STA1101 Normal Distribution and Continuous random variables
CONTINUOUS RANDOM VARIABLES
A random variable whose values are not countable is called a _CONTINUOUS RANDOM VARIABLE._
THE NORMAL DISTRIBUTION
The _NORMAL PROBABILITY DISTRIBUTION_ is given by a bell-shaped(symmetric) curve.
THE STANDARD NORMAL DISTRIBUTION
The normal distribution with
is called the _STANDARD NORMAL DISTRIBUTION._
Example 1: Find the area under the standard normal curve
between z = 0 and z = 1.95
from z = -2.17 to z = 0
Area to the right of z = 2.32
Area to the left of z = -1.54
Example 2a: Find the following probabilities for the standard normal curve.
Example 2b: Find the value of _k_ if
STANDARDIZING A NORMAL DISTRIBUTION
Example 3: Let x be a normal random variable with its mean equal to 40 and standard deviation equal to 5. Find the following probabilities for this normal distribution.
Example 4: Lengths of metal strips produced by a machine are normally distributed with mean length of 150cm and a standard deviation of 10cm.
Find the probability that the length of a randomly selected strip is
(a) shorter than 165cm,
(b) within 5cm of the mean.
Example 5: The time taken by the milkman to deliver to the High Street is normally distributed with a mean of 12 minutes and a standard deviation of 2 minutes. He delivers milk every day. Estimate the number of days during the year when he takes
(a) longer than 17 minutes,
(b) less than ten minutes,
(c) between 9 and 13 minutes.
Example 6: The height of female students at a particular college are normally distributed with a mean of 169cm and a standard deviation of 9 cm.
(a) Given that 80% of these female students have a height less than _h_ cm, find the value of _h._
(b) Given that 60% of these female students have a height greater than _s_ cm, find the value of _s_.
Example 7: The marks of 500 candidates in an examination are normally distributed with a mean of 45 marks and a standard deviation of 20 marks.
(a) Given that the pass mark is 41, estimate the number of candidates who passed the examination.
(b) If 5% of the candidates obtain a distinction by scoring _x_ marks or more, estimate the value of _x_.
Example 8: The lengths of certain item follow a normal distribution with mean
cm and standard deviation 6 cm. It is known that 4.78% of the items have length greater than 82cm. Find the value of the mean
Example 9: The masses of boxes of oranges are normally distributed such that 30% of them are greater than 4.00kg and 20% are greater than 4.53 kg. Estimate the mean and standard deviation of the masses.
Normal probability distribution
1. The total area under the curve is 1 or 100%
2. The curve is symmetric about the mean.
3. Two tails of the curve extend indefinitely.
Standard deviation = � EMBED Equation.3 ���
Mean = � EMBED Equation.3 ���
z Values or z scores
The units marked on the horizontal axis of the standard normal curve are denoted by z and called the z values or z scores.
Converting an x value to a z value
For a normal random variable x, a particular value of x can be converted to its corresponding z value:
� EMBED Equation.3 ���
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