A school administrator has students rate the quality of...

A school administrator has students rate the quality of their education on a scale from 1 (poor) to 7 (exceptional). She claims that $99.7 \%$ of students rated the quality of their education between $3.5$ and $6.5$. If the mean rating is $5.0$, then what is the standard deviation, assuming the data are normally distributed? Hint: Use the empirical rule.

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Key Concepts

Normal Distribution

A probability distribution that is symmetric and bell-shaped, defined by its mean and standard deviation. It is the foundation for many statistical methods because many natural phenomena tend to follow this pattern, making it a core concept in probability and statistics.

Empirical Rule

Also known as the 68-95-99.7 rule, this rule states that for a normal distribution, approximately 68% of the data fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. It is a quick way to estimate the spread of data and understand variability.

Standard Deviation

A statistical measure that quantifies the amount of variation or dispersion in a set of data values. In a normal distribution, the standard deviation is crucial because it determines the width of the curve and thereby the range within which most data points lie.

Mean

The average value of a dataset, representing the central tendency. In a normal distribution, the mean is used as the center point from which distances (measured by standard deviations) are calculated to understand the spread of the data.

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