Copyright ©2011 Nelson Education Limited The Normal Probability Distribution CHAPTER 6

Copyright ©2011 Nelson Education Limited

The Normal Probability Distribution

CHAPTER 6


Properties of ContinuousProbability Distributions

• The area under the curve is equal to 1.• P(a x b) = area under the curve between a and b.

• There is no probability attached to any single value of x. That is, P(x = a) = 0.


Example

Suppose that a random variable X is uniformly distributed on the interval [0,3]. That is, its density is some constant c on [0,3], and is zero otherwise. What is the value of c?


The Normal Distribution

deviation. standard andmean population theare and

1416.3 7183.2

for 2

1)(

2

2

1

e

xexfx

• The shape and location of the normal curve changes as the mean and standard deviation change.

• The formula that generates the normal probability distribution is:

APPLETMY


Using standard normal Table The four digit probability in a particular row and column of Table 3 gives the area under the z curve to the left that particular value of z.

Area for z = 1.36


Finding Probabilities for the General Normal Random Variable

To find an area for a normal random variable x with mean m and standard deviation ,s standardize or rescale the interval in terms of z.

Find the appropriate area using Table 3.

To find an area for a normal random variable x with mean m and standard deviation ,s standardize or rescale the interval in terms of z.

Find the appropriate area using Table 3.

Example: x has a normal distribution with m = 5 and s = 2. Find P(x > 7).

1587.8413.1)1(

)2

57()7(

zP

zPxP

1 z


ExampleThe weights of packages of ground beef are normally distributed with mean 1Kg and standard deviation 0.15Kg. What is the probability that a randomly selected package weighs between 0.7 and 0.775 Kgs?

What is the weight of a package such that only 1% of all packages exceed this weight?


Sampling Distributions

CHAPTER 7


Introduction• Parameters are numerical descriptive measures for

populations.

– For the normal distribution, the location and shape are described by m and s.

– For a binomial distribution consisting of n trials, the location and shape are determined by p.

• Often the values of parameters that specify the exact form of a distribution are unknown.

• You must rely on the sample to learn about these parameters.


Sampling Distributions• Numerical descriptive measures calculated from the

sample are called statistics.• Statistics vary from sample to sample and hence are

random variables.• The probability distributions for statistics are called

sampling distributions.• In repeated sampling, they tell us what values of the

statistics can occur and how often each value occurs.


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X

Example:A population is made up of the numbers 3,5,2,1. You draw a sample of size n=3 without replacement and calculate the sample average. Find the distribution of Xbar, the sample average.


If random samples of n observations are drawn from any population with finite m and standard deviation s , then, when n is large, the sampling distribution of is approximately normal, with mean m and standard deviation . The approximation becomes more accurate as n becomes large.

If random samples of n observations are drawn from any population with finite m and standard deviation s , then, when n is large, the sampling distribution of is approximately normal, with mean m and standard deviation . The approximation becomes more accurate as n becomes large. €

X

n/

Central Limit Theorem

1. any population2. random sampling (independence)3. if start with normal, this is exact!4. how large does n have to be?


Many (most?) statistics used are sums or averages. The CLT gives us their sampling distributions.

Overall, we have that• The average of n measurements is approximately

normal with mean m and variance σ2/n.• The sum of n measurements is approximately

normal with mean nm and variance nσ2.• The sample proportion is approximately normal with

mean p and variance p(1-p)/n.• The binomial is approximately normal with mean np

and variance np(1-p).


How Large does n need to be?• If the original distribution is normal, then any

sample size will do.

• If the sample distribution is approximately normal, then even small n will work.

• If the sample distribution is skewed, then a larger n is needed (eg. n bigger than 30).


The CLT in the real-world:1. Check the random sampling – was it done

properly? Are your observations independent?

2. Look at a histogram of your data to see if it is skewed or symmetric. The more skewed the data, the less credible the approximation. Is your sample size big enough?

Rules of thumb:

a. In general, use n≥30 for skewed distributions

b. Use np, n(1-p)≥ 5 for the binomial/sample proportion.


ExampleSuppose X is a binomial random variable with n = 30 and p = .4. Use the normal approximation to find P(X 10).


ExampleA production line produces AA batteries with a reliability rate of 95%. A sample of n = 200 batteries is selected. Find the probability that at least 195 of the batteries work.


ExampleA bottler of soft drinks packages cans in six-packs. Suppose that the fill per can has an approximate normal distribution with a mean of 355 ml and a standard deviation of 5.91 ml. What is the probability that the total fill for a case is less than 347.79 ml?


ExampleThe soda bottler in the previous example claims that only 5% of the soda cans are under filled. A quality control technician randomly samples 200 cans of soda. What is the probability that more than 10% of the cans are under filled?

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Copyright ©2011 Nelson Education Limited The Normal Probability Distribution CHAPTER 6