Lecture 7

 To understand what a Normal Distribution is  To know how to use the Normal Distribution table  To compute probabilities of events by using the Normal Distribution table 2

(a) Many distributions in biology, geography, economics, and demographic studies are approximately normal. (b) In industrial production, the mass, size and other features of mass-produced products will follow an approximately normal distribution. This is of fundamental importance in quality control. 3

(c) Sets of random errors are approximately normally distributed. This result was much studied by Gauss and the normal distribution is sometimes called the ‘Gaussian distribution’ or the ‘error curve’. (d) It can be used as an approximation to the binomial and Poisson distributions (under certain conditions) and to other distributions. 4

 A very important continuous probability distribution.  Many natural and economic phenomena tend to be approximately normally distributed. Examples: heights, weights, intelligent quotients, physically productions. 5

Normal Distribution Curve Median 6

 Symmetrical & bell-shaped.  It is uni-modal.  The mean, median & mode all lie at the centre of distribution.  The two tails never touch the horizontal axis, although they come close to it.  The area under the curve is 1. 7

 Since the normal distribution shows a distribution of continuous variables, we cannot pinpoint the probability of a particular point on the distribution.  We can, however, compute the probabilities of any intervals on the distribution. 8

 There is a unique relationship between the areas under a normal curve and its standard deviation, σ.  If we draw a vertical line at the centre of the curve, and measure 1 σ away from the curve on either side, then the area enclosed between the two vertical lines at the 1 σ and the curve is about 68% of the total area under the whole curve. 9

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 Statisticians can compute the area of any intervals under the curve in units of σ.  A table of such values has, in fact, been worked out.  As long as we can find out the distance between the centre of the curve and any point in terms of the units of σ, we can obtain the required area. 11

Example In a normal distribution which has a mean of 100 and a standard deviation of 20, what is the area between the centre of the curve and a variable, 130 on the horizontal axis. Solution Z = (130 – 100)/20 =1.5 Then we look for Z = 1.5 from table which corresponds to

The key to working with the Normal distribution is to transform the normal variable X into the standard normal variable Z by the operation Z = Where = Value of a random variable = mean of a normal distribution = standard deviation of the distribution so that the new mean and standard deviation is 0 and 1 respectively. 13

Example For a normal probability distribution with mean µ = 5 and σ =2, find the following probabilities a) P(X < 6.5) b) P(X > 6.5) c) P(4.5 < X < 6.5) d) P(6.5 < X < 8.0) 14

15 Example If x is normally distributed with μ = 20.0 and σ =4.0, determine the following: a) P( x ≥ 20.0) b) P(16.0 ≤ x ≤ 24.0)

16 Example It has been reported that the average hotel check-in time, from curbside to delivery of bags into the room, is 12.1 minutes. An Li has just left the cab that brought her to her hotel. Assuming a normal distribution with a standard deviation of 2.0 minutes, what is the probability that the time required for An Li and her bags to get to the room will be: 1. Greater than 14.1 minutes? 2. Between 10.1 and 14.1 minutes?

17 Explain Normal Distribution. Use the Standard Normal Distribtuon Table. Compute probabilities of events using the standard normal distribution table.