4 assignments in Business Statistics due dates listed within description.
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mod_7_lecture_notes.docxContinuous Probability Distribution
There are different types of distributions and the textbook distinguishes between a uniform distribution, with a rectangular form, and normal distributions, which have the curve that you are accustomed to seeing.
Calculating Z-Scores
The normal distribution is the most important distribution in statistics. Though there are innumerable normal distributions, any one distribution can be represented by the standard normal distribution, where the mean is 0 and the standard deviation is 1. Such a conversion allows us to estimate the relative frequencies of various population values by calculating areas under the normal curve that correspond to those frequencies. Any normal probability distribution can be converted to a standard normal probability distribution by subtracting the mean from each observation and dividing this difference by the standard deviation.
Let us say you scored 90% on an exam. If the class average and standard deviation (σ) can be determined, then z-scores may be considered as the number of standard deviations from the mean value for any given score.
If the class average is 75 and the standard deviation is determined to be 6, we calculate the z-score as:
Figure 1
This is exactly the same as counting the number of standard deviations between scores of 75 and 90. 1σ from the mean is a score of 81, 2σ is a score of 87, and 2.5σ represents a score of 90.
Any normally distributed set of raw population scores can be converted to z-scores using this method.
One fundamental application of this process is in the determination of probability – the likelihood that a given event will occur.
For example:
There are four balls in a bag: one red, one blue, one yellow, and one green. If you put your hand in and select one at random, the probability that you will select a specific color, say red, is 1 out 4, or 0.25, or a 25% chance. If the green ball is replaced with another red ball, then the probability of selecting a red ball is now 2/4 or 0.50 (a 50% chance). In terms of the proportion, the proportion of red balls in the bag is 2 out of 4.
The normal distribution also allows us to convert probability problems into manipulation of z-score problems.
For example:
The mean height of group is found to be μ = 68 inches; σ = 6 inches. We are asked to determine the probability that a person chosen at random is 6 feet tall or taller (6ft = 72 inches) and, thereby, determine what proportion of all adults are 72 inches tall or taller.
The z-score for a height of 72 inches is: (72 - 68) /6 = 0.67.
This represents the area under the normal curve between 68 inches, which is the average, and 72 inches. Given this information, we can now estimate the proportion of the normal distribution above a height of 72. This area represents the proportion of adults who are 72 inches tall or taller.
Using the table in appendix D, we can determine that the proportion of the distribution in the area beyond the z-score of 0.67 is 0.5000 - 0.2486 = 0.2514 or 25.14%.
This table may be manipulated to convert much more complex problems into z-score manipulation.
Please watch this Introduction to Z Scores Video for an explanation of z-scores.
You can follow along with an additional example of how to calculate a z-score by watching this Example of How to Calculate a Z Score Video .
The Empirical Rule
The Empirical Rule, which was introduced back in Chapter 3, tells us that 68% of the area under the curve in a normal distribution will be within ±1 ơ (standard deviations). 95% will be within ±2 and 98% will be within ±3 ơ.
Please watch this Empirical Rule Video for an overview of the empirical rule.
Normal Approximation as a Shortcut
Recall binomial distributions in Chapter 6. Onerous enough for smaller sample sizes, calculating for a larger sample size – 60 is the example in the text – would be extremely time consuming. A more efficient method is to use a normal (continuous) distribution as a substitute for the binomial. The characteristic that allows us to do that is the tendency of binomial distributions to approximate normal distributions as the number of values increases.
Continuity Correction Factor
Remember that when you are using a continuous distribution calculation, you need to adjust the result with the continuity correction factor to handle the discrepancy for a discrete distribution. Another way to think of this discrepancy is the difference between integers and incremental decimals between each integer.
Exponential Distributions
Closely related to the Poisson probability distribution, the exponential distributions provide a different perspective. In the example in the textbook, this calculation is useful for determining the time between arrivals when we would have used a Poisson calculation to determine expected number of arrivals in a given time period. For service industries, such as restaurants and pharmacies, these types of projections allow for estimating the proper level of staffing required to provide adequate customer service without wasting resources by overscheduling labor hours.
Another application in business is felt in maintaining manufacturing equipment. Scheduling production on a set of machines can be a tricky process. The presumption is that if a machine is scheduled to produce a product, it must be effectively functional. There is no inherent, built-in mechanism for downtime due to repair. This conditional probability of the waiting time between breakdowns is best modeled by exponential distribution. Therefore, exponential distributions are used extensively to estimate the time between events, otherwise known as waiting times.
These applications of the z-score show its incredible versatility. By standardizing the distribution, there is no concern over units or other individual characteristics specific to the original dataset. The results generated are universally applicable to the situation in a wide variety of circumstances.
