The Scrapper Elevator Company has 20 sales representatives...

The Scrapper Elevator Company has 20 sales representatives who sell their product throughout the United States and Canada. The number of units sold by each representative is listed below. Assume these sales figures to be the population values. a. Draw a graph showing the population distribution. b. Compute the mean of the population. c. Select five random samples of 5 each. Compute the mean of each sample. Use the methods described in this chapter and Appendix $E$ to determine the items to be included in the sample. d. Compare the mean of the sampling distribution of the sample means to the population mean. Would you expect the two values to be about the same? e. Draw a histogram of the sample means. Do you notice a difference in the shape of the distribution of sample means compared to the shape of the population distribution?

The Scrapper Elevator Company has 20 sales representatives who sell their product throughout the United States and Canada. The number of units sold by each representative is listed below. Assume these sales figures to be the population values.
a. Draw a graph showing the population distribution.
b. Compute the mean of the population.
c. Select five random samples of 5 each. Compute the mean of each sample. Use the methods described in this chapter and Appendix $E$ to determine the items to be included in the sample.
d. Compare the mean of the sampling distribution of the sample means to the population mean. Would you expect the two values to be about the same?
e. Draw a histogram of the sample means. Do you notice a difference in the shape of the distribution of sample means compared to the shape of the population distribution?

Key Concepts

Central Limit Theorem

The Central Limit Theorem is a key statistical principle stating that, for a sufficiently large sample size, the distribution of the sample means will approximate a normal distribution, regardless of the shape of the original population distribution. This theorem underpins many statistical methods and analyses by justifying the use of normal distribution approximations.

Histogram

A histogram is a graphical representation of data distribution using bars to depict frequency of data within specified intervals. It provides a visual summary of how data are spread and clustered, making it easier to identify patterns such as central tendency, variability, and any skewness or outliers present in the data.

Sampling Distribution

A sampling distribution is the probability distribution of a given statistic, such as the mean, calculated from a large number of samples drawn from the same population. It illustrates how the statistic varies from sample to sample and is fundamental for making inferences about the population parameter.

Random Sampling

Random sampling is the process of selecting a subset of individuals from a larger population in such a way that each individual has an equal chance of being chosen. This method is essential for ensuring that the sample is representative of the population, thereby reducing bias and allowing generalizations to be made from the sample to the broader population.

Mean of a Population

The mean of a population is a measure of central tendency that represents the average value of all observations in the set. It is calculated by summing all the individual values and then dividing by the total number of values, serving as a foundational parameter in statistical analyses.

Population Distribution

A population distribution refers to the complete set of values or observations that a variable can take within an entire population. It provides a comprehensive view of the variable's behavior, showing how frequently each distinct value occurs and allowing insights into overall patterns and variability in the data.

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