Multinomial, Hypergeometric, Poisson Distributions

Normal Distribution Dhon G. Dungca, PIE

Normal Distribution • Also known as the Gaussian Distribution • A distribution where the mean, median & mode are equal.

Normal Curve

Mean, Median & Mode

Characteristics of the Normal Distribution • The curve has a single peak. • It is bell-shaped. • The mean lies at the center of the distribution. The distribution is symmetrical about the mean. • Two two tails extend indefinitely in both directions coming closer and closer to the horizontal axis but never quite touching it.

Additional characteristics of the Normal Distribution • The curve has a single maximum at x=. • The curve is symmetrical about the vertical line x= . Thus, the height of the curve at some point say, x= + is the same as the height of the curve at x= -. • The curve is concave downward between x= - and x= +, and concave upward for value of x outside that interval.

Additional characteristics of the Normal Distribution • The total area under the curve is 1.00, and since the curve is symmetrical about x= , it follows that the area on either side of the vertical line x= is 0.50 • As x moves away on either side of the mean , the height of the curve decreases but remains non-negative for all real values of x.

Normal Distribution Formula Z=x- 

for the population

Z=x-x s

for the sample

Normal Distribution Formula • Where:

• Z = the number of standard deviations from x to the mean of this distribution • x = the value of the random variable •  or x = the mean of the distribution •  or s = the standard deviation

Areas under the Normal Curve

P(x) P = 0.50 Z=0

Example 1 A random variable has a normal distribution with the mean = 80 and the standard deviation of 4.8. What are the probabilities that this random variable will take on a value: a) Less than 87.2 b) Greater than 76.4 c) Between 81.2 and 86 0.9332, 0.7734, 0.2957

Example 2 Records show that in a certain hospital the distribution of the length of stay of its patients is normal with a mean of 10.5 days and a standard deviation of 2 days. a) What percentage of the patients stayed 8 days? b) What is the probability that a patient stays in the hospital between 9 and 11 days? 10.56%, 37.21%

Example 3 Given a normal distribution with  = 40 and  = 6, find the value of x that has a) 45% of the area to the left. b) 14% of the area to the right.

39.22, 46.48

Example 4 A certain type of storage battery lasts, on average, 3.0 years with a standard deviation of 0.5 year. Assuming that the battery lives are normally distributed, find the probability that a given battery will last less than 2.3 years. 0.0808

Example 5 In an industrial process the diameter of a ball bearing is an important component part. The buyer sets specifications on the diameter to be 3.0 ± 0.01 cm. The implication is that no part falling outside these specifications will be accepted. It is known that in the process the diameter of a ball bearing has a normal distribution with mean 3.0 and a standard deviation 0.005. On the average, how many manufactured ball bearings will be scrapped? 4.56%

Example 6 Gauges are used to reject all components where a certain dimension is not within the specification 1.50 ± d. It is known that this measurement is normally distributed with mean 1.50 and standard deviation 0.2. Determine the value d such that the specifications “cover” 95% of the measurements. 0.392

Example 7 The average grade for an exam is 74, and the standard deviation is 7. If 12% of the class are given A’s, and the grades are curved to follow a normal distribution, what is the lowest possible A and the highest possible B? lowest A = 83, highest B = 82