You are a marketing manager for a manufacturer of nonperishable products sold in grocery stores. In this role, you need to make various decisions about how much marketing/advertising

Parameter Estimation and Generalizability.html

Parameter Estimation and Generalizability

The most common goal of data analysis is to estimate population parameters of interest from sample data. The parameters of interest are closely related to the decision criteria. Consider the following example.

You are a marketing manager for a brand of ready-to-eat cereal. A part of your responsibilities and one of the most important annual tasks is the setting of your promotional budget. In the budget, you need to allocate your resources among the available marketing-mix vehicles such as general advertising, direct mail, web marketing, and trade promotions. The key to being able to maximize the allocation of financial and other resources among the different vehicles is the assessment of relative productivity. Productivity is measured by incremental sales or profitability of each of the channels. In other words, for each dollar spent on a particular marketing approach, what is the return to the organization (in terms of additional revenues)? How would you arrive at reliable estimates of a marketing vehicle's specific impact?

First, it is important to keep in mind that not all of those vehicles have the same purpose. For example, the goal of general advertising might be to create or enhance brand awareness, while the goal of direct mail might be to stimulate near-term sales of a particular product. The most commonly used measure of the expected impact of general advertising is the gross rating points, which captures the exposure of the brand name generated by the advertising campaign. The success of direct mail campaigns is measured by the rate of response to the flyers (meaning product purchases). Hence, two different data sources would be used to estimate the impact on the population parameters of the different marketing vehicles.

Suppose after having completed your task of promotional budget allocation for your brand, you need to do the same for a sister brand (also ready-to-eat cereal, but selling under a different label). Would you be able to use the same promotional impact parameters you estimated for Brand A in conducting the same activity for Brand B? Are the parameter estimates you computed using Brand A's data generalizable for Brand B? The answer depends on a number of factors, including product similarity, purchaser profile, and prior brand-specific spending allocations. In this example, it is doubtful that estimates derived from Brand A data could be generalized to Brand B. However, there are also cases where parameter estimates are generalizable.

Population vs. Sample.html

Population vs. Sample

The relationship between the population and a sample is one of the key determinants of the quality of insights used in business. If you are interested in conclusions that stretch beyond the available sample, it is important to ensure that the sample closely resembles the population from which it came. The resemblance should be in terms of the factors that are used to describe the sample. For example, if you are interested in an analysis of car purchasing behaviors, both the sample and the population should be similar in vehicle purchase behaviors and the key demographics and other factors influencing that behavior.

In business analysis, you will rarely work with population data, primarily because data on entire populations are too voluminous or expensive to acquire. Nonetheless, your interest is in reaching outside of the limitation of the sample to generalize about the underlying population. For example, you conduct a marketing research study by using relatively small samples of current or potential customers but are ultimately interested in generalizing your conclusions to the population of all current and potential customers. Hence, you need to consider the sample–population relationship.

What if the entire population is available to you for sampling? Would that obviate the need for sampling? Probably not. In order to estimate reliable statistical parameters (discussed in the next lecture) a sample from the population is needed for a couple of reasons. First, the population may contain outlying observations (i.e., extremely atypical values), which might distort measures of parameter values and which typically need to be evaluated. Second, not all cases in the population might be of interest.

To illustrate the difference between a population and a sample taken from that population, consider the frequent shopper card that many grocery stores and drugstores offer.  Through the use of these cards, companies have records on all customers who registered for one, but often are only interested in current or active card holders. It is important to keep in mind that statistical analyses are geared toward decision-guiding insights and the analytical dataset needs to be adapted to the informational needs.  If someone registered for a frequent shopper card, but never used it, that member of the population may not be of interest to the organization. 

Nature of Distributions.html

Nature of Distributions

Even when both a measure of central tendency (like the mean) and a measure of variability (like the range) of the population are provided, the true nature of the distribution may not be known from just those two values. For example, the average value that one can expect to roll with a die (one of a pair of dice) is 3.5. Similarly, a daycare may have a policy of providing services only for children over the age of one and before the child must begin school (at the next start of a school year after they have reached the age of six). Such a daycare facility might very well have an average age of children in attendance that is 3.5. In both cases, the possible values (numbers to come up on the die and the age of an individual child) can be in the range of 1 to 6, but the distribution will likely look quite different.  All sides of the die have an equal probability when the die is rolled, but it is unlikely that the daycare facility has exactly the same number of children of each age.  Even for children whose age is the same, one child may have experienced a birthday last week, while another is expecting another birthday next week.  Thus, even though they are both five years old, their true age is somewhat different.

Many different distributions will be described throughout this course. Familiarizing yourself with the shapes of the distributions that are presented can help you to develop a more intuitive feel as to whether the answers to statistical questions are reasonable. For example, If the average age of the children at the daycare facility was calculated to be 17 years, you might suspect that something was wrong with those calculations.

If, after rolling a die many times, you find that 90% of the time the die lands on a value of one, you might suspect that someone has given you a die that does not fairly provide equal probability of any particular value in the range (one through six) being observed. Thus, knowing what to expect is valuable in determining whether or not your answer is logical. 

Keep in mind that no matter what distribution you are considering, all possibilities that you are considering will constitute 100% of the observations that you will see.  When you begin to learn about probability next week, you will come to realize that, no matter what distribution is under discussion, the probabilities of all observations always add up to a value of exactly one (1.0).  

Additional Materials

Relationships Between Variables Linear or Nonlinear?

Discrete Distributions.html

Discrete Distributions

Discrete distributions are collections of data in which all values are integers.  For example, how many people take this class each session is an example of a discrete distribution.  There may be 10, and there may 15, but there are not 12.3.  

The Binomial Distribution is useful for analyzing random discrete experiments in which there are exactly two possible outcomes for each experiment (hence, the use of the prefix "bi-" in the name of the distribution). For example, each time a coin is flipped there are exactly two possible outcomes that are possible, a "heads" or a "tails" (ignoring the extremely unlikely event of the coin actually landing on its edge when it is flipped). Thus, the Binomial Distribution is useful for analyzing outcomes of experiments in which there are two possible outcomes.

The Poisson Distribution is useful for analyzing random discrete experiments in which there are multiple possible outcomes for each experiment. For example, the number of pieces of mail that are delivered to your home each day is not limited to two possible values (as was the case with the Binomial Distribution). Rather, there could be zero pieces of mail, one piece of mail, two pieces of mail, etc., delivered on a particular day. The Poisson Distribution is used to treat such cases. Obviously, it is extremely unlikely (though not impossible) that you will receive 123 pieces of mail on a particular day. Thus, the Poisson Distribution is useful for analyzing outcomes of experiments that have multiple possible outcomes.

The Hypergeometric Distribution is useful for analyzing random discrete experiments in which outcomes are not necessarily independent from one another. This distribution is useful for considering experiments in which probabilities do not remain constant from one experiment to the next. 

Going back to our example of flipping a coin, each time the coin is flipped, we can assume that there is a 50% chance that the coin will land on "heads" on the next flip and a 50% chance that the coin will land on "tails" on the next flip. Another type of binomial experiment is to draw a single card from a deck of regular playing cards (52 cards in total) and determine whether the card drawn is one of the 13 cards with diamonds printed on it. In this case, there is a 25% chance of a diamond card being drawn, and a 75% chance of a card with some other symbol being drawn.  However, if we leave the card that is drawn out of the deck and draw a second card, the 75%/25% probabilities no longer hold. It is in these types of cases (where the results of one experiment are not independent of previous experiments) that the Hypergeometric Distribution holds and is useful for analysis.