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bstatsmodule5slides.pptxModule 5 continuous random variables
Master for Business Statistics
Dane McGuckian
Topics
5.1 Continuous Random Variables
5.2 The Normal Distribution
5.3 Applications of the Normal Distribution
5.4 Normal Approximation to the Binomial Distribution
5.1
Continuous Random Variables
Continuous Random Variables
Continuous random variables usually result from measuring something like a distance, a weight, a length of time, a volume, or some other similar quantity.
Because they can take on any value inside a particular interval, there are an infinite, uncountable number of possible values for any continuous random variable.
For this reason, when working with a continuous random variable, we will discuss the probability that the random variable is within some specified range.
Continuous Random Variables
Example: A fast-food restaurant manager tracks the length of time his customers wait or their orders.
The random variable is continuous because it consists of measured lengths of time.
We could consider the probability that a customer waits more that five minutes for his or her order, less than five minutes, between four and five minutes, or some other suitable interval of time.
But the probability that a customer waits exactly five minutes for the order is zero.
A consequence of continuous random variables having an infinite, uncountable set of possible set of values is that the probability of any continuous random variable equaling a specific value is always zero.
Continuous Random Variables
The probability distribution for a continuous random variable is usually represented by a function called the probability density function (pdf).
These functions produce smooth curves when graphed, and probability for the random variable is defined as the area under the curve between any two specified points.
Discrete versus Continuous Random Variables
It is common to discuss the probability that a discrete random variable takes on a specific value, but because a continuous random variable has an infinite number of possible values in a particular range, we do not typically discuss the probability that a continuous random variable takes on a specific value.
Continuous probability distribution
Discrete probability distribution
The Area under a Continuous Probability Function
In a continuous probability distribution, the probability than an event, x, is between two numbers is represented by an area, A.
The area under the curve represents all the possible probabilities that can occur from negative infinity to positive infinity.
Because the total probability for all continuous probability distributions is one, the area under the curve must also be one.
Continuous Uniform Distribution
A continuous random variable has a uniform distribution if the graph of its probability distribution is rectangular in shape and can be completely defined by its minimum and maximum values.
Like all continuous distributions, the total area under the graph of a uniform distribution is equal to one, and there is a direct relationship between the area under the curve between two specific points and the probability of the random variable assuming a value between those two points.
The mean for the uniform distribution is
Continuous Uniform Distribution
The standard deviation for the uniform distribution is
where:
is the minimum value for the distribution
is the maximum value for the distribution.
For any uniform distribution, there is a uniform height to its curve.
The height of the uniform distribution for any value such that is given by .
For any value outside of the interval the height of the curve is zero.
Continuous Uniform Distribution
Since the shape of the uniform curve is rectangular and probability corresponds to area under the curve, the probability that for a uniformly distributed random variable defined by the interval is
when .
The Probability of a Uniform Distribution
For a uniform distribution defined on the interval
when .
Probabilities for the Uniform Distribution
The amount of time it takes an accountant to prepare tax returns for her clients is uniformly distributed over the interval between 15 minutes and 60 minutes. What is the probability that she will finish a tax return in 40 minutes or less?
Here a = c = 15; b = 60, d = 40. So
Thus there is a 55.6% probability that she will finish a tax return in 40 minutes or less.
5.2 The Normal Distribution
The Normal Distribution
The normal distribution is a continuous distribution that appears in many applications
Many natural phenomena can be modeled using the normal probability density function.
The formula for the normal curve is
where
Note: and
The Normal Distribution
Notice that the formula for the normal distribution contains the symbols and , which represent the mean and the standard deviation respectively.
The mean determines the location of the curve on the number line, and the standard deviation determines the width or spread of the distribution.
The values of these parameters depend on the population being studied.
For this reason, the formula on the previous slide actually represents a family of normal distributions, not just one curve.
Example: The heights of men have a normal distribution, and men have mean height of 69 inches.
The heights if women are also normally distributed, but the mean for women’s heights is 64 inches.
The two curves also have different standard deviations.
The Normal Distribution
In the illustration below, the two normal curves have different means and different standard deviations.
The difference in the shape of the two curves is a result of the curves having different standard deviations.
The difference in their position on the number line is due to their different means.
The taller and the narrower curve belongs to the distribution with the smaller standard deviation.
The Shape of the Normal Distribution
The graph of the normal probability density function is bell-shaped, but there is not just one normal curve.
There is no limit to the different possible combinations of and , so there are an infinite number of different normal curves.
The particular scale and location of a normal distribution will depend on the distribution’s specific mean and standard deviation.
However, all normal curves are bell-shaped, and they are always perfectly symmetric around their mean.
The Shape of the Normal Distribution
The graph of the normal probability distribution function is bell-shaped and perfectly symmetric around its mean.
This indicates that the left side of the normal distribution is a perfect mirror image of the right side of the distribution.
Since the total area under all normal curves is 1.00 and all normal curves are symmetric around their mean, half of the area (0.50) is below the mean, and half of the area (0.50) is above the mean.
This is very useful information.
Example: If women’s heights are normally distributed and the average height for women is 64 inches, we can say with certainty that half of all women are shorter than 64 inches.
Of course that also implies half of all women are taller than 64 inches.
The Normal Distribution
There is not just one normal curve but there are an unlimited number of normal curves.
Example: Human height is normally distributed, but the heights of men and women form different normal distributions.
IQ scores are also normally distributed, but those scores form a different normal curve than the ones formed by male and female heights.
The list of examples is endless, so when we speak of the normal distribution, we are referring to a family of curves that have the same underlying structure.
The mean and standard deviation, and , allow a single probability density function to produce a family of normal distributions.
Converting Normal Random Variables into Standard Normal Random Variables
When working with normal random variables, we have a need to find areas or probabilities, but the probability density function for the normal distribution is mathematically difficult to work with.
For this reason, when solving problems involving a normally distributed random variable, it would be very helpful to have a table of probabilities for the normal curve.
However, there isn’t just one normal curve to tabulate probabilities for. Because there is no limit to the different possible combinations of and , there are an infinite number of different normal curves. Therefore, we would need an infinite number of normal probability tables to handle every possible application of the normal curve.
Fortunately, there is a way to work around this difficulty.
Converting Normal Random Variables into Standard Normal Random Variables
It is possible to convert any normal random variable with mean () and standard deviation () into a standard normal random variable.
A standard normal random variable is a normally distributed random variable that has a mean equal to zero ( = 0) and a standard deviation equal to one ( = 1).
To convert a normal random variable () into a standard normal random variable (), we use the following formula:
where
is the value of a measurement (or observation) taken from a normally distributed population
is the mean of the distribution for
is the standard deviation of the distribution for
is the standard normal value
Parameters of the Standard Normal Distribution
A standard normal random variable is a normally distributed random variable that has a mean equal to zero ( = 0) and a standard deviation equal to one ( = 1).
Because the standard normal distribution has a mean of zero and a standard deviation of one, values equate to the number of standard deviations above (or below) the mean.
Example: A standard normal value of 1 is the same as one standard deviation above average.
By using a standard normal probability table, it is possible to find the probability that a standard normal value falls between any two points on the –axis.
Z tables and Finding the Areas Under the Standard Normal Curve Between the Mean and a Value
Use the standard normal curve to find
P(
, because
the normal curve is symmetric.
= 0.4429 from the table.
Areas Under the Standard Normal Curve Inside an Interval Surrounding the Mean
Use the standard normal curve to find
because of symmetry
, from the normal table
= 0.7597
Areas Under the Standard Normal Curve between a Positive Z Value and Infinity
Use the standard normal curve to find
, as the total area to the left of the mean (0) is 0.5
, from the table
Areas Under the Standard Normal Curve between Two Values on the Same Side of the Mean
Use the standard normal curve to find
Areas Under the Standard Normal Curve between a Negative Z Value and Infinity
Use the standard normal curve to find
, as the total area to the left of the mean (0) is 0.50.
, from the table
5.3
Applications of the Normal Distribution
The Probability that a Non-Standard Normal Random Variable is Greater than an Above-Average Value
The time it takes a computer chip manufacturer to produce a single chip is normally distributed with a mean of 18.0 seconds and a standard deviation of 1.2 seconds. Find the probability that a chip will take longer than 19.8 seconds to produce.
The Probability that a Non-Standard Normal Random Variable is Less than an Above-Average Value
A large investment bank in Miami released a report on the starting salary offers it made to MBA graduates. The salaries are normally distributed with a mean of $89,200 and a standard deviation of $2,100. Find the probability that a randomly selected MBA graduate was offered a starting salary of less than $92,000.
The Probability that a Non-Standard Normal Random Variable is Between Two Values that Surround the Mean
The containers on a mega-cargo ship in the port of Los Angeles have weights that are normally distributed with a mean of 55,600 pounds and a standard deviation of 2,800 pounds. What is the probability that a randomly selected container from the ship weighs between 53,123 pounds and 60,123 pounds?
The Probability that a Non-Standard Normal Random Variable is Between Two Values that are on the Same Side of the Curve
A manufacturer produces gears for use in an engine’s transmission that have a mean diameter of 10.00mm and a standard deviation of 0.03mm. The lengths of these diameters have a normal distribution. Find the probability that a randomly selected gear has a diameter between 9.94mm and a 9.96mm.
The Value Corresponding to an Upper Percentile of the Normal Distribution
A company in California is concerned about the length of time that its employees spend commuting to work. The one-way commute times for its employees are normally distributed with a mean of 32.1 minutes and a standard deviation of 5.3 minutes. What is the commute time that separates the longest 20% of commutes from the rest?
Here we will work in the “reverse” direction – from % Z (from table) X (using formula).
The z-value such that
is 0.84 (closest
probability in the table being 0.2995.
Now,
So 36.6 minutes is the commute time that separates the longest 20% of commutes from the rest.
The Value Corresponding to a Lower Percentile of the Normal Distribution
A financial services company gives an analytical reasoning test to all job applicants. The completion times for the test are normally distributed with a mean of 50.40 minutes and a standard deviation of 3.10 minutes. What completion time separates the fastest 6% of applicants from the others?
We again work in the “reverse”
direction. Here the z-value such
that
is -1.555 (closest
probability value is 0.4394).
Now
So 45.58 minutes is the completion time that separates the fastest 6% of applicants from the others.
5.4
Normal Approximation to the Binomial Distribution
Using the Normal Distribution to Approximate the Binomial Distribution
When using the normal curve to estimate a binomial probability distribution, we must check two things to confirm the fit is reasonably good:
If either of these is not true, we need to find a different method of approximation.
The Use of the Continuity Correction Factor
Continuity correction is used when using the normal approximation to binomial probability.
Example: The rectangle for x=2 actually goes from 1.5 to 2.5 on the normal distribution.
Therefore we need to add or subtract that extra 0.5 when we are looking at the probability that x is less than or greater than 2.
Know the Reason for the Use of the Continuity Correction Factor
A marketing firm for the movie industry reports that the average film is 128 minutes with a standard deviation of 15 minutes. Assuming these film durations have a bell-shaped distribution, what percent of films have a duration between 158 minutes and 173 minutes?
The area marked in red is required, which is given by:
49.85% - 47.5% = 2.35%
Thus, 2.35% of films have a duration between 158 minutes and 173 minutes.
The Use of the Continuity Correction Factor
Based on prior experience, a car dealership has a 45% chance of selling an extended warranty with each used car that is sold. We want to use the normal approximation to the binomial distribution to find the probability of selling 25 or less extended warranties when 60 cars are sold. Using continuity correction, state the appropriate probability that will need to be found on the normal curve.
Here X is the number of warranties to be sold, so X = 25 or less.
n = 60, p = 0.45, and 1-p = q =0.55.
So, on
the bell-shaped curve (normal) is the
probability that will need to be found.
The Normal Distribution and the Probability that a Binomial Random Variable is Greater than a Value
Thirty percent of visitors to a local toy retailer will make a purchase before exiting the store. Use the normal approximation for binomial probability to determine the probability that more than 50 visitors out of 200 will make a purchase.
Here X is the number of visitors who make a purchase, so X = more than 50. Also n = 200, p = 0.30, and 1-p = q =0.70. It is a binomial distribution because a customer will either make a purchase or not.
Mean = = 200.0.30 = 60
Using the continuity correction factor, we have to find because the problem states “more that 50”. The z-score is:
So, (from table).
The Normal Distribution and the Probability that a Binomial Random Variable is Less than a Value
A small regional airline overbooks its flights because historically only 90% of the reservations will actually show up for the flight. If a flight has 100 available seats, the airline will typically sell 110 reservations for the flight. What is the probability that at most 95 people show up for a flight with 110 reservations?
Here X is the number of people who show up so X = at most 95. Also n = 110, p = 0.90, and 1-p = q =0.10. It is a binomial distribution because a person will either show up for the flight or not.
Mean = n.p = 110.0.90 = 99
Using the continuity correction factor, we have to find because the problem states “at most 95”. The z-score is:
So, (from table).
The Normal Distribution and the Probability that a Binomial Random Variable is Between Two Values
A small regional airline overbooks its flights because historically only 90% of the reservations will actually show up for the flight. If a flight has 100 available seats, the airline will typically sell 110 reservations for the flight. Use the normal approximation for binomial probability to determine the probability that between 100 and 107 people (inclusive) show up for a flight with 110 reservations?
Here X is the number of people who show up so X = between 100 and 107 or [100,107]. Also n = 110, p = 0.90, and 1-p = q =0.10.
Mean = = 110.0.90 = 99
Using the continuity correction factor, we have to find because 100 and 107 are both included”. The z-scores are:
So, (from table).
