (You’ve done it for “the first time.”)

▫It’s really easy. Let’s look at an example…

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•Number 13 Example▫Assume the probability you will make a sale

on any given phone call is 0.11. Find the probability that you A) make your first sale on the third call B) make your first sale on the first, second,

third, fourth, fifth or sixth call. C) Do not make the first sale on one of the

first four calls.

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•Easy in Minitab▫Go to Graph >> Probability Distribution

Plot Once There Click the Fourth Option “View

Probability” and then Click “OK”

(3x9)/W• Now in Probability

Distribution Plot▫Now for this

problem, change the Distribution Dropdown to “Geometric” and input Event probability for this problem (Remember the probability is 0.11)

(3x9)/W• Now click the Shaded Area Tab and pay attention

(3x9)/W• Now in Shaded Area Tab

after choosing Geometric distribution and inputting correct Event Probability▫ Part A) - Make your first

sale on the third call

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•Answer•0.08713 or 0.087

rounded to three decimals

(3x9)/W• Now in Shaded Area Tab

after choosing Geometric distribution and inputting correct Event Probability

Part B) - make your sale on the first, second, third, fourth, fifth or sixth call. (THIS MEANS MAKE YOUR SALE IN ONE OF THE FIRST SIX CALLS – SO IT IS “LEFT TAIL” WITH 6 AS YOUR STOPPING POINT)

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•Answer•0.5030 or 0.0503

rounded to three decimals

(3x9)/W• Now in Shaded Area Tab

after choosing Geometric distribution and inputting correct Event Probability

Part C) - Do not make a sale on the first four calls. (THIS MEANS MAKE YOUR FIRST SALE ON THE FIFTH CALL OR LATER – SO THIS IS A “RIGHT TAIL” STARTING WITH 5.)

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•Answer•0.6274 or 0.627

rounded to three decimals

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•Last Question - Are any of these “Unusual?”

•“NO” – To be considered “Unusual” the probability would have to be less than 0.05

•In other words, 0.0671, 0.124, 0.053 are not unusual. However values like 0.047, 0.003, 0.019 are because they are less than 0.05 or “5%.”

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•Number 15 Example▫On number 15 in the homework, they are

just trying to get you to use the Poisson to find the probability of something happening. Read about the differences in Geometric and Poisson problems, you should be able to spot them pretty easy. With a Poisson, they usually give you an average over a given timeframe.

▫Let’s look at an example…

(3x9)/W• Number 15 Example

▫A major hurricane has really strong winds and if I wanted to confuse you I would tell you how strong the winds have to be, but I don’t want to confuse you. During the last century the mean number of hurricanes to hit Gilligan’s Island per year was about 0.52 (SCREAMS POISSON- “Over the last century the average number of hurricanes per year”) Find the following

Probability exactly one hurricane will hit Gilligan’s Island Probability at most one hurricane will strike Gilligan’s Island Probability more than one hurricane will hit Gilligan’s Island

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•Again back to Minitab (Should sound familiar)▫Go to Graph >> Probability Distribution

Plot Once There Click the Fourth Option “View

Probability” and then Click “OK”

(3x9)/W• Now in Probability

Distribution Plot▫Now for this

problem, change the Distribution Dropdown to “Poisson” and input the mean for this problem (Remember the number of hurricanes was 0.52)

(3x9)/W• Now click the Shaded Area Tab and pay attention

(3x9)/W• Now in Shaded Area Tab

after choosing Geometric distribution and inputting correct Event Probability▫ Part A) – Exactly one

hurricane

(3x9)/W•Answer•0.3092 or 0.309

rounded to three decimals

•This is “NOT UNUSUAL” because it is greater than 0.05

(3x9)/W• Now in Shaded Area

Tab after choosing Geometric distribution and inputting correct Event Probability

Part B) – at most one hurricane

NOTE AT MOST ONE IS ZERO OR ONE, SO IT IS LEFT TAILED WITH AN ENDPOINT AT 1

(3x9)/W• Answer• 0.9037 or 0.904 rounded

to three decimals▫ This is the probability of

at most one hurricane hitting… which means it combines the probabilities of seeing none or zero and one

▫ Again, this is “NOT UNUSUAL” – as a matter of fact there is about a 90% chance of getting either zero or one hurricanes

(3x9)/W• Now in Shaded Area Tab

after choosing Geometric distribution and inputting correct Event Probability

Part C) – More than one hurricane – THIS MEANS “TWO OR MORE” Therefore we have a Right Tail with 2 on the left endpoint.

(3x9)/W• Answer• 0.09633 or 0.096 rounded to

three decimals – again not unusual

• However note that if I add my results from part b and part c together I should get one because I have “covered” all possibilities (In that the probability of seeing 1 or less plus the probability of seeing 2 or more would be everything▫ (THIS IS JUST AN ATTEMPT

TO HELP YOU BETTER UNDERSTAND PROBABILITY DISTRIBUTIONS)

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•Last Question - Are any of these “Unusual?”

•“NO” – To be considered “Unusual” the probability would have to be less than 0.05

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•Hope you enjoyed this… Let me know if these help!

•More examples next week….•Visit me at www.facebook.com/statcave

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