Means Tests for Small Samples

For meaningful statistical analysis, sample size must be sufficiently large to accurately reflect important characteristics of the overall population. Although it is typically desirable to have 30 or more sample observations, this is not always possible. Sometimes, managers must rely on very small samples of data, say n < 30. In such instances, the test statistic formula must be altered slightly.

If the population is normally distributed, the distribution around the small sample mean will be approximately normal. In this situation, the test statistic formula is written

s/wn

MANAGERIAL APPLICATION 3.2

Market Experiments on the Web

In pre-Internet days, companies spent huge amounts of time and money simply trying to measure perceptions about how well customer needs have been met by the firm's products. Now, companies can instantaneously review customer orders and see how well the company is actually satisfying customer needs. Early adopters of Internet-based customer delivery systems have learned (or relearned) a number of fundamental marketing concepts:

• Successful companies define value in terms of product attributes desired by the customer. In old-fashioned terminology, customers are always right.

• Customer value depends upon both physical and situational characteristics of products. What, how, and when are often equally important to the customer.

• Customer value perceptions are dynamic and can change rapidly over time.

The Internet is spawning a revolution in the way things are made and services are delivered. Companies as diverse as BMW, Dell Computer, Levi Strauss, Mattel, McGraw-Hill, and Wells Fargo are all embracing Internet technology as a means for learning and delivering pre cisely what consumers want. In fact, these and a growing list of companies are building customized products designed by millions of customers. Dell led the way by allowing customers to order computers assembled to exact specifications. Now, manufacturers are allowing customers to order computer-fitted apparel, like Levi's cut to fit your body. Men can stop worrying about why 37" pant waist sizes aren't offered; women can stop trying to figure out what the size "petite large" means. Just use the Internet to tell Eddie Bauer, Lands' End, or Levi's how to cut your own perfect fit. Using Internet technology, customers can also buy customized blends of vitamins, music compilations on CDs, and mortgage payment terms. Professors can also assign "textbooks" whose chapters are compiled from diverse material written by a variety of authors. This Internet-spawned revolution is just another step along the path of serving customer needs quicker, better, and cheaper.

See: Martha Francois, "We Need an Education Experiment," The Wall Street Journal Online, March 6, 2002 (http://online.wsj.com).

t statistic

Approximately normal test statistic degrees of freedom

Number of observations beyond the minimum required to calculate a statistic where X is the sample mean, ^ is the known mean of the population, s is the sample standard deviation, and n is sample size. A confidence interval for the true mean ^ can be calculated as X - i(s/Mn) to X + t(s/Mn) where t is from the t table in Appendix C for (n-1) degrees of freedom and the relevant confidence level.

This so-called t statistic is a test statistic that has an approximately normal distribution with a mean of zero and a standard deviation of one. The t statistic (or t value) is normally distributed for large samples, but is less so in the case of small samples. Like the z statistic, it describes the difference between the sample and population means in "standardized units," or by the number of sample standard deviations. Because the t statistic is only approximately normal, the rules of thumb of two standard deviations for the 95 percent confidence interval and three standard deviations for the 99 percent confidence interval hold only for large samples where n > 30. The "hurdle" or critical t value is adjusted upward when sample size is reduced. The amount of upward adjustment depends on the test statistic's degrees of freedom, or the number of observations beyond the absolute minimum required to calculate the statistic. Because at least two observations are necessary before a mean can be calculated, degrees of freedom for a means test are calculated as df = n - 1. The precise critical t value to use in a means test for very small sample sizes is obtained from a t table, such as that found in Appendix C. For example, when sample size is n = 10 observations, the critical t value for a means test with df = 10 - 1 = 9 is 2.262 at the a = 0.05 significance level, and 3.25 at the a = 0.01 significance level. The population mean is expected to be found within ± 2.262 standard deviations of the sample mean with 95 percent confidence, and within ± 3.25 standard deviations of the sample mean with 99 percent confidence.

To this point, measures of central tendency and measures of dispersion have been considered useful for describing populations and samples of data. These measures are very useful to managers who seek a detailed statistical profile of customer characteristics, cost experience, industry profits, and a host of other important economic variables. However, managers are often interested in the central tendency and dispersion of these data and in the extent to which these patterns can be described. For this reason, successful real-world managers devote significant effort to describing the causes and consequences of important economic relations.

regression analysis

Statistical method for describing XY relations

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