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№1 слайд![BBA Applied Statistics Week](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img0.jpg)
Содержание слайда: BBA182 Applied Statistics
Week 9 (1)
Calculating the probability of a continuous random variable – Normal Distribution
Dr Susanne Hansen Saral
Email: susanne.saral@okan.edu.tr
https://piazza.com/class/ixrj5mmox1u2t8?cid=4#
www.khanacademy.org
№2 слайд![Mid-term exam statistics](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img1.jpg)
Содержание слайда: Mid-term exam statistics
№3 слайд![Mid-term exam statistics](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img2.jpg)
Содержание слайда: Mid-term exam statistics
№4 слайд![Continuous random variable A](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img3.jpg)
Содержание слайда: Continuous random variable
A continuous random variable can assume any value in an interval on the real line or in a collection of intervals.
It is not possible to talk about the probability of the random variable assuming a particular value, because the probability will be close to 0.
Instead, we talk about the probability of the random variable assuming a value within a given interval.
№5 слайд![Calculating probabilities of](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img4.jpg)
Содержание слайда: Calculating probabilities of
continuous random variables
№6 слайд![The Standard Normal](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img5.jpg)
Содержание слайда: The Standard Normal Distribution – z-values
Any normal distribution, F(x) (with any mean and standard deviation combination) can be transformed into the standardized normal distribution F(z), with mean 0 and standard deviation 1
We say that Z follows the standard normal distribution.
№7 слайд![Procedure for calculating the](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img6.jpg)
Содержание слайда: Procedure for calculating the probability of x
using the Standard Normal Table
For m = 100, s = 15, find the probability that X is less than 130 = P(x < 130)
Transforming x - random variable into a z - standard random variable:
№8 слайд![Procedure for calculating the](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img7.jpg)
Содержание слайда: Procedure for calculating the probability of x
using the Standard Normal Table (continued)
Step 2
Look up the probability from the table of normal curve areas
Column on the left is Z value
Row at the top has second decimal places for Z values
№9 слайд![Using the Standard Normal](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img8.jpg)
Содержание слайда: Using the Standard Normal Table
№10 слайд![P z lt P z gt - . In](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img9.jpg)
Содержание слайда: P(z < + 2) = P(z > -2) = .9772
In probability terms, a z-score of -2.0 and +2.0 has the same probability, because they are mirror images of each other.
If we look for the z-score 2.0 in the table we find a value of 9772.
№11 слайд![](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img10.jpg)
№12 слайд![Finding the probability of](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img11.jpg)
Содержание слайда: Finding the probability of z-scores with two decimals and graph the probability
P ( z < + 0.55) = 0.7088 or 70.88 %
P (z > + .55) = 1.0 – 0.7088 = 0.2912 or 29.12%
P ( z > - 0.55) = 0.7088 or 70.88 %
P ( z < - 0.55) = 1.0 - .7088 = 0.2912 or 29.12 %
P ( z < + 1.65) = 0.9505 or 95.05 %
P (z > + 1.65) = 1.0 – 0.9505 = 0.0495 or 4.96 %
P( z > - 2.36) = .9909 or 99.09 %
P ( z < + 2.36) = .9909 or 99.09 %
№13 слайд![Determine for shampoo filling](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img12.jpg)
Содержание слайда: Determine for shampoo filling machine 1 the proportion of bottles that:
= 500 ml = 10ml
Contain less than 510 ml P(x < 510)
Contain more than 515 ml P(x > 515)
Contains more than 480 ml P(x > 480)
Contain less than 490 ml P(x < 490)
Contain more than 505 ml P(x > 505)
№14 слайд![Solution Contain more than ml](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img13.jpg)
Содержание слайда: Solution: Contain more than 515 ml
P(x 515ml)
1. Draw the graph to see which area we are looking for:
2. Z –score = = 1.5 = P(z > 1.5)
3. We can find P(z < 1.5) = .9332 directly from the table
P(z 1.5) = 1 - .9332 = .0668
6.68% of the shampoo bottles contain more than 515 ml.
№15 слайд![Solution Contain more than ml](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img14.jpg)
Содержание слайда: Solution: Contain more than 505 ml
P(x > 505) ?
1. Draw the curve so you see which probability area we are looking for.
2. Z –score = = 0.5 = P(z < .5) = .6915
3. P(z 0.5) = 1 - . 6915= .3085
30.85 % of the shampoo bottles contain more than 505ml shampoo.
№16 слайд![Exercise Draw a graph of the](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img15.jpg)
Содержание слайда: Exercise:
Draw a graph of the below probabilities
and find the probability of z in the
standard normal table with = 0, =1
P ( z < + 1.05) =
P (z > -1.05 ) =
P (z < - 3.34) =
P (z > - 3.34) =
P (z > - 2.47) =
P (z < + 1.87) =
P (z > + 2.57) =
P ( z < - 0.32) =
№17 слайд![P z lt . . or . P z gt - . .](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img16.jpg)
Содержание слайда: P ( z < + 1.05) = 0.8531 or 85.31 %
P (z > -1.05 ) = 0.8531 or 85.31 %
P (z < - 3.34) = 1.0 – 0.9996 = 0.0004 or 0.04 %
P (z > - 3.34) = 0.9996 or 99.96 %
P (z > - 2.47) = 0.9932 or 99.32 %
P (z < + 1.87) = 0.9693 or 96.93 %
P (z > + 2.57) = 1.0 – 0.9949 = 0.0054 or 0.054 %
P( z < - 0.32) = 1.0 – 0.6255 = 0.3745 or 37. 45 %
№18 слайд![Haynes Construction Company](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img17.jpg)
Содержание слайда: Haynes Construction Company
Example
Builds three- and four-unit apartment buildings:
Total construction time follows a normal distribution
For triplexes, m = 100 days and = 20 days
Contract calls for completion in 125 days
Late completion will incur a severe penalty fee
Calculate the probability of completing in less than
125 days P(x <125)
№19 слайд![Haynes Construction Company](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img18.jpg)
Содержание слайда: Haynes Construction Company
Compute Z:
№20 слайд![Haynes Construction Company](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img19.jpg)
Содержание слайда: Haynes Construction Company
№21 слайд![Haynes Construction Company](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img20.jpg)
Содержание слайда: Haynes Construction Company
What is the probability that the company will not finish in 125 days and therefore will have to pay a penalty?
m = 100 days and = 20 days
P(z > 1.25) ?
№22 слайд![Haynes Construction Company](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img21.jpg)
Содержание слайда: Haynes Construction Company
What is the probability that the company will not finish in 125 days and therefore will have to pay a penalty?
№23 слайд![Haynes Construction Company](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img22.jpg)
Содержание слайда: Haynes Construction Company
If finished in 75 days or less, Haynes will get a bonus of $5,000
What is the probability of a bonus? P ( x < 75)
№24 слайд![Haynes Construction Company](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img23.jpg)
Содержание слайда: Haynes Construction Company
If finished in 75 days or less, bonus = $5,000
Probability of bonus? P ( x < 75)
№25 слайд![Haynes Construction Company](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img24.jpg)
Содержание слайда: Haynes Construction Company
№26 слайд![Haynes Construction Company](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img25.jpg)
Содержание слайда: Haynes Construction Company
Probability of completing between 110 and 125 days?
№27 слайд![Haynes Construction Company](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img26.jpg)
Содержание слайда: Haynes Construction Company
Probability of completing between 110 and 125 days?
№28 слайд![Haynes Construction Company](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img27.jpg)
Содержание слайда: Haynes Construction Company
Probability of completing between 110 and 125 days?
№29 слайд![Calculation procedure to find](/documents_6/e5fd75817c6ecf2d34e6cf69cd597e18/img28.jpg)
Содержание слайда: Calculation procedure to find the probability of the
area under the normal curve:
1. First draw the normal curve for the problem, to understand what area under the curve we are looking for.
2. Transform x-values to the standardized random variable, z
3. Use the standardized normal distribution table to find the probability of the calculated z-value