the normal probability distribution - yorku math and...

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Copyright ©2011 Nelson Education Limited

The Normal Probability Distribution


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


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

•  The formula that generates the normal probability distribution is:

APPLET MY


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 Copyright ©2011 Nelson Education Limited

 To find an area for a normal random variable x with mean µ and standard deviation σ, standardize or rescale the interval in terms of z.  Find the appropriate area using Table 3.

1 z

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Copyright ©2011 Nelson Education Limited Copyright ©2011 Nelson Education Limited

Sampling Distributions


•  Parameters are numerical descriptive measures for populations. – For the normal distribution, the location and shape

are described by µ and σ. – 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.



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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 µ and standard deviation σ , then, when n is large, the sampling distribution of is approximately normal, with mean µ and standard deviation . The approximation becomes more accurate as n becomes large.

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1.  any population 2.  random sampling (independence) 3.  if start with normal, this is exact! 4.  how large does n have to be?

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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 µ and variance σ2/n.

•  The sum of n measurements is approximately normal with mean nµ 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).


•  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).


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.


Copyright ©2011 Nelson Education Limited Copyright ©2011 Nelson Education Limited

A 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?

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the normal probability distribution - yorku math and...

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