A video-rental chain estimates that annual expenditures of...

A video-rental chain estimates that annual expenditures of members on rentals follow a normal distribution with a mean of $$\$ 100$$. It was also found that $10 \%$ of all members spend more than $$\$ 130$$ in a year. What percentage of members spends more than $$\$ 140$$ in a year?

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Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric around its mean, forming a bell-shaped curve. It is commonly used to model natural phenomena, where most observations cluster around the mean with probabilities tapering off as one moves further from the center.

Z-score

The Z-score is a standardization metric that expresses how many standard deviations an individual data point is from the mean of its distribution. Converting to Z-scores allows for comparison across different normal distributions by referencing a common scale.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of zero and a standard deviation of one. It facilitates the use of standardized tables (Z-tables) for easy calculation of probabilities and percentiles associated with any normally distributed variable.

Percentiles and Tail Probabilities

Percentiles indicate the value below which a given percentage of observations in a distribution falls. Tail probabilities refer to the chance that a value exists beyond a particular threshold on the extreme ends (tails) of the distribution. These concepts are key for calculating the likelihood of extreme outcomes in a normally distributed data set.

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