The average annual amount American households spend for...

The average annual amount American households spend for daily transportation is $\$ 6312$ (Money, August 2001). Assume that the amount spent is normally distributed. a. Suppose you learn that $5 \%$ of American households spend less than $\$ 1000$ for daily transportation. What is the standard deviation of the amount spent? b. What is the probability that a household spends between $\$ 4000$ and $\$ 6000 ?$ c. What is the range of spending for the $3 \%$ of households with the highest daily transportation cost?

The average annual amount American households spend for daily transportation is $\$ 6312$ (Money, August 2001). Assume that the amount spent is normally distributed.
a. Suppose you learn that $5 \%$ of American households spend less than $\$ 1000$ for daily transportation. What is the standard deviation of the amount spent?
b. What is the probability that a household spends between $\$ 4000$ and $\$ 6000 ?$
c. What is the range of spending for the $3 \%$ of households with the highest daily transportation cost?

Key Concepts

Percentiles and Tail Probabilities

Percentiles divide a distribution into 100 equal parts, and tail probabilities refer to the probabilities in the extreme ends (or tails) of the distribution. These concepts are used to determine thresholds such as the 5th percentile or the top 3% threshold in a normally distributed dataset.

Z-Score Transformation

The Z-score transformation converts a value from a normal distribution to the standard normal distribution by subtracting the mean and dividing by the standard deviation. This standardized value allows for the use of standard normal tables to calculate probabilities and percentiles.

Probability Calculation for Intervals

Calculating the probability for an interval under a normal distribution involves finding the area under the curve between two values. This is typically done using Z-scores to transform the values and then consulting the standard normal distribution table or using statistical software.

Mean

The mean is the average or expected value of a data set and is the central point of a normal distribution. It serves as a measure of central tendency and is a key parameter in defining the shape and location of the distribution.

Normal Distribution

A normal distribution is a probability distribution that is symmetric about its mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its mean and standard deviation, and it is often used to model naturally occurring phenomena due to the Central Limit Theorem.

Standard Deviation

The standard deviation is a measure of the dispersion or spread of a set of data points relative to its mean. In a normal distribution, it determines the width of the bell curve, with larger values indicating a wider spread and smaller values indicating a more concentrated distribution.

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