The U.S. Bureau of Labor Statistics reports that the average...

The U.S. Bureau of Labor Statistics reports that the average annual expenditure on food and drink for all families is $\$ 5700$ ( Money, December 2003 ). Assume that annual expenditure on food and drink is normally distributed and that the standard deviation is $\$ 1500$. a. What is the range of expenditures of the $10 \%$ of families with the lowest annual spending on food and drink? b. What percentage of families spend more than $\$ 7000$ annually on food and drink? c. What is the range of expenditures for the $5 \%$ of families with the highest annual spending on food and drink?

The U.S. Bureau of Labor Statistics reports that the average annual expenditure on food and drink for all families is $\$ 5700$ ( Money, December 2003 ). Assume that annual expenditure on food and drink is normally distributed and that the standard deviation is $\$ 1500$.
a. What is the range of expenditures of the $10 \%$ of families with the lowest annual spending on food and drink?
b. What percentage of families spend more than $\$ 7000$ annually on food and drink?
c. What is the range of expenditures for the $5 \%$ of families with the highest annual spending on food and drink?

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve where most of the values cluster around the mean. It is defined by two parameters: the mean (which indicates the center of the distribution) and the standard deviation (which measures the spread or dispersion of the values). Many real-world phenomena, such as spending habits, are assumed to follow a normal distribution, allowing the use of statistical methods to calculate probabilities.

Standard Deviation and Mean

The mean is the average value of a dataset, while the standard deviation quantifies the amount of variation or dispersion around that mean. In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. These concepts are essential in understanding and interpreting where a particular value lies within a distribution.

Z-Scores and Standardization

A z-score is a standardized score that indicates how many standard deviations a particular value is from the mean. Standardizing data by computing z-scores allows for the comparison of values from different normal distributions and is crucial for determining probabilities and percentiles. This process is typically done by subtracting the mean from the value and then dividing by the standard deviation.

Percentiles and Inverse Normal Distribution

Percentiles are used to describe the relative standing of a value within a distribution. For example, the 10th percentile is the value below which 10% of the observations may be found. In problems involving normal distributions, percentiles can be determined by using the inverse of the cumulative distribution function (also known as the quantile function), which finds the z-score corresponding to a given percentile and then converts it back to the original scale using the mean and standard deviation.

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