Recently the United States Department of Agriculture issued...

Recently the United States Department of Agriculture issued a report (http://www.cnpp. usda.gov/sites/default/files/CostoffoodMar2015.pdf) indicating a family of four spent an average of about $\$ 890$ per month on food. Assume the distribution of food expenditures for a family of four follows the normal distribution, with a standard deviation of $\$ 90$ per month. a. What percent of the families spend more than $\$ 430$ but less than $\$ 890$ per month on food? b. What percent of the families spend less than $\$ 830$ per month on food? c. What percent spend between $\$ 830$ and $\$ 1,000$ per month on food? d. What percent spend between $\$ 900$ and $\$ 1,000$ per month on food?

Recently the United States Department of Agriculture issued a report (http://www.cnpp. usda.gov/sites/default/files/CostoffoodMar2015.pdf) indicating a family of four spent an average of about $\$ 890$ per month on food. Assume the distribution of food expenditures for a family of four follows the normal distribution, with a standard deviation of $\$ 90$ per month.
a. What percent of the families spend more than $\$ 430$ but less than $\$ 890$ per month on food?
b. What percent of the families spend less than $\$ 830$ per month on food?
c. What percent spend between $\$ 830$ and $\$ 1,000$ per month on food?
d. What percent spend between $\$ 900$ and $\$ 1,000$ per month on food?

Key Concepts

Cumulative Distribution Function (CDF) and Probability Calculations

The cumulative distribution function (CDF) provides the probability that a normally distributed random variable is less than or equal to a given value. By using the CDF, one can calculate the probability of an interval by finding the difference between CDF values at two points. This method is fundamental for determining percentages of observations that fall within specified ranges.

Standardization (Z-Scores)

Standardization involves converting a normally distributed variable to a standard normal variable by subtracting the mean and dividing by the standard deviation. The resulting value, known as the z-score, indicates how many standard deviations an observation is from the mean. This process is essential for comparing values from different normal distributions and for looking up probabilities using the standard normal distribution table.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean. It is characterized by its bell-shaped curve where most of the observations cluster around the mean and probabilities for values further away decrease exponentially. This distribution is commonly used in statistics to model real-world phenomena where the central limit theorem applies and underpins many inferential techniques.

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