<< Chapter < Page Chapter >> Page >

The shaded area in the following graph indicates the area to the left of x . This area is represented by the probability P ( X < x ). Normal tables, computers, and calculators provide or calculate the probability P ( X < x ).

This is a normal distribution curve. A value, x, is labeled on the horizontal axis, X. A vertical line extends from point x to the curve, and the area under the curve to the left of x is shaded. The area of this shaded section represents the probability that a value of the variable is less than x.

The area to the right is then P ( X > x ) = 1 – P ( X < x ). Remember, P ( X < x ) = Area to the left of the vertical line through x . P ( X < x ) = 1 – P ( X < x ) = Area to the right of the vertical line through x . P ( X < x ) is the same as P ( X x ) and P ( X > x ) is the same as P ( X x ) for continuous distributions.

Calculations of probabilities

Probabilities are calculated using technology. There are instructions given as necessary for the TI-83+ and TI-84 calculators.

Note

To calculate the probability, use the probability tables provided in [link] without the use of technology. The tables include instructions for how to use them.

If the area to the left is 0.0228, then the area to the right is 1 – 0.0228 = 0.9772.

Try it

If the area to the left of x is 0.012, then what is the area to the right?

1 − 0.012 = 0.988

The final exam scores in a statistics class were normally distributed with a mean of 63 and a standard deviation of five.

a. Find the probability that a randomly selected student scored more than 65 on the exam.

a. Let X = a score on the final exam. X ~ N (63, 5), where μ = 63 and σ = 5

Draw a graph.

Then, find P ( x >65).

P ( x >65) = 0.3446

This is a normal distribution curve. The peak of the curve coincides with the point 63 on the horizontal axis. The point 65 is also labeled. A vertical line extends from point 65 to the curve. The probability area to the right of 65 is shaded; it is equal to 0.3446.

The probability that any student selected at random scores more than 65 is 0.3446.

Go into 2nd DISTR .
After pressing 2nd DISTR , press 2:normalcdf .

The syntax for the instructions are as follows:

normalcdf(lower value, upper value, mean, standard deviation) For this problem: normalcdf(65,1E99,63,5) = 0.3446. You get 1E99 (= 10 99 ) by pressing 1 , the EE key (a 2nd key) and then 99 . Or, you can enter 10^99 instead. The number 10 99 is way out in the right tail of the normal curve. We are calculating the area between 65 and 10 99 . In some instances, the lower number of the area might be –1E99 (= –10 99 ). The number –10 99 is way out in the left tail of the normal curve.

Historical note

The TI probability program calculates a z -score and then the probability from the z -score. Before technology, the z -score was looked up in a standard normal probability table (because the math involved is too cumbersome) to find the probability. In this example, a standard normal table with area to the left of the z -score was used. You calculate the z -score and look up the area to the left. The probability is the area to the right.

z = 65  – 63 5 = 0.4

Area to the left is 0.6554.

P ( x >65) = P ( z >0.4) = 1 – 0.6554 = 0.3446

Calculate the z -score:

*Press 2nd Distr
*Press 3:invNorm (
*Enter the area to the left of z followed by )
*Press ENTER .
For this Example, the steps are
2nd Distr
3:invNorm (.6554) ENTER
The answer is 0.3999 which rounds to 0.4.

b. Find the probability that a randomly selected student scored less than 85.

b. Draw a graph.

Then find P ( x <85), and shade the graph.

Using a computer or calculator, find P ( x <85) = 1.

normalcdf(0,85,63,5) = 1 (rounds to one)

The probability that one student scores less than 85 is approximately one (or 100%).

c. Find the 90 th percentile (that is, find the score k that has 90% of the scores below k and 10% of the scores above k ).

c. Find the 90 th percentile. For each problem or part of a problem, draw a new graph. Draw the x -axis. Shade the area that corresponds to the 90 th percentile.

Let k = the 90 th percentile. The variable k is located on the x -axis. P ( x < k ) is the area to the left of k . The 90 th percentile k separates the exam scores into those that are the same or lower than k and those that are the same or higher. Ninety percent of the test scores are the same or lower than k , and ten percent are the same or higher. The variable k is often called a critical value .

k = 69.4

This is a normal distribution curve. The peak of the curve coincides with the point 63 on the horizontal axis. A point, k, is labeled to the right of 63. A vertical line extends from k to the curve. The area under the curve to the left of k is shaded. This represents the probability that x is less than k: P(x < k) = 0.90

The 90 th percentile is 69.4. This means that 90% of the test scores fall at or below 69.4 and 10% fall at or above. To get this answer on the calculator, follow this step:

Questions & Answers

what does it array
Cbdishakur Reply
what are the differences between monopoly and.oligopoly
Onome Reply
what are the difference between monopoly and oligopoly
Cbdishakur
The deference between Monopoly and Oligopoly: Monopoly means:A single-firm-Industry producing and selling a product having no close business and Oligopoly means:A market structure where a few sellers compete with each other and each controls a significant portion of market .
Basanta
so that the price-output policy one affects the other.
Basanta
what is economic
Emakpor Reply
what is economic
Cbdishakur
the word economic was derived from the Greek word oikos (a house)and mein(to manage) which in effect meant managing a household with the limited funds available 🙂.
Basanta
good excample about scarsity
hon
An Enquiry into the nature and causes of wealth Nations, this book clearly defined what economic is🙂🙂🙏🙏 thank you...
Basanta
good example about scarcity: money,time, energy, human or natural resources. Scarcity of resources implies that there supply is very much limited in relation to demand.
Basanta
equilibrium is a situation in which economic forces such as demand and supply are balanced and in the absence of external influences,the value of economic variables will not change
Onome Reply
hmnn
Emakpor
marginal cost and marginal revenue is equilibrium .
Kho
yessss
Basanta
what is equilibrium
Rodrice Reply
policy prescriptions for unemployment
Jeslyne Reply
Am working on it
Blacks
Study
Janelle
study
simeon
what are the factors effecting demand sedule
Kalimu Reply
we should talk about more important topics, you can search it on Google n u will find your answer we should try to focus on how we can improve our society using economics
shubham
so good night
hon
ways of improving human capital
kelly Reply
what is human capital
kelly
Capital can be defined as man made assets use in production .
Abdulai
What is the differences between central Bank And Commercial Bank ?. 2 for each
Abdulai
Two types of bank clearing house.
Abdulai
what are the most durable assets of a bank
Ngongang
What is Opportunity Cost?
Cephas Reply
may be defined as expression of cost in terms of forgone alternative.
Abdulai
Helloo, im new, can i get to know more?
Saniya Reply
You ask questions on any topics you find difficult.
Favour
What is opportunity cost?
Cephas
is price elasticity of demand the same as elasticity of demand
Favour Reply
not really
Victoria
hi
Gh
hello
Bhartendu
i hope everyone be ok
Gh
No
Hassan
please explain
Favour
No
William
explanations please
cleophas
price elasticity of demand is the reaction of customers /demand to price changes(increase or decrease) elasticity of demand is the reaction of prices brought about by the change in demand
Victoria
thank you
Favour
state the laws of demand and supply
William
dd: when price rises demand decreases whereas when price reduces dd rises ss: when ss rises the price rises and when ss decreases price also reduces. There is a positive relationship
Dhoonah
nice
Victoria
Draw a demand curve graph
William
though price elasticity and elasticity are used interchangeably, the demand can respond to income changes and prices of related goods as well.
Gurpalak
explain the difference between merit goods and public goods and show why it is possible for profit to be made in the supply of one of these types of good but not the other
Kavishek
Public goods are defined as products where, for any given output, consumption by additional consumers does not reduce the quantity consumed by existing consumers. Merit goods are, for example, education and to some extent the health-care. They are provided by state as "good for you".
ahmed
The ladies are doing much better than the men
Blacks
what happens when there is a shift in demand curve?
Favour
What is Specialization ? Explain in detail
Muhammad
any one ?
Muhammad
specialisation is a method of production whereby an entity focuses on the production of a limited scope of goods to gain a greater degree of efficiency.
Favour
It's ok
Muhammad
hello
Onome
yah
Abdulai
No. price elasticity of demand refers to the manna in which price of good demanded fluctuate mean while elasticity of demand explains the way consumer change in their willingness as they plan or purchase a good
Ngongang
diffirence between demand and supply
Bonny
what is economic
Seray Reply
It is a social science which studies human behavior as a relationship between ends and scarce which have alternative uses
Obeng
what is norminal wage
Demba Reply
is the wages measured in money as distinct from actual purchasing power
Favour
what is demand curve
Azeez Reply
this is a curve that slop downward from left to rich
Obeng
yes
Basanta
different between capital and wealth
Samuel Reply
Wealth refers to the amount of asset you have, while, capital is the amount of cash money you have with you now and willing to invest in any business.
Favour
Difference between extinct and extici spicies
Amanpreet Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Introduction to statistics i - stat 213 - university of calgary - ver2015revb. OpenStax CNX. Oct 21, 2015 Download for free at http://legacy.cnx.org/content/col11874/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Introduction to statistics i - stat 213 - university of calgary - ver2015revb' conversation and receive update notifications?

Ask