# 6.2 Using the normal distribution  (Page 4/25)

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## References

“Naegele’s rule.” Wikipedia. Available online at http://en.wikipedia.org/wiki/Naegele's_rule (accessed May 14, 2013).

“403: NUMMI.” Chicago Public Media&Ira Glass, 2013. Available online at http://www.thisamericanlife.org/radio-archives/episode/403/nummi (accessed May 14, 2013).

“Scratch-Off Lottery Ticket Playing Tips.” WinAtTheLottery.com, 2013. Available online at http://www.winatthelottery.com/public/department40.cfm (accessed May 14, 2013).

“Smart Phone Users, By The Numbers.” Visual.ly, 2013. Available online at http://visual.ly/smart-phone-users-numbers (accessed May 14, 2013).

## Chapter review

The normal distribution, which is continuous, is the most important of all the probability distributions. Its graph is bell-shaped. This bell-shaped curve is used in almost all disciplines. Since it is a continuous distribution, the total area under the curve is one. The parameters of the normal are the mean µ and the standard deviation σ . A special normal distribution, called the standard normal distribution is the distribution of z -scores. Its mean is zero, and its standard deviation is one.

## Formula review

Normal Distribution: X ~ N ( µ , σ ) where µ is the mean and σ is the standard deviation.

Standard Normal Distribution: Z ~ N (0, 1).

Calculator function for probability: normalcdf (lower x value of the area, upper x value of the area, mean, standard deviation)

Calculator function for the k th percentile: k = invNorm (area to the left of k , mean, standard deviation)

How would you represent the area to the left of one in a probability statement?

P ( x <1)

What is the area to the right of one?

Is P ( x <1) equal to P ( x ≤ 1)? Why?

Yes, because they are the same in a continuous distribution: P ( x = 1) = 0

How would you represent the area to the left of three in a probability statement?

What is the area to the right of three?

1 – P ( x <3) or P ( x >3)

If the area to the left of x in a normal distribution is 0.123, what is the area to the right of x ?

If the area to the right of x in a normal distribution is 0.543, what is the area to the left of x ?

1 – 0.543 = 0.457

Use the following information to answer the next four exercises:

X ~ N (54, 8)

Find the probability that x >56.

Find the probability that x <30.

0.0013

Find the 80 th percentile.

Find the 60 th percentile.

56.03

X ~ N (6, 2)

Find the probability that x is between three and nine.

X ~ N (–3, 4)

Find the probability that x is between one and four.

0.1186

X ~ N (4, 5)

Find the maximum of x in the bottom quartile.

Use the following information to answer the next three exercise: The life of Sunshine CD players is normally distributed with a mean of 4.1 years and a standard deviation of 1.3 years. A CD player is guaranteed for three years. We are interested in the length of time a CD player lasts. Find the probability that a CD player will break down during the guarantee period.

1. Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
2. P (0< x <____________) = ___________ (Use zero for the minimum value of x .)
1. Check student’s solution.
2. 3, 0.1979

Find the probability that a CD player will last between 2.8 and six years.

1. Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
2. P (__________< x <__________) = __________

Find the 70 th percentile of the distribution for the time a CD player lasts.

1. Sketch the situation. Label and scale the axes. Shade the region corresponding to the lower 70%.
2. P ( x < k ) = __________ Therefore, k = _________
1. Check student’s solution.
2. 0.70, 4.78 years

a. l. bowley definition
what are events in statistics
Like a roll of a dice! Or a coin toss. Or a gender reveal party!
what is statistics
can anyone explain it better for me
the science of statistics deal with the collection, analysis, interpretation and presentation of data
saquib
I am also studying statistics
saquib
Correlation regression, explain it to me in short.
guillio
correlation is used to find relationship between two and dependent ), regression used for predicting the future by analyzing past data
Arun
correlation is used to find relationship between two variables
Arun
dependent and independent eg. profit is dependent on sales
Arun
Statistics has been designed as the mathematical science of making decisions and drawing conclusions from data in situations of uncertainty. It includes the designings of experiments, collection, organization, summarization snd interpretation of numerical data.
Aliya
excellent Aliya..... good...Arun....
IRFAN
The degree or strength of relationship(interdependence) between the variables is called "correlation ". Examples: heights and weights of children, ages of husbands and ages of wives at the time of their marriages, marks of students in mathematics and in statistics.
Aliya
The dependence of one variable (dependent variable) one one or more independent variables ( independent variables) is called "regression ".
Aliya
simply regression and multiple regression are the types of regression.
Aliya
IRFAN HAIDER thanks
Aliya
hi
nabil
I need help with a math problem
nabil
shoot
umair
9. The scatterplot below relates wine consumption (in liters of alcohol from wine per person per year) and death rate from heart disease (in deaths per 100,000 people) for 19 developed countries.
nabil
For questions e. and f. use the equation of the Least-Square Regression LSR line is: y = −22.97x+260.56 e. Circle the correct choice and fill in the blank in the following statement: As wine consumption increases by 1 liter of alcohol per person per year, the predicted death
nabil
Rate from heart disease increases/decreases by ______deaths per ________people.
nabil
is a scientific study of collection analysis interpretation and also presenting it by researchers.
Murtala
frequency distribution
noun STATISTICS a mathematical function showing the number of instances in which a variable takes each of its possible values.
Robin
ok
Common language-- taking a bunch of information and seeing if it is related or not to other info
Mandy
Does standard deviation have measuring unit?
Mohamed
yes, the measuring unit of the data you are looking at, for example centimetres for height.
Emma
thanks
Mohamed
is that easy to plot a graph between three axis?
Mohamed
yes we can but we do not have that much effective tools. If the graph is normal or less complicated then it is plotted effectively otherwise it will give you nightmare.
umair
whats the difference between discrete and contineous data
umar
Discrete variables are variables that can assume finite number of values. Continuous variables are variables that can assume infinite number of values
Mike
i will give you an example: {0,4,84} it contains discrete or limited values like it can also contain boolean values{true,false} or {0,1} and continuous are like {1,2,3,4,5......} , {0,0.1,0.2,0.3,0.4...........}
umair
a no. of values which are countable are called discrete variables on the other hand, a no. of values which are not countable are called continuous variables
Aliya
Yup, I would like to support Mr.Umair's argument by saying that it can only apply if we have a 3-D graph,otherwise a plane graph will not apply at all
festus
Aliya and Mike thnks to both of you ❤❤
umar
what's variance
what's case control study?
Shakilla
hi
Noman
?
Sulaiman
what is covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.[1] If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values, (i.e., the variables tend to show simila
Robin
Economics department, faculty of social sciences, NOUN. You are required to calculate: the covariance and State whether the covariance is positive or negative. (11½ marks) Observation E D 1 15 17.24 2 16 15.00 3 8 14.91 4 6 4.50 5 15 18.00 6 12 6.29 7 12 19.23 8 18 18.69 9 12 7.21 10 20 4
Florence
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
Robin
what is the purpose of statistics and why it is important that statistics to be a solo and one complete field?
to organize,analyze and interpret information in order to make decision
Berema
what is noun?
so simple. the name of any person,place or thing.
Edu-info
Using the Chi-square test, two coins were flipped a hundred times. What will be the chances of getting a head and getting a tale? Given observed values is 62 heads and 38 tails. Expected value is 50 heads, 50 tails. Is the difference due to chance or a significant error? a. Draw your hypothesis
how can I win
what is difference between the blocking and confounding
how do you get 2/50 ?
can you explained it for me
korankye
an easier definition of inferential statistics
Inferential statistics makes inferences and predictions about a population based on a sample of data taken from the population in question.
Rukhsana
Inferential statistics helps you to extract insights from a random sample data which then helps you to use specific predictive Modeling/machine learning technic to predict or forecast.
Manish
what is stemplot? can anyone explain?
Javokhirbek
what is statistics
what is collection of data
ernest
no collection data was provided just the mean =14
Leticia
sd=14 describe the position of score to the mean how many points below or above z=1.00 z=1.50
Leticia
I have this sample score 14 18 12 22 14 22 21 20 13 26 13 26 16 21 they want me to.compute the z- score of x= 15 ×=40 and x=9?
Leticia
how do you understand that it is the mean?
Kenedy
fact and figure
hira
factors to consider when using secondary data