A study published in Current Allergy & Clinical Immunology...

A study published in Current Allergy \& Clinical Immunology (Mar. 2004 ) investigated allergic reactions of dental patients to local anesthetics. Based on a survey of dental practitioners, the study reported that the mean number of units (ampoules) of local anesthetics used per week by dentists was $79,$ with a standard deviation of $23 .$ Suppose we want to determine the percentage of dentists who use less than 102 units of local anesthetics per week. a. Assuming that nothing is known about the shape of the distribution for the data, what percentage of dentists use less than 102 units of local anesthetics per week? b. Assuming that the data have a mound-shaped distribution, what percentage of dentists use less than 102 units of local anesthetics per week?

A study published in Current Allergy \& Clinical Immunology (Mar. 2004 ) investigated allergic reactions of dental patients to local anesthetics. Based on a survey of dental practitioners, the study reported that the mean number of units (ampoules) of local anesthetics used per week by dentists was $79,$ with a standard deviation of $23 .$ Suppose we want to determine the percentage of dentists who use less than 102 units of local anesthetics per week.
a. Assuming that nothing is known about the shape of the distribution for the data, what percentage of dentists use less than 102 units of local anesthetics per week?
b. Assuming that the data have a mound-shaped distribution, what percentage of dentists use less than 102 units of local anesthetics per week?

Key Concepts

Normal Distribution

A normal distribution is a symmetric, bell-shaped data distribution characterized by its mean and standard deviation. It is one of the most common distributions in statistics and underpins many methods and rules, such as the Empirical Rule, that assume data is approximately normally distributed, especially in cases of mound-shaped distributions.

Chebyshev's Inequality

Chebyshev's Inequality is a statistical rule that applies to any data distribution regardless of its shape. It provides a conservative bound on the proportion of observations that lie within a specified number of standard deviations from the mean, making it useful when little is known about the distribution of the data.

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, is a guideline that applies specifically to bell-shaped, normally distributed data. It states that approximately 68% of observations lie within one standard deviation of the mean, about 95% lie within two standard deviations, and around 99.7% lie within three standard deviations, helping to quickly estimate probabilities for normally distributed data.

Mean

The mean is a measure of central tendency that represents the average value of a data set. It is calculated by summing all observations and dividing by the number of observations. The mean provides a central reference point around which the variability of the data can be understood.

Standard Deviation

Standard deviation is a measure of the dispersion or spread of a set of data points around the mean. It quantifies how much the individual measurements typically deviate from the mean, with a larger standard deviation indicating greater variability in the data.

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